Gas Laws, Thermal Energy Transfers and Thermodynamics Notes

B.1 Thermal Energy Transfers

• Thermal Energy: Understanding thermal energy and its presence in daily life.

Molecular Theory

• Matter exists in three main states: solid, liquid, and gas.

• Plasma is not discussed in IB physics.

• All matter is composed of individual particles, each possessing its own energy.

• The behavior of these particles dictates the physical properties and behavior of the material.

• Example: The difference between ice and water is the energy of $H_2O$ molecules.

Kinetic Theory of Matter

• States of matter are differentiated by the degree of movement allowed between particles.

• Solids: Particles vibrate but do not translate or rotate.

• Liquids: Particles have weaker interactions, allowing more movement but still experience attraction.

• Gases: Particles move freely without being bound by attraction.

• This theory is fundamental to the discussions in B.1.

Terminology

• Temperature: Measurement of the average kinetic energy per particle.

  • Symbol: T

  • Units: Degrees Celsius (°C) or Kelvin (K)

• Internal Energy ($E_{int}$): Total energy (kinetic + potential) of particles inside a substance.

  • It's a sum, not an average, and is different from temperature.

  • Example: A large quantity of snow can have higher internal energy than a lit match but lower temperature.

  • Symbol: $E_{int}$

  • Units: Joules (J)

• Heat (Q): Energy transferred from one substance to another (change in internal energy).

  • Net energy flows from hotter to cooler objects.

  • Symbol: Q

  • Units: Joules (J)

Specific Heat Capacity ($Q = mc\Delta T$)

• Specific heat capacity is a substance's property indicating the amount of heat energy needed to raise the temperature of a unit mass by one degree Celsius (or one Kelvin).

• It determines how easy or hard a substance is to heat up.

• Q represents the quantity of heat energy transferred, measured in joules (J).

• m is the mass of the substance, in grams (g) or kilograms (kg).

• c is the specific heat capacity, with units of joules per gram per degree Celsius (J/g°C) or joules per kilogram per degree Celsius (J/kg°C).

• $\Delta T$ is the change in temperature in degrees Celsius (°C) or Kelvin (K), calculated as final temperature minus initial temperature.

SHC Example

• A 150g block of iron is cooled from 150°C to 25°C. The specific heat capacity of iron is 0.449 J/g°C. Find the energy released by the iron block.

• $Q=m \cdot c \cdot \Delta T$

• $Q=150 \cdot 0.449 \cdot (25-150)$

• $Q=8,418.75J$

• Answer: The energy released by the iron block is 8,418.75J.

Two Substances

• When two substances come into contact, energy is exchanged.

• Energy flows from the higher temperature object to the lower temperature object until they reach thermal equilibrium, where they have the same temperature.

• The amount of energy lost by one substance equals the amount of energy gained by the other: $Q<em>{lost} = Q</em>{gained}$

Example

• You pour 200g of tea at 95°C into a 150g glass cup, initially at 25°C. What will be the equilibrium temperature of the cup and tea?

• c(tea)=4186 J/kg°C

• c (glass) = 840 J/kg°C

• $Q<em>{lost} = Q</em>{gained}$

• $m<em>{tea} \cdot c</em>{tea} \cdot (T<em>{initial</em>{tea}} - T<em>{eq}) = m</em>{glass} \cdot c<em>{glass} \cdot (T</em>{eq} - T<em>{initial</em>{glass}})$

• $0.2 \cdot 4186 \cdot (95 - T<em>{eq}) = 0.15 \cdot 840 \cdot (T</em>{eq} - 25)$

• $T_{eq} = 85.6°C \approx 86°C$

Latent Heat (Q=mL)

• Specific latent heat is a property of a substance that quantifies the amount of heat needed to change the state of a unit mass of the substance without changing its temperature.

• It's a measure of the hidden energy required or released during a phase change, such as melting or boiling, which is why it's referred to as "latent" (meaning hidden).

• Specific Latent Heat of Fusion ($L_f$): This is the heat required to change a substance from solid to liquid (or vice versa) at its melting point. It's expressed in joules (sometimes kJ) per kilogram or gram (J/kg or J/g).

• Specific Latent Heat of Vaporization ($L_v$): This is the heat required to change a substance from liquid to gas (or vice versa) at its boiling point. It's also expressed in joules (sometimes kJ) per kilogram or gram (J/kg or J/g).

Latent Heat (Q=mL)

• Q is the heat absorbed or released during the phase change, in joules (J).

• m is the mass of the substance undergoing the phase change, in kilograms (kg).

• L is the specific latent heat, which could be $L<em>f$ or $L</em>v$ depending on the process, in joules per kilogram (J/kg).

• Ex. 100g of ice at 0°C needs to be turned into water at 0°C. The latent heat of fusion of ice is 334 J/g. Find the energy required to melt the ice.

• $Q=m \cdot L_f$

• $Q=100 \cdot 334$

• $Q=33,400J$

Thermal equilibrium when phase changes

• What if we take 10 kg of water at 90°C and drop 1 kg of ice at 0°C into it? We know the ice would melt and once melted, the water from the ice and hot water would reach equilibrium.

• The latent heat of fusion for ice is 334000 J/kg, and the specific heat capacity of water if 4184 J/kg°C

Example

• $Q<em>{ice} = -Q</em>{water}$

• $M \cdot L<em>f + M \cdot C</em>w \cdot \Delta T<em>w = -M</em>2 C<em>w \Delta T</em>w$

• $(1kg \cdot 334000 J/kg) + (1kg \cdot 4184 J/kg°C \cdot (T<em>{eq} - 0°C)) = -(10kg \cdot 4184 J/kg°C (T</em>{eq} - 90°C))$

• $334000J + 4184 T<em>{eq} = -41840 T</em>{eq} + 3765600J$

• $4184 T<em>{eq} + 41840 T</em>{eq} = 3765500J - 334000J$

• $46024 T_{eq} = 3431600$

• $T_{eq} = 74.6°C$

Average Kinetic Energy Per Molecule of a Gas

• We can convert between Temperature and average kinetic energy of molecules in a gas using the Boltzmann Constant: $1.38 × 10^{-23} J/K$ (in your data booklet)

• $E<em>k = \frac{3}{2}k</em>bT$

• $E_k$: the average translational kinetic energy of a particle in an ideal gas.

• 3/2 : combination of having 3 dimensions of movement (x/y/z) and the ½ of the kinetic energy formula

• $k_b$ : Boltzmann constant, a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas.

• T: This represents the absolute temperature of the gas measured in kelvins (K)

• Ex. What is the average kinetic energy of gas molecules at room temperature (293K)?

• $E<em>k = \frac{3}{2}k</em>bT$

• $E_k= 1.5<em>1.38</em>10^{-23}*293K$

• $E_k = 6.1*10^{-21} J$

Thermal Conduction

• Conduction is the transfer of energy from one particle to another through contact.

• Variables:

  • $Q/t$ = rate of energy flow from hot to cold (Watts)

  • k = thermal conductivity of a substance (W/mK)

  • A = area of the thermal conductor ($m^2$)

  • T = difference in temperature on either side of the conductor in K or C ($T<em>{hot} - T</em>{cold}$)

  • x = thickness of material that energy is transferring through (m)

• $\frac{\Delta Q}{\Delta t} = -kA \frac{\Delta T}{\Delta x}$

Applications

• Different substances conduct thermal energy differently; air is a great thermal insulator.

• Many windows have a gas layer between the panes for insulation.

• Foam restricts air movement to prevent convection.

How does it work for multiple layers?

• For thermal energy conducted through layers with different thermal properties:

  • Glass, air, and glass layers act like resistors connected in series.

  • Heat is like charge, passing through all layers.

  • Electrical Circuit Analogy:

    • Current (I) = Heat Flow rate ($Q/t$)

    • Voltage (V) = Temperature (T)

    • Electrical Resistance (R) = Thermal Resistance ($x/kA$)

    • Ohm’s Law: V=IR

    • 'Heat Law': $\Delta T = (Q/t)R$ or $\Delta T = (Q/t)(x/kA)$

  • Total thermal resistance is found by adding the ($x/kA$) of each material.

Convection

• Another method of thermal energy transfer (for fluids only), involving high-temperature particles moving to cooler areas, taking their energy with them.

• Heating a fluid increases particle energy and spacing, causing expansion and decreased density.

• If density drops below that of the surrounding fluid, the heated fluid rises as denser fluid pushes underneath it.

Radiation

• Radiation is the movement of energy without moving particles or a medium.

• This energy is in the form of electromagnetic radiation (EMR).

Quantifying Radiation - Stefan-Boltzmann Law

• The energy radiated by an object is proportional to the fourth power of its absolute temperature.

• Stefan-Boltzmann Law: $L = \sigma A T^4$

  • L is the luminosity (power radiated) in Watts (W).

  • $\sigma$ is the Stefan-Boltzmann constant ($5.67 × 10^{-8} W/m^2K^4$).

  • A is the area ($m^2$).

  • T is the absolute temperature in Kelvin (K).

  • All objects emit thermal radiation depending on their temperature.

    • High temperatures (Sun) emit visible and ultraviolet spectra.

    • Moderate Temperatures (Earth) emit infrared radiation.

Quantifying Radiation - Wein’s Displacement Law

• $\lambda_{max} = b/T$

  • $\lambda_{max}$ is the peak wavelength of the emitted radiation

  • b is Wien's displacement constant (approximately $2.897×10^{-3} m \cdot K$)

  • T is the absolute temperature of the black body in kelvins (K)

• Hotter objects emit more radiation, and the peak of their emitted spectrum shifts to shorter wavelengths.

• Hot objects (stars) appear blue or white, while cooler objects (stove heating element) glow red.

B.1 Formulas summarized

• $\rho = \frac{m}{V}$

• $E<em>K = \frac{3}{2}k</em>BT$

• $Q = mc\Delta T$

• $Q = mL$

• $\frac{\Delta Q}{\Delta t} = -kA \frac{\Delta T}{\Delta x}$

• $L = \sigma A T^4$

• $\lambda_{max} = \frac{b}{T}$

  • Density ($\rho = \frac{m}{V}$): Used to calculate the density of a substance, where m is mass and V is volume.

  • Average Kinetic Energy ($E<em>K = \frac{3}{2}k</em>BT$): Relates the average kinetic energy of particles in an ideal gas to the temperature, with $k_B$ as Boltzmann's constant.

  • Heat Transfer ($Q = mc\Delta T$): Describes the energy transferred to or from an object when its temperature changes.

  • Latent Heat ($Q = mL$): Refers to the energy needed for a phase change at constant temperature.

  • Heat Conduction Rate ($\frac{\Delta Q}{\Delta t} = -kA \frac{\Delta T}{\Delta x}$): Gives the rate at which heat is conducted through a material.

  • Radiated Power ($L = \sigma A T^4$): The Stefan-Boltzmann Law, describing the power radiated from a black body in terms of its temperature.

  • Wein's Displacement Law Constant (b): Relates the peak wavelength of emitted radiation from a black body to its temperature.

  • Peak Emission Wavelength ($\lambda_{max}$): Calculates the wavelength at which the emission of radiation from a black body is at its peak.

B.3 Gas Laws

• $P = \frac{F}{A}$

• $n = \frac{N}{N_A}$

• $\frac{P<em>1V</em>1}{T<em>1} = \frac{P</em>2V<em>2}{T</em>2}$

• $PV = NK_BT$ and $PV = nRT$

• $U = \frac{3}{2}NK_BT$ or $U = \frac{3}{2}nRT$

• P = constant (Boyle's law)

• $P<em>1V</em>1 = P<em>2V</em>2$

• Ideal gas law $PV = nRT$

• $\frac{P<em>1V</em>1}{T<em>1} = \frac{P</em>2V<em>2}{T</em>2}$

• Charles' law

Pressure

• Pressure (P), measured in Pascals (Pa), is defined as the force F (N) per unit area A ($m^2$) acting perpendicular to an object's surface.

• Decreasing area increases pressure; decreasing force decreases it.

Atmospheric Pressure

• The atmosphere has a mass of $5 × 10^{18} kg$ and contains roughly $10^{44}$ molecules.

• The pressure at sea level is about $10^5 Pa$.

• Altitude increase leads to fewer particles, less dense air, and lower pressure.

• Your body adapts to the pressure where you live, with internal pressure matching external pressure, achieving barometric equilibrium.

Pressure Example

• A cyclist applies a force of 800N to a pedal of a bicycle. The pedal can be modelled as a rectangle of dimensions 6.5cm × 2.5cm. Calculate the pressure applied between the shoe and the pedal.

Avogadro’s number + the Mole

• $N_A$ = Avogadro Constant = $6.02x10^{23}$

• N = number of molecules/particles

• n = amount of a substance (moles)

• m = mass of a substance (g or kg)

• M = molar mass of a particle (g/mol or kg/mol)

• n = amount of a substance (moles)

Individual Gas Laws

• Boyle’s Law

  • PV = constant

  • For a fixed amount of gas at constant temperature (isothermal), pressure (P) is inversely proportional to volume (V).

• Charles's Law

  • $\frac{V}{T} = constant$

  • For a fixed amount of gas at constant pressure (isobaric), volume (V) is directly proportional to absolute temperature (T).

• Pressure Law (Gay-Lussac’s Law)

  • $\frac{P}{T} = constant$

  • For a fixed amount of gas at constant volume (isovolumetric), pressure (P) is directly proportional to absolute temperature (T).

Individual Gas Laws

• Combined Gas Law

  • $\frac{PV}{T} = Constant$

  • Combines Boyle's Law, Charles's Law, and the Pressure Law

• Avogadro’s Law

  • $\frac{V}{n} = constant$

  • Volume (V) of a gas is directly proportional to the number of moles (n) at constant temperature and pressure.

Ideal Gas Law

• The Ideal Gas Law is a fundamental equation that describes the relationship between pressure, volume, temperature, and the amount of gas.

• P is the pressure (Pa)

• V is the volume ($m^3$)

• n is the number of moles of the gas

• R is the ideal gas constant (8.31 J/K/mol)

• T is the temperature in Kelvin (K)

• $PV=nRT$

B.4 Thermodynamics

• Thermodynamics

  • The branch of physics dealing with the relationships between heat and other forms of energy

• 1st Law of Thermodynamics

  • The internal energy of a system (ΔU) is equal to the heat (Q) added to the system minus the work (W) done by the system on its surroundings

  • $\Delta U=Q−W$

• Entropy (s)

  • A measure of the disorder or randomness of a system

• 2nd Law of Thermodynamics

  • The entropy of an isolated system always increases over time

First Law of TD

• $Q = \Delta U + W$

• $W = P\Delta V$

  • To calculate work done by a gas the following formula is used:

  • W = work in Joules (J)

  • P - pressure in pascals (Pa)

  • $\Delta V$ - Change in volume, Vf-Vi ($m^3$)

  • If work is done by the gas, $\Delta V$ will be positive, as will W. If work is done on the gas causing it to reduce its volume, $\Delta V$ and W will be negative. This can be calculated by observing a PV Graph.

  • Multiplying P and V to get work means taking the area of the PV graph

1st Law as it relates to Temperature

• $\Delta U=\frac{3}{2}N k_B \Delta T$

  • We can use a version of Boltzmann’s formula for average kinetic energy of a gas to figure out the change of temperature that results from the change of energy in a gas.

  • $\Delta U$ - Change of internal energy (J)

  • N - number of gas particles

  • $k_B$ - Boltzmann constant $1.38x10^{-23} J/K$

  • $\Delta T$ - Change of temperature (K)

Ideal Gas Law

• The Ideal Gas Law is a fundamental equation that describes the relationship between pressure, volume, temperature, and the amount of gas.

  • P is the pressure (Pa)

  • V is the volume ($m^3$)

  • n is the number of moles of the gas

  • R is the ideal gas constant (8.31 J/K/mol)

  • T is the temperature in Kelvin (K)

  • $PV=nRT$

Four Thermodynamic Processes

• Isobaric process: Q = ΔU + W

  • An isobaric process is one in which the pressure remains constant. This constant pressure allows for the work done by or on the system during expansion or compression to be directly proportional to the change in volume. In an isobaric process, the heat transferred to the system does work but also changes the internal energy of the system, following the first law of thermodynamics

• Isothermal process: Q = 0 + W or Q = W

  • In an isothermal process, the temperature of the system remains constant. For an ideal gas, this implies that the internal energy of the system remains unchanged Thus, any heat added to the system is used entirely to do work

• Isovolumetric process: Q = ΔU + 0 or Q = ΔU

  • An isovolumetric process, also known as an isochoric process, occurs at a constant volume. This means there is no work done by the system ( W = 0 ), as there is no volume change. Any heat transfer into or out of the system changes only the internal energy, not the external work, making the heat added equal to the change in internal energy

• Adiabatic process: 0 = ΔU + W, ΔU = −W for a compression, −ΔU = W for an expansion

  • An adiabatic process occurs without any heat transfer between a system and its surroundings (Q = 0). In adiabatic processes, changes in internal energy of the system are reflected as work done by or on the system. For an ideal gas undergoing an adiabatic process, the relationship between pressure and volume is described by $PV^{\frac{5}{3}} = constant$

Isovolumetric Processes

• 
  Let’s think about isovolumetric processes in terms of the ideal gas law: PV=nRT
  V, n, and R, are all unchanging in this equation, meaning: P/T = constant
  Why do we care?
  This means that P/T before the change occurs will be equal to P/T after the change occurs,
  meaning:
  P1/T1 = P2/T2
  On a P-V diagram, isochoric processes are represented by a vertical line

Isothermal Processes

In an isothermal process, the temperature of the system remains constant.
According to the ideal gas law PV=nRT, if T, n, and R, are constant, the
relationship PV=constant must hold.
Why It's Important: Maintaining constant temperature implies that any heat
added to the system goes into doing work or expanding the gas without
increasing its internal energy.
Graphical Interpretation: On a P-V diagram, isothermal processes are
represented by hyperbolic curves.
Equation: PV=constant or P1V1=P2V2

Isobaric Processes

Isobaric refers to processes occurring at a constant pressure. When pressure remains
unchanged, the volume and temperature relationship is direct; as one increases, so does
the other.
Why It's Important: This is common in everyday applications where the pressure of the
container cannot change significantly, such as in a balloon.
Graphical Interpretation: On a P-V diagram, isobaric processes are represented by
horizontal lines.
Equation: V/T = Constant or V1/T1 = V2/T2

Adiabatic Processes

Adiabatic processes occur without any heat exchange with the environment. This means all
the energy in the system is used in doing work or changing the internal energy.
Why It's Important: These processes are idealized scenarios that help in understanding the
limits of energy conservation and efficiency in thermodynamic cycles.
Formula: For ideal gases, the relationship between pressure and volume during an adiabatic
process can often be expressed as PVγ =constant, where γ is a constant that is based on heat
capacity and depends on the type of gas. This constant will be provided for the time being.
PVγ =constant
Carnot

Carnot

Carnot Engine - A Carnot engine is a theoretical construct used in
thermodynamics to define the maximum possible
efficiency a heat engine can achieve operating between two temperatures.
It operates on the Carnot cycle, which consists of four reversible processes: two
isothermal and two adiabatic.
How Does it Work?
Isothermal Expansion: The gas expands, absorbs heat from the hot reservoir, and
performs work on the surroundings.
Adiabatic Expansion: The gas expands further without heat exchange, temperature
decreases.
Isothermal Compression: The gas is compressed, releases heat to the cold
reservoir, and work is done on the gas.
Adiabatic Compression: The gas is compressed further, no heat exchange, and
temperature increases to the initial state.

Carnot - Efficiency

We can calculate the efficiency of of a Carnot Engine with the formula to above.
Tcold and Thot are the absolute temperatures of the cold and hot reservoirs respectively.
Why is it Important?
The Carnot cycle provides an upper limit to the efficiency of all heat engines. Real
engines have efficiencies lower than the Carnot efficiency due to practical limitations
like friction and non-reversible processes.
Limitations: The Carnot cycle is an idealized process; no real engine can perform at Carnot efficiency
because real processes are not completely reversible and involve energy losses.

The 2nd Law of Thermodynamics
The Second Law states that the total entropy of an isolated system can never decrease
over time.
Entropy is a measure of the disorder or randomness in a system. As a system’s energy
becomes more spread out and less available for doing work, it’s entropy increases.
It's a fundamental concept in thermodynamics reflecting the number of ways a system
can be arranged without changing its macroscopic properties.
For reversible processes, entropy can be described as ΔS=ΔQ/T
S represents entropy, and its units are Joules per Kelvin (J/K).
Q is the heat transferred in Joules (J).
T is the temperature in Kelvin (K).
Thermodynamic Entropy
This is the ‘big picture’ or macroscopic view of entropy. The equation describes how
much the entropy of a system changes when a quantity of heat ΔQ is added or removed
at a constant temperature. It is used primarily in macroscopic thermodynamic analysis,
especially in reversible processes.
For reversible processes, entropy can be described as ΔS=ΔQ/T
S represents entropy, and its units are Joules per Kelvin (J/K).
Q is the heat transferred in Joules (J).
T is the temperature at which heat is transferred in Kelvin (K).
Statistical Entropy
This is the microscopic view of entropy. This equation connects entropy
to the number of microscopic configurations that correspond to its
larger, macroscopic state. This is the main idea explaining why entropy
tends to increase.
S is the entropy (J/K).
kB is the Boltzmann constant (1.38×10−23 J/K)
Ω is the number of possible microstates of the system
LN stands for natural logarithm
Implications of the 2nd law
Because entropy always increases, no process involving energy transfer can be 100%
efficient. Some energy is always spread out towards less usable forms (heat).
Some consider this to be a defining factor in determining the direction of the flow of
time. From the big bang all the way to the heat death of the universe