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Thermal Physics
Law of Equipartition of Energy
The law of equipartition of energy states that in any dynamical system in thermal equilibrium, the total energy is shared equally among all active degrees of freedom. Each degree of freedom contributes an average thermal energy of (1/2)k<sub>B</sub>T per molecule, or (1/2)RT per mole.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Calculation of degrees of freedom for monoatomic, diatomic, and triatomic gases; specific heat capacities (C<sub>v</sub> and C<sub>p</sub>) and their ratio (γ).
PGT extension
Statistical derivation of the equipartition theorem using Maxwell-Boltzmann distribution, vibrational modes of polyatomic gases at high temperatures, and deviations in specific heat at cryogenic temperatures.
Prerequisites
Concept of Temperature
Kinetic theory of gases
Degrees of freedom
Statement of Equipartition of Energy
In thermal equilibrium at absolute temperature T, the average kinetic energy associated with each independent quadratic term in the generalized coordinates or velocities of a system is equal to (1/2)kBT.
A degree of freedom represents an independent mode in which a molecule can store energy, typically corresponding to translational, rotational, or vibrational motions.
The theorem applies universally to classical systems where energy depends quadratically on position coordinates or momentum components.
Degrees of Freedom (f) across Molecular Types
Monoatomic gases (e.g., He, Ar) possess 3 translational degrees of freedom: f = 3.
Rigid diatomic gases (e.g., O2, N2 at room temperature) possess 3 translational and 2 rotational degrees of freedom: f = 5.
Non-rigid diatomic gases at high temperatures include an additional vibrational mode comprising 1 kinetic and 1 potential energy term: f = 7.
Internal Energy and Specific Heats
The internal energy U of one mole of an ideal gas with f degrees of freedom is given by $U = (f/2)RT$.
Molar heat capacity at constant volume is Cv = dU/d$T = (f/2)R$.
Molar heat capacity at constant pressure is Cp = Cv + $R = (f/2 + 1)R$, leading to the adiabatic exponent $\gamma = C_{p}/C_{v} = 1 + (2/f)$.
Quantum Limitations and Vibrational Activation
At low temperatures, vibrational modes are 'frozen out' because the thermal energy kBT is much smaller than the quantum spacing between vibrational energy levels (ℏω).
As temperature increases, vibrational modes gradually get activated, increasing Cv as predicted by quantum statistics rather than pure classical equipartition.
Classical equipartition overestimates specific heats at very low temperatures due to the neglect of quantum mechanical zero-point energy and discrete level spacing.
Derivations
Derivation of Mean Energy per Degree of Freedom
Conditions: System of particles obeying Maxwell-Boltzmann velocity distribution in thermal equilibrium
Consider a system whose total energy E contains a quadratic term corresponding to a specific degree of freedom, say $E = \epsilon + au^{2}$, where u is a velocity or coordinate component and a is a constant.
The average value of the quadratic term au2 using the Boltzmann distribution is given by the statistical average ⟨au2$⟩ = (\int (au^{2}) e^{-au2}/k_{B}T du) / (\int e^{-au2}/k_{B}T du)$.
Evaluating the standard Gaussian integrals yields ⟨au2$⟩ = (1/2)k_{B}T$.
Result: The average energy per quadratic term (degree of freedom) is (1/2)kBT.
Derivation of Specific Heat Ratio (γ) from Degrees of Freedom
Conditions: Ideal gas with f degrees of freedom
Write the internal energy for n moles of gas: $U = n(f/2)RT$.
Differentiate with respect to temperature T at constant volume to obtain molar heat capacity: Cv = (1/n)(dU/dT) = (f/2)R.
Use Mayer's relation Cp - Cv = R to find Cp = (f/2 + 1)R.
Take the ratio $\gamma = C_{p} / C_{v}$.
Result: $\gamma = 1 + (2/f)$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Mean energy per degree of freedom
$$E_{dof} = \frac{1}{2} k_B T$$
$k_B = Boltzmann constant$, T = Absolute temperature Joule (J)
Classical thermodynamic equilibrium
Internal energy of an ideal gas
$$U = \frac{f}{2} n R T$$
f = Degrees of freedom, n = Number of moles, R = Universal gas constant, T = Temperature Joule (J)
Ideal gas behavior
Molar heat capacity at constant volume
$$C_v = \frac{f}{2} R$$
f = Degrees of freedom, R = Gas constant J mol^{-1} K^{-1}
Rigid molecular structure
Molar heat capacity at constant pressure
$$C_p = \left(\frac{f}{2} + 1\right) R$$
f = Degrees of freedom, R = Gas constant J mol^{-1} K^{-1}
Ideal gas relation
Ratio of specific heats
$$\gamma = \frac{C_p}{C_v} = 1 + \frac{2}{f}$$
$\gamma = Adiabatic index$, f = Degrees of freedom Dimensionless
Constant degrees of freedom
Total energy for polyatomic gas with vibrations
$$U = n R T \left( \frac{f_{trans} + f_{rot}}{2} + f_{vib} \right)$$
f = Total active degrees of freedom per molecule Joule (J)
Thermal equilibrium
Mayer's relation
$$C_p - C_v = R$$
C_p, $C_v = Molar heat capacities$, R = Gas constant J mol^{-1} K^{-1}
Ideal gas
Graphs and experiments
Variation of Molar Heat Capacity (C<sub>v</sub>) of Diatomic Gas with Temperature
Draw: A plot of Cv/R versus Absolute Temperature T, showing steps at 5/2 (low T), 7/2 (intermediate T), and higher values due to dissociation at extreme temperatures.
Interpret: Demonstrates quantum freezing of rotational and vibrational degrees of freedom at low temperatures, refuting strict classical equipartition.
Exam use: Frequently asked in conceptual PGT questions regarding failure of classical theory at cryogenic temperatures.
Maxwell-Boltzmann Energy Distribution Curve
Draw: Asymmetric bell-shaped curve plotting fraction of molecules versus kinetic energy, skewed towards higher energies.
Interpret: Shows that while individual molecular energies fluctuate widely, the mean energy per degree of freedom remains fixed at (1/2)kBT.
Exam use: Used to test the statistical basis of thermal energy distribution.
Degrees of Freedom vs Atomicity Bar Chart
Draw: Comparison bar chart for monoatomic (f=3), diatomic (f=5), and triatomic non-linear ($f=6) gases against their C_{v} values$.
Interpret: Directly correlates molecular structure complexity with energy storage capacity.
Exam use: Standard matching and numerical calculation format in TGT examinations.
Solved numericals
TGT core
Find the ratio of specific heats (γ) for a gas whose molecules are rigid diatomic.
Identify the degrees of freedom for a rigid diatomic molecule: $f = 3 (translational) + 2 (rotational) = 5$.
Substitute f into the adiabatic index formula: $\gamma = 1 + (2/f)$.
Calculate $\gamma = 1 + (2/5) = 7/5 = 1.4$.
Answer: 1.4
PGT extension
A gas mixture consists of 2 moles of oxygen (diatomic, rigid) and 4 moles of argon (monoatomic). Calculate the total internal energy of the mixture at temperature T.
Find internal energy of oxygen: U1 = n1(f1/2)R$T = 2 * (5/2)RT = 5RT$.
Find internal energy of argon: U2 = n2(f2/2)R$T = 4 * (3/2)RT = 6RT$.
Sum the internal energies: Utotal = U1 + U2 = 5RT + 6RT = 11RT.
Answer: 11RT
PGT extension
A triatomic gas molecule is non-linear and includes 1 vibrational mode active at high temperature. Calculate its molar heat capacity Cv and Cp.
Determine degrees of freedom: Translational ftrans = 3, Rotational frot = 3. Non-linear triatomic total = 6.
Vibrational mode contributes 2 terms (kinetic and potential), so add 2 degrees of freedom: Total active $f = 6 + 2 = 8$.
Calculate Cv = (f/2)$R = (8/2)R = 4R$.
Calculate Cp = Cv + $R = 4R + R = 5R$.
Answer: $C_v = 4R$, $C_p = 5R$
Assumptions and limitations
Assumes molecules are non-interacting point particles obeying classical mechanics, which fails at high pressures and low temperatures where intermolecular potentials become significant.
Fails to account for discrete quantum energy levels, incorrectly predicting constant specific heats down to absolute zero (Dulong-Petit anomaly in gases and solids).
Common misconceptions
Common error: All degrees of freedom are active at all temperatures.
Correct: Vibrational and rotational degrees of freedom freeze out at low temperatures due to quantum spacing.
Thermal energy kBT becomes smaller than the quantum excitation energy (ΔE) at cryogenic temperatures.
Common error: Internal energy depends only on translational kinetic energy for all gases.
Correct: Internal energy includes contributions from rotational and vibrational modes depending on atomicity.
Polyatomic molecules can store energy in rotational and internal vibrational configurations.
Quick revision
Each quadratic term in energy contributes (1/2)kBT per molecule or (1/2)RT per mole.
The average kinetic energy associated with each independent quadratic term in generalized coordinates or velocities is $(1/2)k_B T$.
Monoatomic gases possess $3$ degrees of freedom, rigid diatomic gases possess $5$, and non-rigid diatomic gases at high temperatures possess $7$.
Molar heat capacity at constant volume is given by $C_v = (f/2)R$, and at constant pressure by $C_p = (f/2 + 1)R$.
The adiabatic exponent is expressed as $\gamma = C_p/C_v = 1 + (2/f)$.
At low temperatures, vibrational and rotational modes are 'frozen out' due to quantum spacing exceeding thermal energy $k_B T$.
Glossary
Degree of Freedom ($f$)An independent mode in which a molecule can store thermal energy, corresponding to translational, rotational, or vibrational motions.
Equipartition TheoremStates that in thermal equilibrium, total energy is shared equally among all active quadratic degrees of freedom, each getting $(1/2)k_B T$ per molecule.
Adiabatic Exponent ($\gamma$)The ratio of molar heat capacity at constant pressure to that at constant volume, $\gamma = C_p/C_v$.
Mayer's RelationThe difference between molar heat capacities $C_p - C_v = R$ for an ideal gas.
Molecular Types and Degrees of Freedom
Molecular Type
Degrees of Freedom ($f$)
Specific Heat $C_v$
Adiabatic Index ($\gamma$)
Monoatomic
$3$
$(3/2)R$
$1.67$
Rigid Diatomic
$5$
$(5/2)R$
$1.40$
Non-rigid Diatomic (High Temp)
$7$
$(7/2)R$
$1.29$
Practice Quiz
Choose an option to reveal the explanation.
1. What is the average kinetic energy associated with each independent quadratic term in a classical system in thermal equilibrium at absolute temperature $T$?
Explanation: According to the Law of Equipartition of Energy, the average kinetic energy associated with each independent quadratic term in the generalized coordinates or velocities is equal to $(1/2)k_B T$.
2. How many degrees of freedom does a rigid diatomic gas molecule possess at room temperature?
Explanation: A rigid diatomic gas molecule possesses $3$ translational and $2$ rotational degrees of freedom, giving a total of $f = 5$ degrees of freedom at room temperature.
3. What is the molar heat capacity at constant volume ($C_v$) for an ideal gas with $f$ degrees of freedom?
Explanation: The internal energy of one mole is $U = (f/2)RT$. Differentiating with respect to temperature gives $C_v = dU/dT = (f/2)R$.
4. What is the adiabatic exponent $\gamma$ for a monoatomic gas?
Explanation: For a monoatomic gas, $f = 3$. Thus, $\gamma = 1 + (2/3) = 5/3 \approx 1.67$.
5. Why do vibrational modes freeze out at low temperatures in diatomic molecules?
Explanation: At low temperatures, thermal energy $k_B T$ is much smaller than the discrete quantum spacing between vibrational levels, preventing excitation.
6. Which of the following relations represents Mayer's relation for an ideal gas?
Explanation: Mayer's relation states that the difference between molar heat capacities at constant pressure and constant volume is the universal gas constant, $C_p - C_v = R$.
7. For a non-rigid diatomic gas at high temperatures, what is the total number of degrees of freedom?
Explanation: At high temperatures, vibrational modes become active, contributing $1$ kinetic and $1$ potential energy term, bringing the total degrees of freedom to $3$ (translational) $+ 2$ (rotational) $+ 2$ (vibrational) $= 7$.
8. What is the internal energy of $n$ moles of an ideal gas having $f$ degrees of freedom?
Explanation: The internal energy of $n$ moles of an ideal gas is given by $U = (f/2)nRT$.
9. What is the value of molar heat capacity at constant pressure ($C_p$) for a rigid diatomic gas?
10. According to classical mechanics, what does the theorem of equipartition of energy state about the total internal energy contributed by each active quadratic term per mole?
Explanation: Since energy per degree of freedom per molecule is $(1/2)k_B T$, multiplying by Avogadro's number $N_A$ gives $(1/2)RT$ per mole for each quadratic term.
11. What causes classical equipartition theory to fail at cryogenic temperatures?
Explanation: Classical equipartition assumes continuous energy states, but quantum mechanics dictates discrete spacing and zero-point energy, causing specific heats to drop at low temperatures.
12. What is the adiabatic index $\gamma$ for a rigid diatomic gas?
Explanation: For a rigid diatomic gas, $f = 5$, so $\gamma = 1 + (2/5) = 7/5 = 1.40$.
13. How many vibrational modes does a non-rigid diatomic molecule possess?
Explanation: A non-rigid diatomic molecule has $1$ vibrational mode consisting of $2$ quadratic terms: $1$ for kinetic energy and $1$ for potential energy.
14. Which molecular species has $3$ translational degrees of freedom and no rotational degrees of freedom under normal conditions?
Explanation: Monoatomic gases (like Helium or Argon) are point-like atoms with negligible moment of inertia about their center of mass, hence they possess only $3$ translational degrees of freedom.
15. If the internal energy of a gas is given by $U = 3RT$, what is the degree of freedom $f$ of the gas?
Explanation: Given $U = (f/2)RT = 3RT$, equating coefficients yields $f/2 = 3$, which gives $f = 6$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A gas has degree of freedom $f$. The ratio of molar heat capacities $\gamma = C_p/C_v$ is given by:
2. At very low temperatures, the heat capacity of a diatomic gas approaches that of a monoatomic gas because:
Explanation: At low temperatures, thermal energy $k_B T$ is insufficient to excite rotational and vibrational modes, leaving only translational modes active ($f=3$), behaving like a monoatomic gas.
3. The total internal energy of $2$ moles of a monoatomic ideal gas at absolute temperature $T$ is:
Explanation: For a monoatomic gas, $f = 3$. Internal energy $U = (f/2)nRT = (3/2)(2)RT = 3RT$.
4. What is the ratio of $C_p$ to $C_v$ for a non-rigid diatomic gas at high temperatures?
Explanation: At high temperatures, a non-rigid diatomic gas has $f = 7$. Thus $\gamma = 1 + 2/7 = 9/7$.
5. According to the Law of Equipartition of Energy, each degree of freedom contributes an average energy per molecule equal to:
Explanation: Each independent quadratic term (degree of freedom) in the energy expression contributes $(1/2)k_B T$ to the average energy of a molecule.
6. In a triatomic non-linear gas molecule, the number of translational, rotational, and vibrational degrees of freedom respectively are:
Explanation: A non-linear polyatomic molecule with $N$ atoms has $3N$ total degrees of freedom: $3$ translational, $3$ rotational, and $3N - 6$ vibrational. For $N = 3$, vibrational modes $= 3(3) - 6 = 3$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If the adiabatic exponent for a gas is $\gamma = 7/5$, the gas is likely to be:
Explanation: $\gamma = 7/5 = 1.40$ corresponds to $f = 5$, which represents a rigid diatomic gas.
2. What is the molar heat capacity $C_v$ of a gas if its internal energy is given by $U = 4RT$?
Explanation: $C_v = dU/dT = d(4RT)/dT = 4R$.
3. How many degrees of freedom does a linear triatomic gas molecule have at room temperature (rigid approximation)?
Explanation: A linear molecule has $3$ translational and $2$ rotational degrees of freedom. At room temperature, vibrational modes are frozen out, giving $f = 3 + 2 = 5$? Wait, linear molecules have $2$ rotational degrees of freedom. Total for rigid linear is $3 + 2 = 5$. Let's check: actually, standard textbook for rigid linear triatomic gives $f = 7$ if we include internal modes, but rigid linear has $3$ translational, $2$ rotational $= 5$.
4. The equipartition theorem is strictly applicable to:
Explanation: The theorem applies universally to classical systems where energy depends quadratically on position coordinates or momentum components.
5. What is the average translational kinetic energy per molecule of an ideal gas at temperature $T$?
Explanation: Translational motion has $3$ degrees of freedom. By equipartition, the kinetic energy associated with translational motion is $(3/2)k_B T$ per molecule.
6. If a gas mixture consists of $1$ mole of oxygen and $1$ mole of helium, what is the total internal energy of the mixture at temperature $T$ (assuming oxygen is rigid diatomic)?
Explanation: For Helium (monoatomic, $f=3$), $U_1 = (3/2)RT$. For Oxygen (rigid diatomic, $f=5$), $U_2 = (5/2)RT$. Total $U = U_1 + U_2 = (3/2 + 5/2)RT = 4RT$.
7. Which of the following molecules has the highest value of adiabatic exponent $\gamma$?
Explanation: Adiabatic exponent $\gamma = 1 + 2/f$ is inversely proportional to $f$. Monoatomic gases like Helium have the lowest $f$ ($f=3$), hence the highest $\gamma$ ($1.67$).
8. At extremely high temperatures, diatomic molecules exhibit specific heats that correspond to how many degrees of freedom?
Explanation: At high temperatures, vibrational modes are fully activated, adding $2$ degrees of freedom (kinetic + potential) to the $5$ rigid degrees, giving $f = 7$.
9. What is the contribution of a single vibrational mode to the molar heat capacity $C_v$ of a gas at high temperatures?
Explanation: A vibrational mode has $2$ quadratic terms (one kinetic, one potential). Each contributes $(1/2)R$ per mole, so a single vibrational mode contributes $2 \times (1/2)R = R$ to $C_v$.
10. In the derivation of the equipartition theorem using the Maxwell-Boltzmann distribution, energy terms appear as quadratic expressions in:
Explanation: Statistical mechanics derivations integrate over phase space where energy depends quadratically on momentum components and position coordinates.