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Thermal Physics
Internal Energy
Internal energy is the total microscopic energy of a system comprising the kinetic and potential energies of its constituent molecules, atoms, or subatomic particles. It is a state function representing an extensive property of thermodynamic systems.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of internal energy, its state function nature, dependence on temperature for ideal gases, and relation to the First Law of Thermodynamics.
PGT extension
Statistical interpretation via degrees of freedom, Maxwell-Boltzmann energy distribution, internal energy of real gases using van der Waals forces, and configurational internal energy.
Prerequisites
Concept of Temperature and Zeroth Law of Thermodynamics
Kinetic Theory of Gases
Degrees of Freedom and Law of Equipartition of Energy
Microscopic Basis of Internal Energy
Internal energy (U) includes translational, rotational, and vibrational kinetic energies of molecules, along with intermolecular potential energy.
It excludes macroscopic kinetic energy of the system as a whole and potential energy arising from external gravitational or electrostatic fields.
For an ideal gas, intermolecular forces are assumed zero, implying internal energy depends exclusively on temperature.
Internal Energy as a State Function
U is a thermodynamic state function; its change dU between two equilibrium states depends only on the initial and final states, not the path.
Cyclic integrals of internal energy changes are identically zero: oint dU = 0.
Exact differential property allows dU to be integrated directly without path specification.
Equpartition of Energy and Internal Energy
Each active degree of freedom contributes (1/2)k_BT per molecule or (1/2)RT per mole to the internal energy.
Monatomic gases possess 3 translational degrees of freedom, diatomic gases possess 5 (at room temperature) or 7 (at high temperature) degrees of freedom.
Total internal energy for an ideal gas is given by $U = f/2 n RT$ where f is the total degrees of freedom.
Internal Energy of Real Gases
Real gases possess intermolecular attractive and repulsive potentials, making internal energy a function of both temperature and volume (U = U(T, V)).
Joule's free expansion experiment shows that for a real gas, internal energy varies slightly with volume due to internal work against attractive forces.
Van der Waals correction introduces a potential energy term proportional to the square of density: $U_{real} = (f/2)nRT - (a n^2 / V)$.
Derivations
Internal Energy of an Ideal Gas using Law of Equipartition
Conditions: Ideal gas in thermal equilibrium with f degrees of freedom
State the mean energy associated with a single degree of freedom per molecule as (1/2)k_BT based on Maxwell-Boltzmann distribution.
Multiply the energy per degree of freedom by the number of degrees of freedom f to get total energy per molecule: $E = (f/2)k_BT$.
Scale up to N molecules (or n moles) by multiplying by Avogadro number N_A, yielding $U = (f/2)N k_BT = (f/2)n RT$.
Result: U = \frac{f}{2} n R T
Relation between Internal Energy Change and Heat Capacity
Conditions: Constant volume process (isochoric process) for any pure substance
Write the First Law of Thermodynamics: d$Q = dU + dW$, where dW = P dV.
For a constant volume process, dV = 0, so dW = 0, leading to dQ = dU.
Express molar heat capacity at constant volume as $C_v = (1/n)(dQ/dT)_V$.
Substitute dQ = dU to obtain d$U = n C_v dT$, and integrate for finite temperature changes.
Result: Delta $U = \int_{T_1}^{T_2} n C_v dT$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Internal Energy of Ideal Gas
$$U = \frac{f}{2} n R T$$
U = Internal energy, f = degrees of freedom, n = number of moles, R = universal gas constant, T = absolute temperature Joules (J)
Ideal gas
Differential Change in Internal Energy
$$dU = n C_v dT$$
dU = Change in internal energy, n = moles, $C_v = molar heat capacity at constant volume$, dT = temperature change Joules (J)
Valid for any substance at constant volume, and strictly for ideal gas at any volume
First Law of Thermodynamics (Energy Conservation)
$$dU = dQ - dW$$
dU = change in internal energy, dQ = heat added to system, dW = work done by system Joules (J)
General closed system
Internal Energy of Monatomic Gas
$$U = \frac{3}{2} n R T$$
n = moles, R = gas constant, T = temperature Joules (J)
Monatomic ideal gas
Internal Energy of Diatomic Gas (Room Temp)
$$U = \frac{5}{2} n R T$$
n = moles, R = gas constant, T = temperature Joules (J)
Diatomic ideal gas with 3 translational and 2 rotational degrees of freedom
Mayer's Relation
$$C_p - C_v = R$$
$C_p = molar heat capacity at constant pressure$, $C_v = molar heat capacity at constant volume$, R = universal gas constant J/(mol K)
Ideal gas
Real Gas Internal Energy Correction
$$U_{real} = \frac{f}{2} n R T - \frac{a n^2}{V}$$
a = van der Waals constant for intermolecular attraction, V = volume, n = moles Joules (J)
Van der Waals gas
Equipartition Energy per Molecule
$$E_{molecule} = \frac{f}{2} k_B T$$
$k_B = Boltzmann constant$, f = degrees of freedom, T = temperature Joules (J)
Equilibrium statistical ensemble
Graphs and experiments
Internal Energy vs Temperature for an Ideal Gas
Draw: A straight line passing through the origin on a U vs T plot, where the slope represents (f/2)nR.
Interpret: Shows direct proportionality between internal energy and absolute temperature for an ideal gas.
Exam use: Used to test understanding of linear dependence and identification of slopes for different gases (monatomic vs diatomic).
Joule's Free Expansion Experiment
Draw: Two insulated bulbs connected by a closed valve; one filled with gas, the other evacuated, submerged in a water bath.
Interpret: When the valve opens, gas expands into vacuum (dW = 0, dQ = 0), and temperature change is measured to test internal energy dependence on volume.
Exam use: Demonstrates that internal energy of an ideal gas is independent of volume ((partial U / partial V)_T = 0).
Isothermal vs Adiabatic Curves on P-V Diagram
Draw: Two curves on a P-V plane starting from the same point, with the adiabatic curve steeper than the isothermal curve.
Interpret: In an adiabatic expansion, work is done at the expense of internal energy, leading to a drop in temperature.
Exam use: Frequently tested in calculations involving internal energy change during adiabatic processes (Delta $U = -W)$.
Solved numericals
TGT core
Calculate the change in internal energy of 2 moles of a monoatomic ideal gas when its temperature is increased from 200 K to 300 K. (Given $R = 8.314 J/(mol K))$
Identify the gas type: Monoatomic gas has 3 degrees of freedom (f = 3).
Write the formula for internal energy change: Delta $U = (f/2) n R \delta T$.
A rigid container holds 1 mole of a diatomic gas at temperature T_0. Due to electrical heating, heat Q is supplied to the gas, and 20% of this heat escapes through the container walls. Find the rise in temperature of the gas.
Net heat retained by the gas: $Q_{net} = Q - 0.20 Q = 0.80 Q$.
For a rigid container, volume is constant (dV = 0), so all retained heat goes into changing internal energy: Delta $U = Q_{net} = 0.80 Q$.
Relate internal energy change to temperature rise for a diatomic gas (f = 5, $C_v = 5R/2): \delta U = n C_v \delta T = (5/2) R \delta T$.
Equate and solve for Delta T: (5/2) R Delta T = 0.80 Q => Delta T = \frac{1.6 Q}{5 R} = 0.32 \frac{Q}{R}.
Answer: 0.32 Q / R
PGT extension
Using the van der Waals equation of state, find the change in internal energy when 1 mole of a real gas undergoes isothermal expansion from volume V_1 to V_2 at temperature T.
Start from the thermodynamic relation: (partial U / partial V)_$T = T (\partial P / \partial T)_V - P$.
Write the van der Waals equation: $P = \frac{RT}{V-b} - \frac{a}{V^2}$.
Calculate partial derivative (partial P / partial T)_$V = \frac{R}{V-b}$.
Substitute into the thermodynamic relation: (partial U / partial V)_$T = T(\frac{R}{V-b}) - (\frac{RT}{V-b} - \frac{a}{V^2}) = \frac{a}{V^2}$.
Integrate with respect to volume from V_1 to V_2: Delta $U = \int_{V_1}^{V_2} \frac{a}{V^2} dV = a (\frac{1}{V_1} - \frac{1}{V_2})$.
Answer: a (\frac{1}{V_1} - \frac{1}{V_2})
Assumptions and limitations
Ideal gas internal energy formulations assume negligible intermolecular forces, which breaks down at high pressures and low temperatures.
Classical equipartition theorem fails at very low temperatures (cryogenic regimes) due to quantum freeze-out of vibrational and rotational degrees of freedom.
Common misconceptions
Common error: Internal energy depends on the path taken by the system during a thermodynamic process.
Correct: Internal energy is a state function and depends only on the current thermodynamic state of the system.
Path dependence applies to path functions like heat (Q) and work (W), not state functions like internal energy (U).
Common error: Temperature is a measure of the total internal energy of a body.
Correct: Temperature is a measure of the average translational kinetic energy per molecule, whereas internal energy is the total energy summed over all molecules.
A small cup of boiling water has a high temperature but lower total internal energy compared to a swimming pool full of lukewarm water.
Common error: For an isothermal expansion of an ideal gas, the internal energy changes because work is done.
Correct: Internal energy remains constant during an isothermal process for an ideal gas.
Since temperature is constant, Delta T = 0 implies Delta U = 0; the heat absorbed is entirely converted into work.
Quick revision
Internal energy U is an extensive state function dependent solely on temperature for ideal gases.
For an ideal gas, $U = (f/2) n R T$.
In an isothermal process for an ideal gas, Delta U = 0.
In an adiabatic process, Delta $U = -W$.
For real gases, internal energy depends on both temperature and volume due to intermolecular potentials.
Must remember
Internal energy ($U$) is a thermodynamic state function whose change depends only on initial and final states, not on the path.
For an ideal gas, internal energy depends exclusively on temperature ($U = \frac{f}{2} n R T$).
Each active degree of freedom contributes $\frac{1}{2} k_B T$ per molecule or $\frac{1}{2} R T$ per mole.
Real gas internal energy includes a correction term for intermolecular attraction: $U_{real} = \frac{f}{2} n R T - \frac{a n^2}{V}$.
The cyclic integral of internal energy change for any thermodynamic cycle is identically zero: $\oint dU = 0$.
Glossary
Internal Energy ($U$)The total microscopic kinetic and potential energy of the molecules comprising a system.
State FunctionA property whose value depends only on the current state of the system and not on the path taken to reach it.
Degrees of Freedom ($f$)The number of independent coordinates required to specify the position and configuration of a dynamical system.
Van der Waals Constant ($a$)A parameter representing the strength of intermolecular attractive forces in a real gas.
Ideal Gas vs Real Gas Internal Energy
Feature
Ideal Gas
Real Gas
Dependence
Depends only on temperature ($T$)
Depends on both temperature and volume ($T, V$)
Intermolecular Potential
Assumed zero
Accounted for via van der Waals $a$ term
Equation
$U = \frac{f}{2} n R T$
$U = \frac{f}{2} n R T - \frac{a n^2}{V}$
Practice Quiz
Choose an option to reveal the explanation.
1. Which of the following quantities is a thermodynamic state function?
Explanation: Internal energy ($U$) is a state function depending only on the initial and final states, whereas heat and work are path functions.
2. For an ideal gas, internal energy depends exclusively on:
Explanation: Since intermolecular forces are zero in an ideal gas, internal energy has no potential energy component and depends solely on temperature.
3. What is the internal energy of $2$ moles of a monatomic ideal gas at temperature $T$?
Explanation: For a monatomic gas, $f = 3$. Thus, $U = \frac{3}{2} n R T = \frac{3}{2}(2)R T = 3 R T$.
4. What is the cyclic integral of internal energy change ($\oint dU$) for any closed thermodynamic cycle?
Explanation: Because internal energy is a state function, its net change over any complete cycle is zero.
5. According to the equipartition theorem, each active degree of freedom contributes how much energy per mole?
Explanation: Each active degree of freedom contributes $\frac{1}{2} k_B T$ per molecule, which corresponds to $\frac{1}{2} R T$ per mole.
6. How many degrees of freedom does a rigid diatomic gas molecule possess at room temperature?
Explanation: At room temperature, a diatomic molecule has $3$ translational and $2$ rotational degrees of freedom, totaling $5$.
7. The internal energy of a real gas differs from that of an ideal gas due to:
Explanation: Real gas internal energy incorporates intermolecular attractive and repulsive potentials, notably represented by the van der Waals correction term.
8. During the isothermal expansion of an ideal gas, its internal energy:
Explanation: Since temperature remains constant during an isothermal process for an ideal gas, $\Delta T = 0$, implying $\Delta U = 0$.
9. What is the differential change in internal energy expressed in terms of molar heat capacity at constant volume?
Explanation: By definition, the internal energy change for $n$ moles is $dU = n C_v dT$, valid strictly for ideal gases and generally at constant volume.
10. Mayer's relation connecting $C_p$, $C_v$, and $R$ is given by:
Explanation: Mayer's relation for an ideal gas states that $C_p - C_v = R$.
11. Which of the following is excluded from the microscopic internal energy of a system?
Explanation: Internal energy excludes macroscopic mechanical kinetic and potential energies associated with the system's motion or position in external fields.
12. What is the correct expression for the van der Waals real gas internal energy?
Explanation: The attractive intermolecular forces reduce the internal energy by an amount proportional to the square of density, giving $U_{real} = \frac{f}{2} n R T - \frac{a n^2}{V}$.
13. At high temperatures, diatomic gases acquire additional degrees of freedom due to:
Explanation: At high temperatures, vibrational modes become active, increasing the total degrees of freedom from $5$ to $7$ for diatomic molecules.
14. According to the First Law of Thermodynamics, the change in internal energy is given by:
Explanation: The First Law expressed in differential form for a closed system is $dU = dQ - dW$, where $dW$ is work done *by* the system.
15. Why does Joule's free expansion of a real gas show a slight change in internal energy with volume?
Explanation: In free expansion into a vacuum ($Q=0, W=0$), the internal energy remains constant overall, but real gas molecules perform internal work against attractive forces, altering local energy distributions.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. If $3$ moles of an ideal monatomic gas are heated from $200\text{ K}$ to $400\text{ K}$, the change in internal energy is approximately:
Explanation: $\Delta U = \frac{3}{2} n R \Delta T = \frac{3}{2} \times 3 \times 8.314 \times (400 - 200) = 4.5 \times 8.314 \times 200 \approx 7482.6\text{ J} = 7.48\text{ kJ}$.
2. Which of the following molecules has a total of $7$ degrees of freedom at high temperature?
Explanation: Diatomic molecules like oxygen ($O_2$) possess $3$ translational, $2$ rotational, and $2$ vibrational degrees of freedom at high temperatures ($3 + 2 + 2 = 7$).
3. The internal energy of a fixed mass of an ideal gas depends on:
Explanation: For an ideal gas, internal energy is strictly a function of absolute temperature.
4. In a cyclic process, the change in internal energy of the system is:
Explanation: Since the system returns to its initial state in a cycle, state functions like internal energy show zero net change.
5. The molar heat capacity at constant volume $C_v$ of a diatomic gas with $5$ degrees of freedom is:
Explanation: Since $U = \frac{f}{2} n R T$, differentiating with respect to $T$ for $n=1$ gives $C_v = \frac{f}{2} R = \frac{5}{2} R$.
6. What is the correction term added to volume in van der Waals equation, and how does internal energy depend on volume?
Explanation: The potential energy correction term in real gas internal energy is $-\frac{a n^2}{V}$, which varies inversely with volume squared.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. A system undergoes a thermodynamic process in which $200\text{ J}$ of heat is supplied and $50\text{ J}$ of work is done by the system. The change in internal energy is:
2. The ratio of molar heat capacities $\gamma = C_p / C_v$ for a monatomic ideal gas is:
Explanation: For monatomic gas, $C_v = \frac{3}{2} R$ and $C_p = \frac{5}{2} R$, so $\gamma = 5/3 \approx 1.67$.
3. Which of the following statements is INCORRECT regarding internal energy?
Explanation: Only changes in internal energy ($\Delta U$) can be measured thermodynamically, not its absolute value.
4. If $1$ mole of a triatomic non-linear gas is considered at room temperature, its internal energy is:
Explanation: A non-linear triatomic molecule has $3$ translational and $3$ rotational degrees of freedom ($f = 6$). Thus, $U = \frac{6}{2} n R T = 3 R T$.
5. In an adiabatic expansion of an ideal gas, the internal energy:
Explanation: In an adiabatic process ($dQ = 0$), work is done by the gas ($dW > 0$), so $dU = -dW < 0$, meaning internal energy decreases.
6. What is the internal energy of a mixture of $1$ mole of monatomic gas and $1$ mole of diatomic gas (room temp) at temperature $T$?
Explanation: $U_{total} = U_{mono} + U_{di} = \frac{3}{2} R T + \frac{5}{2} R T = \frac{8}{2} R T = 4 R T$.
7. Temperature is fundamentally a measure of:
Explanation: Statistical mechanics defines absolute temperature directly proportional to the average translational kinetic energy per molecule.
8. The exact differential property of internal energy ($dU$) implies that:
Explanation: Because $dU$ is an exact differential, its line integral between two states is independent of the path taken.
9. For a van der Waals gas, if the volume is allowed to approach infinity at constant temperature, its internal energy approaches:
Explanation: As $V \to \infty$, the correction term $\frac{a n^2}{V} \to 0$, and the real gas internal energy approaches that of an ideal gas.
10. The equipartition theorem applies strictly to systems in:
Explanation: The equipartition theorem is derived for classical statistical ensembles in thermal equilibrium.