AP Physics C: Thermodynamics

Physics-centered review and inquiry covering thermodynamic laws, kinetic theory, ideal gases, work, entropy, p–V graphs, and absolute zero.

What You Should Be Able To Do

• Explain thermodynamics as a physics model connecting energy transfer, particle motion, and macroscopic variables.
• Use \(PV=nRT\) and \(PV=Nk_BT\) correctly with SI units.
• Interpret temperature as a measure of average translational kinetic energy.
• Calculate work from a p–V process using area under a curve or an integral.
• Apply the first and second laws of thermodynamics to simple systems.
• Calculate entropy change for straightforward reversible processes.
• Use pressure-temperature data to estimate absolute zero by extrapolation.
• Distinguish conduction, convection, and radiation as mechanisms of heat transfer and apply Fourier’s law.
• Use \(Q=mL\) to calculate energy involved in phase changes (fusion and vaporization).
• Interpret heating curves and identify regions of phase change versus temperature change.
• Calculate linear and area thermal expansion using \(\Delta L=\alpha L_0 \Delta T\) and \(\Delta A=2\alpha A_0 \Delta T\).
• Solve thermal equilibrium problems where two or more objects exchange heat until reaching a common final temperature.

Glossary of Thermodynamics Terms

The Laws of Thermodynamics

Zeroth Law

If system A is in thermal equilibrium with system B, and B is in thermal equilibrium with system C, then A and C are in thermal equilibrium with each other. This is why thermometers work.

First Law

This version uses \(W_{\text{on}}\), the work done on the system. If work is done by the system, then \(W_{\text{on}}\) is negative.

Second Law

For any real process, the total entropy of the universe does not decrease:

Third Law

As a perfect crystal approaches absolute zero, its entropy approaches a minimum value. In practice, no physical process can cool a system all the way to \(0\ \text{K}\).

The Ideal Gas Law from a Physics Perspective

The chemistry form of the ideal gas law is:

The physics form is often more useful because it describes particles directly:

Connecting Moles and Particles

where \(N_A=6.022\times10^{23}\ \text{particles/mol}\). Since \(R=N_Ak_B\), the two forms of the ideal gas law are the same physics written at different scales.

Common Physics Mistakes

• Using Celsius in \(PV=nRT\). Temperature must be in kelvin.
• Using liters instead of cubic meters without converting.
• Thinking gas particles push continuously on the walls. The force comes from many tiny impulse collisions.
• Thinking ideal gas particles have no energy. They have kinetic energy; they are assumed to have negligible intermolecular potential energy.

Kinetic Theory of Gases

Kinetic theory explains thermodynamics using Newtonian mechanics and statistics. For an ideal monatomic gas:

The total internal energy is:

A deeper kinetic theory result connects pressure to molecular speed:

This says pressure increases when particles are more massive, more numerous, moving faster, or confined to a smaller volume.

Calculus-Based Work in Thermodynamics

For a quasi-static expansion or compression, the work done by a gas is the area under the \(P\)-versus-\(V\) graph:

Using work done on the gas:

Entropy and the Second Law

Entropy change for a reversible process is calculated using:

For a constant-temperature reversible process:

For heating or cooling a substance with approximately constant specific heat:

Heat Transfer

Heat flows from hotter objects to cooler objects through three mechanisms: conduction, convection, and radiation. The quantity of heat transferred to change the temperature of a substance (without a phase change) is:

where \(m\) is the mass, \(c\) is the specific heat capacity, and \(\Delta T = T_f - T_i\).

Conduction

Conduction is heat transfer through direct molecular contact. Energy passes from faster-vibrating molecules to slower ones. Fourier’s law gives the rate of heat flow through a material:

where \(k\) is the thermal conductivity of the material (W/(m·K)), \(A\) is the cross-sectional area, \(\Delta T\) is the temperature difference across the material, and \(L\) is the thickness.

Convection

Convection is heat transfer by the bulk motion of a fluid. Warmer fluid is less dense and rises, while cooler fluid sinks, setting up convection currents. Convection can be natural (driven by buoyancy) or forced (driven by a fan or pump).

Radiation

All objects with temperature above absolute zero emit electromagnetic radiation. The power radiated by an object is given by the Stefan–Boltzmann law:

where \(\varepsilon\) is emissivity (0 to 1), \(\sigma = 5.67\times10^{-8}\ \text{W/(m}^2\text{K}^4)\) is the Stefan–Boltzmann constant, \(A\) is the surface area, and \(T\) is the absolute temperature in kelvin. Dark, rough surfaces have higher emissivity than shiny, polished surfaces.

Phase Changes and Latent Heat

Matter exists in three common phases: solid, liquid, and gas. Transitions between these phases are called phase changes. During a phase change, energy is absorbed or released while the temperature remains constant. The energy goes into breaking or forming intermolecular bonds rather than changing molecular kinetic energy.

Key Phase Transitions

Latent Heat

The energy required for a phase change is calculated using:

where \(m\) is the mass and \(L\) is the latent heat (specific to the substance and the transition). The temperature does not change during this process.

Heat of Fusion (\(L_f\))

The latent heat of fusion is the energy per unit mass required to change a substance from solid to liquid (or released going from liquid to solid) at its melting point:

For water, \(L_f=3.34\times10^5\ \text{J/kg}\) at \(0^\circ\text{C}\). This means it takes \(334\ \text{kJ}\) to melt \(1\ \text{kg}\) of ice at \(0^\circ\text{C}\) into water at \(0^\circ\text{C}\).

Heat of Vaporization (\(L_v\))

The latent heat of vaporization is the energy per unit mass required to change a substance from liquid to gas (or released going from gas to liquid) at its boiling point:

For water, \(L_v=2.26\times10^6\ \text{J/kg}\) at \(100^\circ\text{C}\). This is nearly seven times larger than \(L_f\), because converting liquid to gas requires completely separating molecules from their neighbors.

Heating Curves

A heating curve plots temperature versus heat added for a substance. It has five distinct regions for a substance going from solid to gas:

1. Solid heating: Temperature rises. Use \(Q=mc_{\text{solid}}\Delta T\).
2. Melting plateau: Temperature stays at the melting point. Use \(Q=mL_f\).
3. Liquid heating: Temperature rises. Use \(Q=mc_{\text{liquid}}\Delta T\).
4. Boiling plateau: Temperature stays at the boiling point. Use \(Q=mL_v\).
5. Gas heating: Temperature rises. Use \(Q=mc_{\text{gas}}\Delta T\).

Thermal Expansion

Most materials expand when heated because increased thermal energy causes atoms to vibrate with larger amplitude around their equilibrium positions. Thermal expansion is important in engineering: bridges have expansion joints, railroad tracks are laid with gaps, and concrete highways have relief joints.

Linear Expansion (One Dimension)

For a solid object with initial length \(L_0\), the change in length due to a temperature change \(\Delta T\) is:

where \(\alpha\) is the coefficient of linear expansion in units of \(1/\text{K}\) or \(1/{}^\circ\text{C}\). The new length is:

Area Expansion (Two Dimensions)

For a flat surface or sheet with initial area \(A_0\), the change in area is approximately:

This approximation comes from expanding both length and width: if \(L\) and \(W\) each grow by a factor of \((1+\alpha\,\Delta T)\), then:

The \(\alpha^2(\Delta T)^2\) term is negligibly small for typical temperature changes.

Constants and Known Values

Reference tables for thermodynamics calculations. Values are approximate and at standard conditions unless noted.

Fundamental Constants

Specific Heat Capacities

Latent Heats

Thermal Conductivities

Coefficients of Linear Expansion

Thermal Equilibrium Problems

When two or more objects at different temperatures are placed in thermal contact (and insulated from the surroundings), heat flows from the hotter object to the cooler one until they reach a common final temperature \(T_f\). By conservation of energy:

or equivalently, the total heat gained by cooler objects equals the total heat lost by warmer objects:

Graph Interpretation

Inquiry Lab: Determining Absolute Zero

Purpose

Use pressure-temperature data for a fixed-volume gas to estimate absolute zero. You will generate your own graph, determine a best-fit line, and extrapolate to the temperature where pressure would become zero.

Physics Background

For a fixed amount of gas at constant volume:

Since kelvin temperature is related to Celsius temperature by:

a graph of pressure \(P\) versus Celsius temperature \(T_C\) should be approximately linear. If the line is extended until \(P=0\), the x-intercept estimates absolute zero.

Provided Data Set

Student Tasks

1. Plot pressure \(P\) on the vertical axis and Celsius temperature \(T_C\) on the horizontal axis.
2. Draw or calculate a best-fit line.
3. Write the equation of your line in the form \(P=mT_C+b\).
4. Set \(P=0\) and solve for the x-intercept.
5. Use the x-intercept as your experimental estimate of absolute zero.
6. Calculate percent error using \(-273.15^\circ\text{C}\) as the accepted value.
7. Explain why this experiment supports using the kelvin temperature scale.

Blank Graph Space

Use this space if printing, or recreate it in your notebook or spreadsheet.

Analysis Questions

1. What physical variable was held constant in this experiment?
2. Why should pressure decrease as temperature decreases?
3. Why is it reasonable to extrapolate the trend even though no data was collected near \(-273^\circ\text{C}\)?
4. Why would a real gas stop behaving ideally before reaching absolute zero?
5. How would random measurement error affect the x-intercept?

Practice Problems with Hidden Solutions

Check Understanding

Use these quick checks to test the main ideas. Your score is not saved.