Thermal Physics Notes
Temperature
Temperature is a quantitative measure of hotness or coldness.
It provides a basis for comparing the thermal states of different objects.
Traditional clinical thermometers use mercury's thermal expansion to indicate temperature.
Any physical property that changes with temperature can be used to create a temperature scale. Examples include:
Volume of a liquid
Resistance of a wire
Voltage of a thermocouple
Volume of a fixed mass of gas at constant pressure
Pressure of a fixed mass of gas at constant volume
Color of a filament
Length of a solid
Thermodynamic equilibrium is reached when thermophysical properties no longer change over time.
Zeroth law of thermodynamics: If two systems are each in thermal equilibrium with a third, then all three are in thermal equilibrium with each other.
Common Temperature Scales
Celsius and Fahrenheit scales are common, defined by the ice point and steam point of water.
Celsius: Ice point is 0 °C, steam point is 100 °C.
Fahrenheit: Ice point is 32 °F, steam point is 212 °F.
The zero points on these scales are arbitrarily selected and have no fundamental significance.
The Kelvin Temperature Scale
Defines an absolute zero, which has fundamental significance.
Derived from studying pressure and volume changes in gases with temperature.
A constant volume gas thermometer measures gas pressure at different temperatures.
Plots of gas pressure vs. Celsius temperature are straight lines that extrapolate to the same point at zero pressure.
Absolute zero is found to be -273.15 °C. A negative pressure has no meaning, suggesting the temperature cannot go lower than -273.15 °C
The Kelvin scale has its zero at absolute zero, with the same degree size as Celsius.
Conversion between Kelvin (T) and Celsius (t): T = t + 273.15
SI base unit for temperature is the kelvin (K).
For an ideal gas at constant volume, the relationship between temperature and pressure is: \frac{T2}{T1} = \frac{P2}{P1}
The Kelvin temperature can be defined using the triple point of water (where solid, liquid, and vapor coexist) at 0.01 °C and a water-vapor pressure of 610 Pa.
The triple-point temperature is defined as T_{triple} = 273.16 K.
Kelvin temperature is defined as: T = 273.16 \frac{P}{P_{triple}} at constant volume.
Thermal expansion of matter
Most materials expand when heated and contract when cooled.
Thermal expansion is related to changes in the average separation between atoms or molecules.
The interatomic potential energy curve determines the mean separation between constituent particles.
In the vicinity of the equilibrium position r_0, the potential curve is parabolic, leading to simple harmonic oscillations.
Thermal expansion occurs because the real potential energy curve deviates from a parabolic shape.
As temperature rises, atomic vibrations increase in amplitude, and atoms spend more time at larger separations, leading to expansion.
Linear Expansion of Solids
The increase in one dimension of a solid is called linear expansion.
Factors affecting expansion/contraction:
Magnitude of the temperature change \Delta T
Initial linear dimension of interest l_0
Change in length is proportional to both initial length and temperature change: \Delta l \propto l0 (T - T0)
Introducing the mean coefficient of linear expansion \alpha:
\Delta l = \alpha l0 (T - T0)
\alpha = \frac{\Delta l}{l0 (T - T0)} (units: °C⁻¹)
The relationship between length and temperature is expressed as: l = l0 [1 + \alpha (T - T0)]
This result is valid for moderate temperature changes.
Expansion of Holes
For a homogeneous body with holes, the change in length is strictly proportional to the original length l.
The sizes of holes or cavities expand or contract as if the holes were filled with the material of the body.
The Binomial Theorem
Used to expand the quantity (1 + x)^n where x \ll 1
Series expansion: (1 + x)^n = 1 + nx + \frac{n(n - 1)}{2!}x^2 + \frac{n(n - 1)(n - 2)}{3!}x^3 + \cdots
Approximation for x \ll 1: (1 + x)^n \approx 1 + nx
Cubical (or Volume) Expansion
Substances expand in all three dimensions when heated.
The increase in volume is known as cubical (or volume) expansion.
If the temperature change \Delta T is not too great, the increase in volume \Delta V is proportional to \Delta T and the original volume V_0.
The coefficient of cubical (or volume) expansion is defined as:
\beta = \frac{\Delta V}{V_0 \Delta T}
The mean coefficient of cubical expansion is defined as the fractional change in volume per degree Celsius change in temperature.
The coefficient of cubical expansion \beta is defined for all three states of matter: solids, liquids, and gases.
The term linear expansion has no meaning for a liquid or a gas, since the fluid always takes the shape of its container.
It follows from the definition that: V = V0[1 + \beta(T - T0)]
Where V is the volume at some temperature T.
Thermal Stress
Occurs when the ends of a rod or slab of material are rigidly fixed, preventing thermal expansion or contraction.
Thermal stress is calculated in two steps:
Calculate the amount the rod/slab would expand (or contract): \Delta l = \alpha l_0 \Delta T
Calculate the force required to compress (or expand) the material back to its original length.
The force per unit area (or stress) is related to the fractional change of length (the strain) by: \frac{F}{A} = Y \frac{\Delta l}{l_0}, where Y is Young’s modulus for the material.
This gives rise to an equation for the thermal stress as:\frac{F}{A} = Y \alpha \Delta T
For a three-dimensional object, the stress exerted on the object is the pressure exerted over the entire surface of the object.
The bulk modulus (B) of a material is defined as: B = \frac{\Delta P}{\Delta V / V_0}.
Expansion of Liquids
Liquids have no shape and therefore no fixed dimensions, so cubical expansion is the only concern.
Liquids must be held in some containing vessel, so the apparent expansion of the liquid is less than the real value due to the expansion of the vessel.
Real expansion of liquid = apparent expansion of liquid + real expansion of vessel.
\betar V0 \Delta T = \betaa V0 \Delta T + \beta{ves} V0 \Delta T
\betar = \betaa + \beta{ves}, or \betaa = \betar - \beta{ves}.
Expansion of Gases
The volume occupied by a gas is very sensitive to both temperature and pressure.
To measure the thermal expansion of a gas, the pressure must be kept constant.
The zero coefficient of cubical expansion of a gas at constant pressure {beta}0 is defined as: \beta0 = \frac{V - V0}{V0 T}, where V_0 is the volume of a given mass of gas at 0 °C and V is the volume at a temperature T
The result of many careful experiments indicates that \beta_0 is the same for all gases provided the pressure is low enough so that the gas may be regarded as ideal.
This value of \beta0 is \beta0 = \frac{1}{273.15} °C⁻¹ (
Relative Thermal Expansion
Typical orders of magnitude of thermal expansion:
Solid: \beta between 10^{-5} and 10^{-6} °C⁻¹
Liquid: \beta between 10^{-4} and 10^{-5} °C⁻¹
Gas: \beta_0 approximately 10^{-3} °C⁻¹
The thermal expansion of a gas is about a thousand times greater than that of a typical solid for a given temperature change.
Important Relationship Between α and β
Comparison of volume expansion equations yields the relationship: \beta = 3\alpha
This result only applies to isotropic solids, where α is the same in all directions. Many solids (especially crystals) are anisotropic.
Effect of Expansion on Density
When a substance is heated, its volume increases but its mass remains constant, so density decreases.
\rhoT = \frac{\rho0}{(1 + \beta \Delta T)}
Anomalous Behavior of Water
Water's volume decreases when heated from 0°C to 4°C. Above 4°C, it expands normally.
Maximum density occurs at 4°C.
This behavior influences how lakes freeze: surface water cools, sinks until the entire lake reaches 4°C. Further cooling makes surface water less dense, leading to ice formation on top, insulating the lake and allowing aquatic life to survive.
Heat
Introduction to Heat
Heat and temperature are related but distinct concepts.
Temperature indicates how hot or cold an object is, while heat refers to the energy transferred due to temperature differences.
Internal energy is the sum of molecular energies (translational, rotational, vibrational kinetic energies, and potential energy).
Heat is energy in transit due to a temperature difference; substances contain internal energy, not heat.
Heat Capacity
Heat capacity (C) is the amount of heat (Q) required to raise the temperature of a body by \Delta T: C = \frac{Q}{\Delta T} (SI units: J °C⁻¹).
Specific heat capacity (c) is the amount of heat (Q) required to raise the temperature of mass (m) by \Delta T: c = \frac{Q}{m \Delta T} (SI units: J kg⁻¹ °C⁻¹).
Changes of Phase
During a phase change, the temperature remains constant despite the addition or removal of heat (e.g., melting ice at 0 °C or boiling water at 100 °C under standard atmospheric pressure).
Specific heat of transformation (L) is defined as the heat (Q) required to change the phase of mass (m) without changing the temperature: L = \frac{Q}{m} (SI units: J kg⁻¹).
The equations Q = mc\Delta T and Q = mL are important for calorimetry calculations.
Key points:
Transformation can be melting/fusion, freezing, vaporization, condensation, or sublimation.
Specific heats of transformation are also called ‘specific latent heats’ or ‘latent heats’.
Energy is required during melting and vaporization, and energy is released during freezing and condensation.
During melting Q = mLf (f denotes fusion).
During vaporization Q = mLv (v denotes vaporization).
Heat Units
SI unit of heat is the joule (J).
Dietitians and nutritionists use the Calorie (with a capital C) to specify the energy content of foods: 1 \text{ Calorie} = 4.1868 \text{ J}.
Distinctions Between Heat and Temperature
Care should be taken to distinguish between the terms heat and temperature.
Heat is the net energy transferred spontaneously from regions of high temperature to low temperature.
Temperature indicates whether or not heat will flow, and in which direction.
Objects at the same temperature are in thermal equilibrium, and no heat will flow.
A quantity of heat that is transferred depends on the temperature difference between the materials, as well as their masses and specific heat capacities.
Calorimetry calculations
Introduction to Calorimetry Calculations
Calorimetry involves ‘measuring heat’ and performing calculations with heat.
When substances at different temperatures are brought into contact, they reach thermal equilibrium after mixing.
The fundamental principle is the law of conservation of energy: Heat 'lost' by hot substance(s) = Heat 'gained' by cold substance(s).
\text{Heat 'lost'} = \text{Heat 'gained'}
It is often assumed that heat transferred to the surroundings is negligible, which can be minimized practically.
Mechanisms of Heat Transfer
Introduction to Heat Transfer Mechanisms
Heat is transmitted from one place to another through three primary processes: conduction, convection, and radiation.
Conduction: Energy transfer through a body (or between bodies in contact) due to interatomic or intermolecular collisions.
Convection: Energy transfer from one place to another by the bulk motion of a fluid.
Radiation: Heat transfer by electromagnetic radiation, with no need for matter between bodies.
Conduction
Suppose a metal rod of length \Delta x and cross-sectional area A has one end held in a flame. Heat reaches the other end by conduction.
In the steady state: the rate of heat flow along the rod Q/t (where Q is the heat conducted in time t) is proportional to:
The cross-sectional area A, i.e., \frac{Q}{t} \propto A
The temperature gradient \frac{\Delta T}{\Delta x} across the ends of the rod, where \Delta T is the temperature difference.
Combining these results: \frac{Q}{t} = \lambda A \frac{\Delta T}{\Delta x} , where \lambda is the thermal conductivity of the material.
In this equation, the temperature gradient \frac{\Delta T}{\Delta x} is taken as a positive quantity.
The heat power \frac{Q}{t} is measured in J s^(−1) which equals a Watt (W).
Therefore, the SI unit of \lambda is [λ] = [Q/t] / [A][∆T/∆x] = W / (m² · °C/m) = W m^(-1) °C^(-1)
Convection
Convection is the transfer of heat by mass motion of a fluid from one region of space to another.
Familiar examples include heaters, car engine cooling systems, and blood flow in the body.
If fluid is circulated by a blower or pump, it's called forced convection; if by density differences due to thermal effects, it's natural convection.
Experimental facts:
Rate of heat transfer is proportional to surface area.
Viscosity slows natural convection near surfaces.
Forced convection decreases the thickness of the surface film, increasing the rate of heat transfer. This leads to the wind-chill factor.
The rate of heat transfer is approximately proportional to the 5/4 power of the temperature difference between the surface and the main body of fluid.
Radiation
The transfer of energy by radiation does not require the intervention of material media.
All bodies above absolute zero (0 K) radiate energy, with the rate dependent on the body's temperature and surface.
A black body absorbs all incident radiant energy.
The absorptivity α of a surface for radiant energy is defined as the fraction of the total incident radiation absorbed by the surface; for a black body, α = 1.
Bodies that are good absorbers are also good emitters, and poor absorbers are poor emitters.
The rate at which energy is radiated (i.e., the power) from a surface A at an absolute temperature T is: \frac{Q}{t} = \sigma \varepsilon A T^4,
where \sigma is the Stefan-Boltzmann constant (\sigma = 5.670 × 10⁻⁸ W m⁻² K⁻⁴), and \varepsilon is the emissivity of the body.
The emissivity is defined as the rate at which energy is emitted from the body relative to that emitted by an identical black body at the same temperature.
α = ε for any body (Kirchhoff’s Law).
Any object radiates and absorbs energy from other bodies. The net rate of radiant energy transfer is given by: \frac{Q}{t} = \sigma \varepsilon A (T1^4 - T2^4), assuming the object is at temperature T₁ and the environment is at temperature T₂.
For radiation from the sun, note that 1350 J of energy strikes Earth's atmosphere from the sun per second per square meter (solar constant).
An object facing the Sun absorbs heat at: \frac{Q}{t} = I \varepsilon A \cos \theta, where \theta is the angle between the sun's rays and a line perpendicular to area A.
Spectral Distribution of Black Body Radiation
Examines emitted power per unit wavelength interval for black body radiation across different temperatures.
For a given temperature T1, there's a wavelength \lambda1 at which intensity is maximum.
Wien's Displacement Law: \lambda_{max}T = constant
Experimentally, \lambda_{max}T = 2.897 \times 10^{-3} K m
Pyrometers use radiation to measure temperature, such as determining the temperature of distant celestial objects.
Radiation emitted due to temperature is called thermal radiation; objects emit and absorb radiation from surroundings.
The Gas Laws
Revision of Basic Terminology
Atomic and Molecular Mass
The unified atomic mass unit (u) is defined as one-twelfth of the mass of an unbound neutral atom of carbon-12 in its nuclear and electronic ground state (at rest).
The atomic mass of an element is the mass of an atom of that element on a scale on which the mass of an atom of ^{12}_6C = 12 u.
Conversion: 1 u = 1.660 538 921(73) × 10^{-27} kg
The molecular mass is the sum of the atomic masses of its constituent atoms.
The Mole and Avogadro’s Number
One mole of any substance contains as many particles as there are atoms in 12 g of the isotope ^{12}C.
12 g of ^{12}C contains 6.022 \times 10^{23} atoms (Avogadro’s number, N_A).
One mole of any substance has a mass in grams equal to its relative atomic or molecular mass.
The number of moles n is given by: n= \frac{\text{mass (g)}}{\text{molar mass (g)}} = \frac{m}{M}
Standard Temperature and Pressure (STP)
Standard temperature = 273.15 K.
Standard pressure = 101 325 Pa = 1 Atm = 760 mmHg.
One mole of any gas at STP occupies 22.4 litres.
Absolute and Gauge Pressure
The pressure at a given depth h below the surface of a fluid of density \rho is given by: P = P0 + h\rho g, where P0 is the atmospheric pressure at the surface of the fluid
The pressure P in Equation (30) is the total or absolute pressure at the depth h.
The pressure (h \rho g) due only to the liquid is the gauge pressure
Equation of State of an Ideal Gas
For a fixed mass of gas at constant temperature: P1V1 = P2V2 or \frac{P}{V} = \text{constant} (Boyle’s Law).
For a fixed mass of gas at constant pressure: \frac{V1}{T1} = \frac{V2}{T2} or \frac{V}{T} = \text{constant} (Charles’ Law), where T is the absolute temperature.
For a fixed mass of gas at constant volume: \frac{P1}{T1} = \frac{P2}{T2} or \frac{P}{T} = \text{constant} (Gay–Lussac’s Law), where T is the absolute temperature
Combining the three laws yields:
\frac{P1V1}{T1} = \frac{P2V2}{T2} or \frac{PV}{T} = \text{constant,} where T is the absolute temperature
Since the mass of the gas m is fixed (and constant), Equation (32) can be written in the form \frac{PV}{T}=mr, where r is a constant for a particular gas.
From experiment that for different gases r = \frac{R}{M}, where R is a constant which is the same for all gases and M is the relative atomic or molecular mass
\frac{PV}{T} = m \frac{R}{M}
From Equation (29) (n = \frac{m}{M}) Equation (34) leads to \frac{PV}{T} = nR or PV = nRT, where T is the absolute temperature.
The numerical value of the universal gas constantR = 8.314 J mol^{−1} K^{−1}.
The behaviour of real gases conforms closely to Equation (35) except at high densities.
Equation (35) gives the relationship between the variables n, P, T and V and it is called the equation of state of an ideal gas (or the ideal gas equation).
Dalton’s Law of Partial Pressures
Consider a container of volume V filled with two non-interacting ideal gases A and B. The partial pressures of each gas are PA = \frac{nART}{V} and PB = \frac{nBRT}{V}.
The total pressure of the mixture is P = \frac{nRT}{V}, where n = nA + nB, and so
P = \frac{nRT}{V} = (nA + nB)\frac{RT}{V} = \frac{nART}{V}+ \frac{nBRT}{V} = PA + PB
Dalton’s law of partial pressures:
The total pressure of a mixture of non-interacting gases is equal to the sum of the partial pressures of the component gases: P{total} = P1 + P2 + P3 + …
Pressure Due to Vapor
When a vapor is compressed, the variation of pressure with volume is described in several steps that can be shown on a graph of pressure vs. volume
When the vapor is unsaturated, a small isothermal compression does not result in any condensation of liquid.
Once the vapor becomes saturated and the liquid appears in the cylinder, a decrease in volume does not lead to an increase in pressure, but only the condensation of more liquid. The fixed pressure which is observed when a saturated vapor is in equilibrium with its fluid is called the saturation vapor pressure (S.V.P.) of the substance for the particular temperature.
A liquid boils when its S.V.P. is equal to the pressure on its free surface. Thus if a liquid is open to the atmosphere it boils at the temperature for which the S.V.P. equals the atmospheric pressure
Pressure Due to Mixture of Gases and a Vapor
Boyle’s Law may be used to describe the behavior of a gas and an unsaturated vapor. Once the vapor is saturated and an excess of liquid is present, though, it can no longer be used.
If a gas and saturated vapor are compressed at constant temperature the vapor continues to exert a partial pressure equal to the constant S.V.P. at the prevailing temperature, but the partial pressure of the gas will obey Boyle’s law.
If P is the total pressure of the mixture, V its volume, and f the constant S.V.P. at the prevailing temperature, then (P-f)V=constant6
Elementary kinetic theory
Basic Assumptions
The assumptions of the kinetic theory of gases are:
A gas consists of a large number of identical particles which are in continual, random motion.
The particles move in straight lines and obey Newton’s laws of motion.
The particles exert no forces on each other except at the moment of collision.
The collisions are perfectly elastic and take negligible time.
The volume of the particles is negligible compared with the volume of the container.
Pressure Exerted by Ideal Gas
The pressure of a gas results from the bombardment of particles on the container walls.
Force on the wall is the rate of change of the momentum of particles bouncing off the wall.
For a cubical vessel with N molecules of mass m, the pressure P is given by
P = \frac{Nm \langle v_x^2 \rangle}{l^3} = \frac{Nm \langle v^2 \rangle}{3V}
Where \langle vx^2 \rangle is the average value of vx^2 for all N molecules, \langle v^2 \rangle is the average value of v^2=vx^2+vy^2+v_z^2, and V is the volume of the container
Thus PV = \frac{1}{3}Nm \langle v^2 \rangle
Root-Mean-Square Speed
The root-mean-square speed v_{rms} is defined as the square root of the mean-square speed:
v_{rms} = \sqrt{\langle v^2 \rangle}
From experimental observations: PV = nRT (35)
From kinetic theory: PV = \frac{1}{3}Nm \langle v^2 \rangle = \frac{1}{3}Nmv_{rms}^2 (39 and 40)
Equating (35) with (39) and (40) gives, for one mole (n=1) RT = \frac{1}{3}NA m v{rms}^2
The molar mass M = NA m, thus v{rms} = \sqrt{\frac{3RT}{M}}
Kinetic energy of a single gas particle: \frac{1}{2}m v{rms}^2 = \frac{3R}{2NA} \times T.
Average KE depends only on absolute temperature (where kb is the Boltzmann constant: kb=\frac{R}{NA}).
Distribution of Molecular Speeds
In a container filled with gas, molecules do not all have the same speed; some move faster, some slower.
The most probable speed is the speed at which more molecules travel than any other speed, but the majority of molecules travel near the most probable speed
The Maxwell distribution is a graph plotting the distribution of speeds in a sample of molecular at a given temperature. In other words, it plots the number of molecules per unit speed interval as a function of molecular speed
Properties of the Maxwell distribution:
Most molecules travel at the most probable speed v{mp} (Note that the most probable speed v{mp} is close to the root-mean-square speed v_{rms}, but they are not equal)
Some molecules travel at very high-speeds or very low-speeds but they compose a minority of the population of gas molecules
Increasing temperature causes the most probable speed to shift to right, and the height of the peak to decrease
The area under the curve is the same regardless of temperature. This area represents the total number of molecules present in the container
Thermodynamics
Introduction to Thermodynamics
Thermodynamics studies processes converting heat into work/energy and vice versa.
Internal energy, U, is central, comprising kinetic and potential energies of molecules (translational, vibrational, rotational modes).
It is challenging to evaluate the internal energies of substances via this approach.
Changes in internal energy can be found through changes in temperature, pressure, volume, or state.
These changes are caused by heat transfer or work on the system.
Internal Energy of Ideal Monatomic Gas
For a monatomic ideal gas (e.g., inert gas), composed of non-interacting single atoms, all energy is kinetic.
Internal energy (U) is the total kinetic energy of N atoms:
U = N(\frac{1}{2} m v_{rms}^2)
U = \frac{3}{2}PV = \frac{3}{2}nRT
\Delta U = \frac{3}{2}nR\Delta T
First Law of Thermodynamics
Statement of energy conservation:
Q > 0: heat added to the system.
Q < 0: heat removed from the system.
W < 0: work done on the system.
W > 0: work done by the system.
\Delta U = Q - W
Internal energy increases if heat is added and decreases if the system does work.
Note: Work definition in thermodynamics is opposite to mechanics.
Work Done by a Gas
Consider a gas confined in an air-tight cylinder fitted with a frictionless piston.
For a quasi-static expansion (system remains in equilibrium), work done is dW = F ds = P Ads = P dV
Total work done by a gas expanding from Vi to Vf: W = \int{Vi}^{V_f} P dV
Isobaric Process (constant pressure):
W = P \Delta V = P (Vf - Vi)
W = P \Delta V = nR\Delta T
Isochoric Process (constant volume):
W = 0
Isothermal Process (constant temperature):
W = \int{V1}^{V2} P dV = \int{V1}^{V2} \frac{nRT}{V} dV = nRT \ln \frac{V2}{V1}
\Delta U = 0\implies Q = W
Molar Specific Heat Capacities of Monatomic Gases
Changes in temperature, \Delta T, are related to heat, Q, by Q = mc\Delta T, where c is specific heat capacity.
For gases, use moles, n: Q = nC\Delta T, where C is molar specific heat capacity.
Important difference for heating gas at constant pressure (Cp) and volume (Cv).
Constant Volume
Heat added increases the internal energy. By 1st Law of Thermodynamics and from Equation (43), \Delta U = Q -W = Q \implies Q dV = \frac{3}{2}nR\Delta
But Q = nCV\Delta T \implies nCV\Delta T = \frac{3}{2}nR \Delta T implying CV = \frac{3}{2}R
Constant Pressure
Some heat does work = (W=\Delta V/ =nR\Delta T/)
So additional heat is required to bring about the same change in internal energy Q = nCP ∆ = ∆U + W = (3)/(2)nR∆T+= nR∆T
which gives nCP \Delta T=\frac{5}{2}nr/\Delta T \implies CP=\frac{5}{2}R
From equations previously mentioned, CP-CV= R
Equations are only valid for inter gases that act (mostly) as monatomic gases, so need sophisticated treatment for molecules in more atoms
Temperature
Temperature is a quantitative measure of hotness or coldness, indicating the average kinetic energy of the particles within a system.
It provides a basis for comparing the thermal states of different objects, determining which will transfer heat to the other when they are brought into thermal contact.
Traditional clinical thermometers use mercury's thermal expansion to indicate temperature, relying on the consistent and measurable property of mercury's volume change with temperature.
Any physical property that changes with temperature can be used to create a temperature scale. Examples include:
Volume of a liquid: As temperature increases, most liquids expand. This principle is used in liquid-in-glass thermometers.
Resistance of a wire: The electrical resistance of a metal typically increases with temperature, a principle used in resistance thermometers.
Voltage of a thermocouple: Different metals generate a voltage related to temperature at the junction between them, used in thermocouples.
Volume of a fixed mass of gas at constant pressure: Gases expand when heated if pressure is kept constant, as observed in gas thermometers.
Pressure of a fixed mass of gas at constant volume: The pressure increases with temperature when volume is constant, used in constant-volume gas thermometers.
Color of a filament: The color of a glowing filament changes with temperature, as seen in incandescent light bulbs.
Length of a solid: Solids expand linearly with temperature, a principle used in bimetallic strips for thermostats.
Thermodynamic equilibrium is reached when thermophysical properties no longer change over time, indicating a stable thermal state within a system.
Zeroth law of thermodynamics: If two systems are each in thermal equilibrium with a third, then all three are in thermal equilibrium with each other. This law allows for the definition of temperature in a self-consistent way; if two systems are separately in equilibrium with a third, they must be in equilibrium with each other, implying they have the same temperature.
Common Temperature Scales
Celsius and Fahrenheit scales are common, defined by the ice point and steam point of water, providing familiar reference points for everyday temperature measurements.
Celsius: Ice point is 0 °C (the temperature at which water freezes), steam point is 100 °C (the temperature at which water boils at standard atmospheric pressure).
Fahrenheit: Ice point is 32 °F, steam point is 212 °F. The Fahrenheit scale is commonly used in the United States.
The zero points on these scales are arbitrarily selected and have no fundamental significance, meaning they do not correspond to a state of zero energy or molecular motion.
The Kelvin Temperature Scale
Defines an absolute zero, which has fundamental significance as the point at which all molecular motion ceases (in theory).
Derived from studying pressure and volume changes in gases with temperature, based on the behavior of gases at different temperatures to extrapolate to the point of zero pressure.
A constant volume gas thermometer measures gas pressure at different temperatures, utilizing the principle that gas pressure is directly proportional to temperature when volume is held constant.
Plots of gas pressure vs. Celsius temperature are straight lines that extrapolate to the same point at zero pressure, demonstrating a linear relationship and a common origin.
Absolute zero is found to be -273.15 °C. A negative pressure has no meaning, suggesting the temperature cannot go lower than -273.15 °C, indicating a theoretical lower limit to temperature.
The Kelvin scale has its zero at absolute zero, with the same degree size as Celsius, making it convenient for scientific calculations involving temperature changes.
Conversion between Kelvin (T) and Celsius (t): T = t + 273.15
SI base unit for temperature is the kelvin (K). It is used in virtually all scientific contexts due to its absolute nature.
For an ideal gas at constant volume, the relationship between temperature and pressure is: \frac{T2}{T1} = \frac{P2}{P1}
The Kelvin temperature can be defined using the triple point of water (where solid, liquid, and vapor coexist) at 0.01 °C and a water-vapor pressure of 610 Pa. This provides a highly reproducible and precise reference point.
The triple-point temperature is defined as T {triple} = 273.16 K. This is the temperature at which water can exist in all three phases in equilibrium.
Kelvin temperature is defined as: T = 273.16 \frac{P}{P {triple}} at constant volume. This equation allows for the precise measurement of temperature based on pressure readings at a constant volume.
Thermal expansion of matter
Most materials expand when heated and contract when cooled, a phenomenon exploited in various engineering applications.
Thermal expansion is related to changes in the average separation between atoms or molecules, driven by increased kinetic energy at higher temperatures.
The interatomic potential energy curve determines the mean separation between constituent particles, showing how potential energy varies with the distance between atoms.
In the vicinity of the equilibrium position r_0, the potential curve is parabolic, leading to simple harmonic oscillations. This is an approximation valid for small deviations from equilibrium.
Thermal expansion occurs because the real potential energy curve deviates from a parabolic shape. Anharmonicity in the potential energy curve leads to asymmetric vibrations.
As temperature rises, atomic vibrations increase in amplitude, and atoms spend more time at larger separations, leading to expansion. This is because the potential energy curve is not symmetric, so the average separation increases with energy.
Linear Expansion of Solids
The increase in one dimension of a solid is called linear expansion, crucial for designing structures to withstand temperature changes.
Factors affecting expansion/contraction:
Magnitude of the temperature change \Delta T: Larger temperature changes result in greater expansion or contraction.
Initial linear dimension of interest l_0: Longer objects expand more for the same temperature change.
Change in length is proportional to both initial length and temperature change: \Delta l \propto l0 (T - T0)
Introducing the mean coefficient of linear expansion \alpha:
\Delta l = \alpha l0 (T - T0)
\alpha = \frac{\Delta l}{l0 (T - T0)} (units: °C⁻¹). This coefficient is material-specific and indicates how much a material expands per degree Celsius.
The relationship between length and temperature is expressed as: l = l0 [1 + \alpha (T - T0)]
This result is valid for moderate temperature changes, where the coefficient of linear expansion can be considered constant.
Expansion of Holes
For a homogeneous body with holes, the change in length is strictly proportional to the original length l. Holes expand or contract just like the solid material.
The sizes of holes or cavities expand or contract as if the holes were filled with the material of the body. This is important in engineering design where holes are present in structures.
The Binomial Theorem
Used to expand the quantity (1 + x)^n where x \ll 1, often used in approximations for thermal expansion calculations.
Series expansion: (1 + x)^n = 1 + nx + \frac{n(n - 1)}{2!}x^2 + \frac{n(n - 1)(n - 2)}{3!}x^3 + \cdots
Approximation for x \ll 1: (1 + x)^n \approx 1 + nx
Cubical (or Volume) Expansion
Substances expand in all three dimensions when heated, crucial for understanding the behavior of solids, liquids, and gases.
The increase in volume is known as cubical (or volume) expansion. It influences density and fluid behavior.
If the temperature change \Delta T is not too great, the increase in volume \Delta V is proportional to \Delta T and the original volume V_0.
The coefficient of cubical (or volume) expansion is defined as:
\beta = \frac{\Delta V}{V_0 \Delta T}
The mean coefficient of cubical expansion is defined as the fractional change in volume per degree Celsius change in temperature. It is a measure of how much a substance's volume changes with temperature.
The coefficient of cubical expansion \beta is defined for all three states of matter: solids, liquids, and gases, although its magnitude varies significantly.
The term linear expansion has no meaning for a liquid or a gas, since the fluid always takes the shape of its container. Volume expansion is the relevant parameter for fluids.
It follows from the definition that: V = V0[1 + \beta(T - T0)]
Where V is the volume at some temperature T.
Thermal Stress
Occurs when the ends of a rod or slab of material are rigidly fixed, preventing thermal expansion or contraction, leading to internal stresses.
Thermal stress is calculated in two steps:
Calculate the amount the rod/slab would expand (or contract): \Delta l = \alpha l_0 \Delta T
Calculate the force required to compress (or expand) the material back to its original length. This force is what causes thermal stress.
The force per unit area (or stress) is related to the fractional change of length (the strain) by: \frac{F}{A} = Y \frac{\Delta l}{l_0}, where Y is Young’s modulus for the material. Young's modulus is a measure of stiffness.
This gives rise to an equation for the thermal stress as:\frac{F}{A} = Y \alpha \Delta T
For a three-dimensional object, the stress exerted on the object is the pressure exerted over the entire surface of the object. Thermal stress can lead to significant pressure on the object.
The bulk modulus (B) of a material is defined as: B = \frac{\Delta P}{\Delta V / V_0}. It relates pressure change to volume change.
Expansion of Liquids
Liquids have no shape and therefore no fixed dimensions, so cubical expansion is the only concern. They take the shape of their container.
Liquids must be held in some containing vessel, so the apparent expansion of the liquid is less than the real value due to the expansion of the vessel. The vessel itself expands, affecting the observed liquid expansion.
Real expansion of liquid = apparent expansion of liquid + real expansion of vessel.
\betar V0 \Delta T = \betaa V0 \Delta T + \beta {ves} V_0 \Delta T
\betar = \betaa + \beta {ves}, or \betaa = \betar - \beta {ves}.
Expansion of Gases
The volume occupied by a gas is very sensitive to both temperature and pressure. Gases are highly compressible and expansible.
To measure the thermal expansion of a gas, the pressure must be kept constant. Otherwise, both temperature and pressure changes affect volume.
The zero coefficient of cubical expansion of a gas at constant pressure \beta0 is defined as: \beta0 = \frac{V - V0}{V0 T}, where V_0 is the volume of a given mass of gas at 0 °C and V is the volume at a temperature T
The result of many careful experiments indicates that \beta_0 is the same for all gases provided the pressure is low enough so that the gas may be regarded as ideal. This is a key result from experiments on gas behavior.
This value of \beta0 is \beta0 = \frac{1}{273.15} °C⁻¹ (
Relative Thermal Expansion
Typical orders of magnitude of thermal expansion:
Solid: \beta between 10^{-5} and 10^{-6} °C⁻¹
Liquid: \beta between 10^{-4} and 10^{-5} °C⁻¹
Gas: \beta_0 approximately 10^{-3} °C⁻¹
The thermal expansion of a gas is about a thousand times greater than that of a typical solid for a given temperature change. Gases are much more responsive to temperature changes.
Important Relationship Between α and β
Comparison of volume expansion equations yields the relationship: \beta = 3\alpha
This result only applies to isotropic solids, where α is the same in all directions. Many solids (especially crystals) are anisotropic. For anisotropic materials, the relationship is more complex.
Effect of Expansion on Density
When a substance is heated, its volume increases but its mass remains constant, so density decreases. Density is inversely proportional to volume.
\rhoT = \frac{\rho0}{(1 + \beta \Delta T)}
Anomalous Behavior of Water
Water's volume decreases when heated from 0°C to 4°C. Above 4°C, it expands normally. This is unusual behavior compared to most substances.
Maximum density occurs at 4°C. This is a critical temperature for aquatic life.
This behavior influences how lakes freeze: surface water cools, sinks until the entire lake reaches 4°C. Further cooling makes surface water less dense, leading to ice formation on top, insulating the lake and allowing aquatic life to survive. This is crucial for the survival of aquatic ecosystems in cold climates.
Heat
Introduction to Heat
Heat and temperature are related but distinct concepts. Temperature is a measure of the average kinetic energy of molecules, while heat is energy transfer due to temperature differences.
Temperature indicates how hot or cold an object is, while heat refers to the energy transferred due to temperature differences. Heat always flows from a hotter object to a colder one.
Internal energy is the sum of molecular energies (translational, rotational, vibrational kinetic energies, and potential energy). It represents the total energy within a system.
Heat is energy in transit due to a temperature difference; substances contain internal energy, not heat. Heat is only the energy being transferred.
Heat Capacity
Heat capacity (C) is the amount of heat (Q) required to raise the temperature of a body by \Delta T: C = \frac{Q}{\Delta T} (SI units: J °C⁻¹). Heat capacity depends on the mass and material of the object.
Specific heat capacity (c) is the amount of heat (Q) required to raise the temperature of mass (m) by \Delta T: c = \frac{Q}{m \Delta T} (SI units: J kg⁻¹ °C⁻¹). Specific heat capacity is an intensive property, depending only on the material.
Changes of Phase
During a phase change, the temperature remains constant despite the addition or removal of heat (e.g., melting ice at 0 °C or boiling water at 100 °C under standard atmospheric pressure). The energy goes into changing the state of the substance.
Specific heat of transformation (L) is defined as the heat (Q) required to change the phase of mass (m) without changing the temperature: L = \frac{Q}{m} (SI units: J kg⁻¹). This is also known as latent heat.
The equations Q = mc\Delta T and Q = mL are important for calorimetry calculations. They are used to calculate heat transfer in various processes.
Key points:
Transformation can be melting/fusion, freezing, vaporization, condensation, or sublimation. These are all examples of phase changes.
Specific heats of transformation are also called ‘specific latent heats’ or ‘latent heats’. The term 'latent' implies that the heat is hidden or stored during the phase change.
Energy is required during melting and vaporization, and energy is released during freezing and condensation. These processes involve breaking or forming intermolecular bonds.
During melting Q = mLf (f denotes fusion). Lf is the latent heat of fusion.
During vaporization Q = mLv (v denotes vaporization). Lv is the latent heat of vaporization.
Heat Units
SI unit of heat is the joule (J). It is the standard unit of energy.
Dietitians and nutritionists use the Calorie (with a capital C) to specify the energy content of foods: 1 \text{ Calorie} = 4.1868 \text{ J}. This is also sometimes referred to as a kilocalorie (kcal).
Distinctions Between Heat and Temperature
Care should be taken to distinguish between the terms heat and temperature. They are often confused but have distinct meanings.
Heat is the net energy transferred spontaneously from regions of high temperature to low temperature. It is energy in transit.
Temperature indicates whether or not heat will flow, and in which direction. Temperature is a state variable.
Objects at the same temperature are in thermal equilibrium, and no heat will flow. There is no net energy transfer between them.
A quantity of heat that is transferred depends on the temperature difference between the materials, as well as their masses and specific heat capacities. These factors determine the amount of energy transferred.
Calorimetry calculations
Introduction to Calorimetry Calculations
Calorimetry involves ‘measuring heat’ and performing calculations with heat. It is essential for determining thermal properties of substances.
When substances at different temperatures are brought into contact, they reach thermal equilibrium after mixing. The final temperature depends on the masses, specific heats, and initial temperatures of the substances.
The fundamental principle is the law of conservation of energy: Heat 'lost' by hot substance(s) = Heat 'gained' by cold substance(s).
\text{Heat 'lost'} = \text{Heat 'gained'}
It is often assumed that heat transferred to the surroundings is negligible, which can be minimized practically. Calorimeters are designed to minimize heat loss to the environment.
Mechanisms of Heat Transfer
Introduction to Heat Transfer Mechanisms
Heat is transmitted from one place to another through three primary processes: conduction, convection, and radiation.
Conduction: Energy transfer through a body (or between bodies in contact) due to interatomic or intermolecular collisions. It requires physical contact.
Convection: Energy transfer from one place to another by the bulk motion of a fluid. It involves the movement of heated fluids.
Radiation: Heat transfer by electromagnetic radiation, with no need for matter between bodies. It can occur in a vacuum.
Conduction
Suppose a metal rod of length \Delta x and cross-sectional area A has one end held in a flame. Heat reaches the other end by conduction. This is a typical setup to illustrate heat conduction.
In the steady state: the rate of heat flow along the rod Q/t (where Q is the heat conducted in time t) is proportional to:
The cross-sectional area A, i.e., \frac{Q}{t} \propto A
The temperature gradient \frac{\Delta T}{\Delta x} across the ends of the rod, where \Delta T is the temperature difference.
Combining these results: \frac{Q}{t} = \lambda A \frac{\Delta T}{\Delta x} , where \lambda is the thermal conductivity of the material. Thermal conductivity indicates how well a material conducts heat.
In this equation, the temperature gradient \frac{\Delta T}{\Delta x} is taken as a positive quantity. The temperature gradient drives the heat flow.
The heat power \frac{Q}{t} is measured in J s^(−1) which equals a Watt (W). Watt is the unit of power.
Therefore, the SI unit of \lambda is [λ] = [Q/t] / [A][∆T/∆x] = W / (m² · °C/m) = W m^(-1) °C^(-1)
Convection
Convection is the transfer of heat by mass motion of a fluid from one region of space to another. It involves the movement of heated fluids.
Familiar examples include heaters, car engine cooling systems, and blood flow in the body. These systems rely on convection to transfer heat.
If fluid is circulated by a blower or pump, it's called forced convection; if by density differences due to thermal effects, it's natural convection. Forced convection is more efficient than natural convection.
Experimental facts:
Rate of heat transfer is proportional to surface area. Larger surface area allows for more heat transfer.
Viscosity slows natural convection near surfaces. Viscosity affects the fluid's ability to move.
Forced convection decreases the thickness of the surface film, increasing the rate of heat transfer. This leads to the wind-chill factor. The wind-chill factor is the perceived decrease in air temperature felt by the body on exposed skin due to the flow of air.
The rate of heat transfer is approximately proportional to the 5/4 power of the temperature difference between the surface and the main body of fluid. This is an empirical relationship.
Radiation
The transfer of energy by radiation does not require the intervention of material media. It can occur in a vacuum, like space.
All bodies above absolute zero (0 K) radiate energy, with the rate dependent on the body's temperature and surface. Even objects that appear cold emit radiation.
A black body absorbs all incident radiant energy. It is a perfect absorber and emitter.
The absorptivity α of a surface for radiant energy is defined as the fraction of the total incident radiation absorbed by the surface; for a black body, α = 1.
Bodies that are good absorbers are also good emitters, and poor absorbers are poor emitters. This is known as Kirchhoff's Law.
The rate at which energy is radiated (i.e., the power) from a surface A at an absolute temperature T is: \frac{Q}{t} = \sigma \varepsilon A T^4,
where \sigma is the Stefan-Boltzmann constant (\sigma = 5.670 × 10⁻⁸ W m⁻² K⁻⁴), and \varepsilon is the emissivity of the body.
The emissivity is defined as the rate at which energy is emitted from the body relative to that emitted by an identical black body at the same temperature. It ranges from 0 to 1.
α = ε for any body (Kirchhoff’s Law). It relates absorptivity and emissivity.
Any object radiates and absorbs energy from other bodies. The net rate of radiant energy transfer is given by: \frac{Q}{t} = \sigma \varepsilon A (T1^4 - T2^4), assuming the object is at temperature T₁ and the environment is at temperature T₂.
For radiation from the sun, note that 1350 J of energy strikes Earth's atmosphere from the sun per second per square meter (solar constant). The solar constant is a measure of solar radiation intensity.
An object facing the Sun absorbs heat at: \frac{Q}{t} = I \varepsilon A \cos \theta, where \theta is the angle between the sun's rays and a line perpendicular to area A.
Spectral Distribution of Black Body Radiation
Examines emitted power per unit wavelength interval for black body radiation across different temperatures. The distribution changes with temperature.
For a given temperature T1, there's a wavelength \lambda1 at which intensity is maximum. This wavelength shifts with temperature.
Wien's Displacement Law: \lambda {max}T = constant
Experimentally, \lambda {max}T = 2.897 \times 10^{-3} K m
Pyrometers use radiation to measure temperature, such as determining the temperature of distant celestial objects. They measure infrared radiation.
Radiation emitted due to temperature is called thermal radiation; objects emit and absorb radiation from surroundings. This is a fundamental aspect of thermal physics.
The Gas Laws
Revision of Basic Terminology
Atomic and Molecular Mass
The unified atomic mass unit (u) is defined as one-twelfth of the mass of an unbound neutral atom of carbon-12 in its nuclear and electronic ground state (at rest). It is a standard unit for atomic masses.
The atomic mass of an element is the mass of an atom of that element on a scale on which the mass of an atom of ^{12}_6C = 12 u.
Conversion: 1 u = 1.660 538 921(73) \times 10^{-27} kg
The molecular mass is the sum of the atomic masses of its constituent atoms. It is used for molecules.
The Mole and Avogadro’s Number
One mole of any substance contains as many particles as there are atoms in 12 g of the isotope ^{12}C. This is a fundamental definition in chemistry.
12 g of ^{12}C contains 6.022 \times 10^{23} atoms (Avogadro’s number, N_A).
One mole of any substance has a mass in grams equal to its relative atomic or molecular mass. This is a key relationship for converting between mass and moles.
The number of moles n is given by: n= \frac{\text{mass (g)}}{\text{molar mass (g)}} = \frac{m}{M}
Standard Temperature and Pressure (STP)
Standard temperature = 273.15 K. It is used as a reference point for gas volumes.
Standard pressure = 101 325 Pa = 1 Atm = 760 mmHg. These are equivalent pressure units.
One mole of any gas at STP occupies 22.4 litres. This is the molar volume of a gas at STP.
Absolute and Gauge Pressure
The pressure at a given depth h below the surface of a fluid of density \rho is given by: P = P0 + h\rho g, where P0 is the atmospheric pressure at the surface of the fluid. This is the hydrostatic pressure equation.
The pressure P in Equation (30) is the total or absolute pressure at the depth h.
The pressure (h \rho g) due only to the liquid is the gauge pressure. It is the pressure relative to atmospheric pressure.
Equation of State of an Ideal Gas
For a fixed mass of gas at constant temperature: P1V1 = P2V2 or \frac{P}{V} = \text{constant} (Boyle’s Law). This law relates pressure and volume at constant temperature.
For a fixed mass of gas at constant pressure: \frac{V1}{T1} = \frac{V2}{T2} or \frac{V}{T} = \text{constant} (Charles’ Law), where T is the absolute temperature. This law relates volume and temperature at constant pressure.
For a fixed mass of gas at constant volume: \frac{P1}{T1} = \frac{P2}{T2} or \frac{P}{T} = \text{constant} (Gay–Lussac’s Law), where T is the absolute temperature. This law relates pressure and temperature at constant volume.
Combining the three laws yields:
\frac{P1V1}{T1} = \frac{P2V2}{T2} or \frac{PV}{T} = \text{constant,} where T is the absolute temperature
Since the mass of the gas m is fixed (and constant), Equation (32) can be written in the form \frac{PV}{T}=mr, where r is a constant for a particular gas.
From experiment that for different gases r = \frac{R}{M}, where R is a constant which is the same for all gases and M is the relative atomic or molecular mass
\frac{PV}{T} = m \frac{R}{M}
From Equation (29) (n = \frac{m}{M}) Equation (34) leads to \frac{PV}{T} = nR or PV = nRT, where T is the absolute temperature.
The numerical value of the universal gas constantR = 8.314 J mol^{−1} K^{−1}.
The behaviour of real gases conforms closely to Equation (35) except at high densities. Real gases deviate from ideal behavior at high pressures and low temperatures.
Equation (35) gives the relationship between the variables n, P, T and V and it is called the equation of state of an ideal gas (or the ideal gas equation).
Dalton’s Law of Partial Pressures
Consider a container of volume V filled with two non-interacting ideal gases A and B. The partial pressures of each gas are PA = \frac{nART}{V} and PB = \frac{nBRT}{V}.
The total pressure of the mixture is P = \frac{nRT}{V}, where n = nA + nB, and so
P = \frac{nRT}{V} = (nA + nB)\frac{RT}{V} = \frac{nART}{V}+ \frac{nBRT}{V} = PA + PB
Dalton’s law of partial pressures:
The total pressure of a mixture of non-interacting gases is equal to the sum of the partial pressures of the component gases: P\text{total} = P1 + P2 + P3 + …
Pressure Due to Vapor
When a vapor is compressed, the variation of pressure with volume is described in several steps that can be shown on a graph of pressure vs. volume. The behavior depends on whether the vapor is saturated or unsaturated.
When the vapor is unsaturated, a small isothermal compression does not result in any condensation of liquid. The vapor behaves like an ideal gas.
Once the vapor becomes saturated and the liquid appears in the cylinder, a decrease in volume does not lead to an increase in pressure, but only the condensation of more liquid. The fixed pressure which is observed when a saturated vapor is in equilibrium with its fluid is called the saturation vapor pressure (S.V.P.) of the substance for the particular temperature.
A liquid boils when its S.V.P. is equal to the pressure on its free surface. Thus if a liquid is open to the atmosphere it boils at the temperature for which the S.V.P. equals the atmospheric pressure
Pressure Due to Mixture of Gases and a Vapor
Boyle’s Law may be used to describe the behavior of a gas and an unsaturated vapor. Once the vapor is saturated and an excess of liquid is present, though, it can no longer be used.
If a gas and saturated vapor are compressed at constant temperature the vapor continues to exert a partial pressure equal to the constant S.V.P. at the prevailing temperature, but the partial pressure of the gas will obey Boyle’s law.
If P is the total pressure of the mixture, V its volume, and f the constant S.V.P. at the prevailing temperature, then (P-f)V = constant
Elementary kinetic theory
Basic Assumptions
The assumptions of the kinetic theory of gases are:
A gas consists of a large number of identical particles which are in continual, random motion. The particles move in all directions with varying speeds.
The particles move in straight lines and obey Newton’s laws of motion. They move freely until they collide with another particle or the container walls.
The particles exert no forces on each other except at the moment of collision. There are no intermolecular forces except during collisions.
The collisions are perfectly elastic and take negligible time. Kinetic energy is conserved during collisions, and the collision time is much shorter than the time between collisions.
The volume of the particles is negligible compared with the volume of the container. This is a key assumption for ideal gas behavior.
Pressure Exerted by Ideal Gas
The pressure of a gas results from the bombardment of particles on the container walls. The pressure is due to the force exerted by the particles on the walls.
Force on the wall is the rate of change of the momentum of particles bouncing off the wall. Each collision exerts a force on the wall.
For a cubical vessel with N molecules of mass m, the pressure P is given by
P = \frac{Nm \langle v_x^2 \rangle}{l^3} = \frac{Nm \langle v^2 \rangle}{3V}
Where \langle vx^2 \rangle is the average value of vx^2 for all N molecules, \langle v^2 \rangle is the average value of v^2=vx^2+vy^2+v_z^2, and V is the volume of the container
Thus PV = \frac{1}{3}Nm \langle v^2 \rangle
Root-Mean-Square Speed
The root-mean-square speed v_\text{rms} is defined as the square root of the mean-square speed:
$$v_\text{rms} = \sqrt