Thermal Energy, Specific Heat, and Entropy in Gases and Solids

Thermal Energy and Specific Heat

Thermal Energy in Monatomic Gases

The thermal energy of a monatomic ideal gas is determined by the translational kinetic energy of its atoms. Each atom moves randomly, and the total energy is the sum of the kinetic energies of all atoms.

• Thermal Energy Formula: For N atoms at temperature T, the total thermal energy is given by: $E_{th} = N \epsilon_{avg}$ where $\epsilon_{avg}$ is the average kinetic energy per atom.

• Monatomic Gas: Atoms have only translational kinetic energy, with no rotational or vibrational energy.

• Specific Heat: The molar specific heat at constant volume for a monatomic gas is: $C_v = \frac{3}{2}R \approx 12.5\ \mathrm{J/(mol\cdot K)}$

Degrees of Freedom

Degrees of freedom refer to the independent ways in which a system can store energy. For a monatomic gas, these are the three translational directions (x, y, z).

• Translational Energy: $\epsilon = \epsilon_x + \epsilon_y + \epsilon_z$

• Other Modes: Diatomic and polyatomic molecules can also store energy in rotational and vibrational modes.

• Definition: The number of degrees of freedom is the number of independent energy storage modes.

Equipartition Theorem

The equipartition theorem states that thermal energy is equally distributed among all degrees of freedom. For each degree of freedom, the energy per mole is $\frac{1}{2}nRT$ or per particle $\frac{1}{2}Nk_BT$.

• Monatomic Gas: Three degrees of freedom (all translational): $E_{th} = \frac{3}{2}Nk_BT$

Thermal Energy in Solids

Degrees of Freedom in Solids

In solids, atoms are bound in a lattice and can vibrate about their equilibrium positions. Each atom has three translational and three vibrational (potential) degrees of freedom, totaling six.

• Thermal Energy: $E_{th} = 3Nk_BT = 3nRT$

• Potential and Kinetic Energy: Both contribute to the total energy in solids.

Dulong–Petit Law

The Dulong–Petit law predicts that the molar heat capacity at constant volume for many solids is approximately $3R$ at high temperatures. However, deviations occur at low temperatures or for certain materials.

• Heat Capacity: $C_v \approx 3R$ for most solids at room temperature.

• Deviations: Materials like diamond and silicon show lower heat capacities at low temperatures due to quantum effects.

Heat Transfer and Thermal Equilibrium

Thermal Interaction Between Systems

When two systems at different temperatures are separated by a thin barrier, energy is transferred via molecular collisions until thermal equilibrium is reached.

• Energy Transfer: Heat flows from the hotter to the cooler system until both reach the same temperature.

• Thermal Equilibrium: Achieved when the average kinetic energy (and thus temperature) is equal on both sides.

Irreversible and Reversible Processes

Irreversibility in Macroscopic Processes

Many natural processes, such as melting ice or mixing cream into coffee, are irreversible. The second law of thermodynamics explains this irreversibility through the concept of entropy.

• Second Law of Thermodynamics: The entropy of an isolated system never decreases; it increases until equilibrium is reached.

• Entropy: A measure of the dispersal of thermal energy.

Microscopic Reversibility vs. Macroscopic Irreversibility

While molecular collisions are reversible, the overall dispersal of energy in a system is not. Energy spontaneously spreads out and does not reconcentrate in one place.

Entropy and Its Statistical Interpretation

Microstates, Macrostates, and Multiplicity

Entropy quantifies the number of microscopic arrangements (microstates) corresponding to a macroscopic state (macrostate). The multiplicity $\Omega$ is the number of microstates for a given macrostate.

• Entropy Formula: $S = k_B \ln \Omega$

• Most Probable State: The equilibrium macrostate has the highest multiplicity and thus the highest entropy.

Entropy Change in Processes

The change in entropy for a reversible process is given by:

• $dS = \frac{dQ}{T}$

• For an isothermal process: $\Delta S = \frac{Q}{T}$

Entropy in Phase Changes and Heating

• Phase Change (e.g., melting ice): $\Delta S = \frac{Q}{T}$, where $Q$ is the heat absorbed at constant temperature.

• Heating at Constant Specific Heat: $\Delta S = Mc \ln\left(\frac{T_f}{T_i}\right)$

Entropy of an Ideal Gas

The entropy change for an ideal gas can be calculated as:

• $\Delta S_{gas} = nC_V \ln\left(\frac{T_f}{T_i}\right) + nR \ln\left(\frac{V_f}{V_i}\right)$

• For constant pressure: $\Delta S_{gas} = nC_P \ln\left(\frac{T_f}{T_i}\right)$

Entropy and the Universe

Entropy in Isolated and Non-Isolated Systems

For an isolated system, the total entropy change is always non-negative:

• $\Delta S_{isolated} \geq 0$

• For a system and its environment: $\Delta S_{universe} = \Delta S_{sys} + \Delta S_{env} \geq 0$

Free Expansion and Entropy

When a gas expands freely into a larger volume, its entropy increases even though its thermal energy does not change. This is an example of an irreversible process.

Summary Table: Degrees of Freedom and Specific Heat

System	Degrees of Freedom	Thermal Energy	Specific Heat $C_v$
Monatomic Gas	3 (translational)	$\frac{3}{2}nRT$	$\frac{3}{2}R$
Diatomic Gas	5 (3 translational, 2 rotational)	$\frac{5}{2}nRT$	$\frac{5}{2}R$
Solid	6 (3 translational, 3 vibrational)	$3nRT$	$3R$

Key Equations

• Thermal Energy (Monatomic Gas): $E_{th} = \frac{3}{2}nRT$

• Equipartition Theorem: $E_{th} = \frac{f}{2}nRT$ (where $f$ is degrees of freedom)

• Entropy (Statistical): $S = k_B \ln \Omega$

• Entropy Change (Reversible): $dS = \frac{dQ}{T}$

• Entropy Change (Heating): $\Delta S = Mc \ln\left(\frac{T_f}{T_i}\right)$

• Entropy Change (Ideal Gas): $\Delta S_{gas} = nC_V \ln\left(\frac{T_f}{T_i}\right) + nR \ln\left(\frac{V_f}{V_i}\right)$

Additional info: The notes above integrate the main textbook content, relevant equations, and the most directly relevant images to reinforce key concepts in thermal energy, specific heat, and entropy, as covered in college-level physics courses.