Physics Chapter 17: Temperature, Thermal Expansion, and the Ideal Gas Law Notes

Thermodynamics and the Atomic Theory of Matter

Thermodynamics is defined as the branch of physics that studies thermal effects using macroscopic quantities. These macroscopic quantities include:

• Pressure ($P$)
• Temperature ($T$)
• Volume ($V$)
• Internal energy ($U$)

Atomic Theory of Matter is based on experiments that led Dalton to develop a theory regarding the structure of matter. This theory consists of four main concepts:

• All matter is composed of tiny, indivisible particles called atoms.
• Atoms of each element are exactly alike and possess the same mass.
• An atom of one element cannot be changed into an atom of a different element.
• Atoms of different elements can join together to form compounds.

Atomic and molecular masses are measured in unified atomic mass units ($u$). This unit is defined such that the carbon-12 atom has a mass of exactly $12.0000\,u$. The conversion to kilograms is:

• $1\,u = 1.6605 \times 10^{-27}\,kg$

Brownian motion refers to the jittery motion of tiny flecks observed in water. This motion is the result of collisions between these flecks and individual water molecules. On a microscopic scale, the arrangement of molecules differs significantly between the three states of matter:

• Solids
• Liquids
• Gases

Temperature and Thermometers

Temperature is defined as a measure of how hot or cold an object is. A physical property on which a particular thermometer is based is called a thermometric property. The material used in the thermometer is known as the thermometric substance. For a property to be useful in a thermometer, it should vary linearly with temperature over a reasonable range, known as the thermometric range.

Physical properties used to measure temperature include:

• Change in volume (e.g., the expansion of a liquid).
• Change in the length of a mercury column.
• Change in the resistance of a wire.
• Change in the pressure of a gas at a constant volume.

Common instruments for temperature measurement include:

• Liquid-in-glass thermometers.
• Bimetallic strips.
• Constant-volume gas thermometers: These depend on the properties of an ideal gas, which remain consistent across wide temperature ranges. Consequently, they are used to calibrate other types of thermometers.

Three temperature scales are commonly used: Celsius ($^{\circ}C$), Fahrenheit ($^{\circ}F$), and Kelvin ($K$). Key reference points for water include:

• Freezing Point: $0^{\circ}C$ or $32^{\circ}F$
• Boiling Point: $100^{\circ}C$ or $212^{\circ}F$

Conversion formulas between these scales are:

• $T(^{\circ}C) = \frac{5}{9} [T(^{\circ}F) - 32]$
• $T(^{\circ}F) = \frac{9}{5} T(^{\circ}C) + 32$
• $T(K) = 273.15 + T(^{\circ}C)$

Specific examples for scale conversion and assessment:

• Normal body temperature: $98.6^{\circ}F$ (calculated as approximately $37^{\circ}C$).
• Room temperature: Often taken to be $68^{\circ}F$.
• Light bulb filament: Approximately $1900^{\circ}C$.
• Fever threshold: $38.9^{\circ}C$.
• Numerical equivalence: The temperature where $T_C = T_F$ is $-40$.

Types of Pressure

There are three distinct types of pressure discussed:

• Absolute Pressure: The sum of the pressure due to a fluid and the pressure due to the atmosphere.
• Gauge Pressure: The difference between the absolute pressure and the atmospheric pressure.
• Vacuum Pressure.

Equations and constants for pressure:

• $\text{Absolute Pressure} = \text{Gauge Pressure} + 1\,atm$
• $1\,atm = 101.3\,kPa = 1.013 \times 10^{5}\,N/m^2$
• Example: If $AP = 196\,kPa$, it represents absolute pressure.

Thermal Equilibrium and the Zeroth Law of Thermodynamics

When two objects are placed in thermal contact, they will eventually reach the same temperature, a state known as thermal equilibrium. In this condition, there is no net flow of energy between the objects.

The Zeroth Law of Thermodynamics states that if two objects are each in equilibrium with a third object, then they are in thermal equilibrium with each other.

Internal energy is the sum of the random distribution of kinetic and potential energies of the atoms or molecules within a system. Internal energy can be increased through heating or compression.

Thermal Expansion

Most materials expand when heated. Linear expansion describes the change in length of a solid:

• $\Delta L = \alpha L_0 \Delta T$
• Here, $\alpha$ is the coefficient of linear expansion.

Coefficients of Expansion at approximately $20^{\circ}C$ ($(C^{\circ})^{-1}$):

• Aluminum: $\alpha = 25 \times 10^{-6}$, $\beta = 75 \times 10^{-6}$
• Brass: $\alpha = 19 \times 10^{-6}$, $\beta = 56 \times 10^{-6}$
• Copper: $\alpha = 17 \times 10^{-6}$, $\beta = 50 \times 10^{-6}$
• Gold: $\alpha = 14 \times 10^{-6}$, $\beta = 42 \times 10^{-6}$
• Iron or Steel: $\alpha = 12 \times 10^{-6}$, $\beta = 35 \times 10^{-6}$
• Lead: $\alpha = 29 \times 10^{-6}$, $\beta = 87 \times 10^{-6}$
• Glass (Pyrex): $\alpha = 3 \times 10^{-6}$, $\beta = 9 \times 10^{-6}$
• Glass (ordinary): $\alpha = 9 \times 10^{-6}$, $\beta = 27 \times 10^{-6}$
• Quartz: $\alpha = 0.4 \times 10^{-6}$, $\beta = 1 \times 10^{-6}$
• Concrete and brick: $\alpha \approx 12 \times 10^{-6}$, $\beta \approx 36 \times 10^{-6}$
• Marble: $\alpha = 1.4 \text{ to } 3.5 \times 10^{-6}$, $\beta = 4 \text{ to } 10 \times 10^{-6}$
• Gasoline: $\beta = 950 \times 10^{-6}$
• Mercury: $\beta = 180 \times 10^{-6}$
• Ethyl alcohol: $\beta = 1100 \times 10^{-6}$
• Glycerin: $\beta = 500 \times 10^{-6}$
• Water: $\beta = 210 \times 10^{-6}$
• Air (and other gases at 1 atm): $\beta = 3400 \times 10^{-6}$

Volume expansion is given by:

• $\Delta V = \beta V_0 \Delta T$
• Here, $\beta$ is the coefficient of volume expansion. For uniform solids, $\beta \approx 3\alpha$.

Specific Scenarios and Examples:

• Bridge expansion: A steel bridge bed is $200\,m$ long at $20^{\circ}C$. Given a temperature range of $-30^{\circ}C$ to $+40^{\circ}C$, the contraction and expansion can be calculated using the structural length and $\alpha$ for steel.
• Circular rings: When a thin, circular ring is heated in an oven, the hole in the ring actually gets larger.
• Ring on a rod: An iron ring at $20^{\circ}C$ has an inside diameter of $6.420\,cm$, while the rod diameter is $6.445\,cm$. To fit, the ring hole must expand to be slightly larger than the rod ($6.445 + 0.008\,cm$). The required temperature can be found using the expansion formula.
• Tight jar lid: Holding a metal lid under hot water causes the lid (usually metal) to expand more than the glass jar, making it easier to open.
• Gas tank overflow: A $70\,L$ steel tank is filled with gasoline at $20^{\circ}C$. When the temperature reaches $40^{\circ}C$, the gasoline expands more than the steel tank ($\beta_{gasoline} > \beta_{steel}$), causing overflow.

Thermal Anomaly of Water:

• Water behaves differently than most substances; its minimum volume (maximum density) occurs at $4^{\circ}C$. As it cools below $4^{\circ}C$ toward its freezing point, it expands.

Thermal Stresses

Thermal stress occur when a material is fixed at its ends and cannot expand or contract during temperature changes. This results in large compressive or tensile forces.

The force ($F$) required to keep a material from expanding is:

• $F = E A \alpha \Delta T$
• Where $E$ is Young’s modulus, $A$ is the cross-sectional area, $\alpha$ is the linear expansion coefficient, and $\Delta T$ is the change in temperature.

The resulting stress is:

• $\text{Stress} = \frac{F}{A} = E \alpha \Delta T$

Example 17-8: Concrete blocks on a highway are $10\,m$ long with no space between them. If placed at $10^{\circ}C$ and the temperature rises to $40^{\circ}C$, the compressive stress is calculated using the contact area of $0.20\,m^2$ to determine if fracture will occur.

The Gas Laws and Absolute Temperature Scale

Gases have no fixed volume and no fixed shape. The relationship between volume, pressure, temperature, and mass is described by an equation of state.

Boyle’s Law: The volume of a given amount of gas is inversely proportional to the pressure if the temperature remains constant ($V \propto \frac{1}{P}$).

Charles’s Law / Volume-Temperature relationship: Volume is linearly proportional to temperature at constant pressure, provided the temperature is significantly above the condensation point.

• Extrapolating this relationship, volume theoretically becomes zero at $-273.15^{\circ}C$, defined as absolute zero.
• This leads to the Kelvin (Absolute) scale where $0\,K = -273.15^{\circ}C$.
• Water freezing point: $273.15\,K$.
• Water boiling point: $373.15\,K$.

Pressure-Temperature relationship: When volume is constant, pressure is directly proportional to temperature.

Conceptual Warning: A closed glass jar should not be thrown into a campfire because the increasing temperature will cause the internal pressure to rise until the jar explodes.

The Ideal Gas Law

Combining the physical relations for volume, pressure, temperature, and quantity results in the Ideal Gas Law:

• $PV = nRT$
• In this equation, $n$ is the number of moles and $R$ is the universal gas constant.

A mole ($mol$) is the number of grams of a substance equal to its molecular mass:

• $1\,mol$ of $H_2$ has a mass of $2\,g$.
• $1\,mol$ of $Ne$ has a mass of $20\,g$.
• $1\,mol$ of $CO_2$ has a mass of $44\,g$.
• Number of moles calculation: $n = \frac{\text{mass (g)}}{\text{molecular mass (g/mol)}}$

Standard Temperature and Pressure (STP):

• $T = 273\,K$ ($0^{\circ}C$)
• $P = 1.00\,atm = 1.013 \times 10^5\,N/m^2 = 101.3\,kPa$
• The volume of $1.00\,mol$ of any ideal gas at STP is $22.4\,L$.

Problem Solving guidelines:

• Always measure temperature ($T$) in Kelvins.
• Pressure ($P$) must be the absolute pressure.

Example 17-11 (Helium balloon): A spherical balloon with a radius of $18.0\,cm$ at $20^{\circ}C$ has an internal pressure of $1.05\,atm$. The number of moles and mass of helium can be found using the Ideal Gas Law. Example 17-12 (Air in a room): Estimating the mass of air in a room ($5\,m \times 3\,m \times 2.5\,m$) at STP. Example 17-13 (Automobile tire): A tire is filled to a gauge pressure of $200\,kPa$ at $10^{\circ}C$. After driving $100\,km$, the temperature reaches $40^{\circ}C$. The new pressure is found by using absolute pressure in the gas law calculation.

Ideal Gas Law in Terms of Molecules and Avogadro’s Number

The number of molecules in one mole is constant for all gases. This is known as Avogadro’s number ($N_A$):

• $N_A = 6.022 \times 10^{23}\,mol^{-1}$

The total number of molecules ($N$) in a gas is $N = n N_A$. The Ideal Gas Law can then be written as:

• $PV = NkT$
• Here, $k$ is Boltzmann’s constant ($k = \frac{R}{N_A}$).

Calculations using molecular perspectives:

• Mass of a hydrogen atom: Determined using Avogadro’s number.
• Molecules in a breath: Estimating the number of molecules in a $1.0\,L$ breath of air at sea level.

Ideal Gas Temperature Scale—A Standard

This standard utilizes the constant-volume gas thermometer and the ideal gas law. It relies on two fixed points:

• Absolute zero: The point where pressure is zero ($0\,K$).
• The triple point of water: The specific temperature and pressure where all three phases (solid, liquid, gas) coexist. It is defined as $273.16\,K$. The pressure at this point is $4.58\,torr$.

The temperature is defined proportionally to pressure:

• $T = (273.16\,K) \frac{P}{P_{tp}}$
• To determine temperature accurately using a real gas, the pressure must be kept as low as possible.