understand that internal energy is determined by the state of the system and that it can be expressed as the sum of a random distribution of kinetic and potential energies associated with the molecules of a system

Internal Energy – Section 16.1

Learning Objective

Explain why internal energy \(U\) is a state function and show that it can be expressed as the sum of the microscopic kinetic and potential energies of the molecules that constitute the system.

1. What is Internal Energy?

  • State function: \(U\) depends only on the instantaneous state of the system (e.g. temperature \(T\), volume \(V\), pressure \(p\)). The value of \(U\) is independent of the path taken to reach that state.

  • Microscopic origin: \(U\) is the total energy of all the molecules, consisting of

    • Translational kinetic energy

    • Rotational kinetic energy

    • Vibrational kinetic energy

    • Inter‑molecular (potential) energy – includes vibrational potential energy and any other bonding energy.

  • Random distribution: Molecular motions are random; the macroscopic value of \(U\) is the statistical average of these microscopic contributions.

2. General Mathematical Form

For a system containing \(N\) molecules

\[

U = \sum{i=1}^{N}\Bigl(Ki^{\text{trans}}+Ki^{\text{rot}}+Ki^{\text{vib}}+V_i\Bigr)

\]

  • \(K_i^{\text{trans}}\) – translational kinetic energy of molecule \(i\)

  • \(K_i^{\text{rot}}\) – rotational kinetic energy of molecule \(i\)

  • \(K_i^{\text{vib}}\) – vibrational kinetic energy of molecule \(i\)

  • \(V_i\) – intermolecular (potential) energy of molecule \(i\) (includes vibrational potential energy).

For an ideal gas the potential term is taken as zero; for real gases, liquids and solids it can be large, especially during phase changes.

3. Contributions from Different Types of Molecules

Gas / PhaseDegrees of Freedom (DoF)Kinetic‑energy contribution to \(U\) (per mole)Potential‑energy contribution to \(U\)
Monatomic ideal gas3 translational (no rotation, no vibration)\(\displaystyle U_{\text{kin}} = \frac{3}{2}RT\)Negligible (ideal‑gas assumption)
Linear diatomic ideal gas3 translational + 2 rotational = 5 DoF
Vibrational modes frozen when \(T \ll \Theta_{\text{vib}}\)		\(\displaystyle U{\text{kin}} = \frac{5}{2}RT\) (vibration frozen)
When one vibrational mode is active: \(\displaystyle U{\text{kin}} = \frac{5}{2}RT + \frac{1}{2}RT = 3RT\)		Zero when vibration frozen; each active vibrational mode adds \(\tfrac{1}{2}RT\) as potential energy.
Non‑linear polyatomic ideal gas3 translational + 3 rotational = 6 DoF
\(f = 3N_{\text{atoms}}-6\) vibrational modes		\(\displaystyle U{\text{kin}} = 3RT\) (vibrations frozen)
With \(f\) active vibrational modes: \(\displaystyle U{\text{kin}} = 3RT + \frac{f}{2}RT\)		Zero for frozen vibrations; each active mode contributes an additional \(\tfrac{1}{2}RT\) as potential energy.
Real gas / liquid / solidAll translational, rotational and vibrational DoF are present; strong intermolecular forces.Depends on temperature; vibrational modes become progressively excited as \(T\) rises.Significant – accounts for cohesion, latent heat of melting/boiling, and the large heat capacities of condensed phases.

4. Internal Energy of an Ideal Gas – Explicit Temperature Dependence

  • For any ideal gas (vibrations frozen) the total internal energy is

    \[

    U = \frac{f}{2}\,nRT

    \]

    where \(f\) is the total number of active kinetic degrees of freedom (translational + rotational) and \(n\) is the amount of substance in moles.

  • Because \(U\) depends only on temperature, the change in internal energy at constant volume is

    \[

    \Delta U = CV\,\Delta T\qquad\text{with}\qquad CV = \left(\frac{\partial U}{\partial T}\right)_V = \frac{f}{2}nR.

    \]

  • Example: 2 mol of a monatomic ideal gas heated from 300 K to 350 K.

    \(f = 3\) → \(C_V = \tfrac{3}{2}nR = 3R\).

    \(\Delta U = C_V\Delta T = 3R(50\;\text{K}) = 3(8.314)(50) \approx 1.25\times10^3\;\text{J}.\)

5. First Law of Thermodynamics – Sign Conventions

The Cambridge syllabus uses the convention

\[

\Delta U = q + w

\]

  • \(q\) – heat added to the system (positive when heat flows into the system).

  • \(w\) – work done on the system (positive when the surroundings do work on the system).

    Note: Some textbooks write \(\Delta U = Q - W\) (where \(W\) is work done by the system). Both are equivalent; the syllabus adopts the \(q + w\) form.

  • Because \(U\) is a state function, \(\Delta U\) depends only on the initial and final states, not on the path taken.

6. Illustrating Path‑Independence

Two different processes that start and finish at the same temperature and volume give the same \(\Delta U\):

  1. Isothermal expansion of an ideal gas from \((T1,V1)\) to \((T1,V2)\). Since \(T\) is constant, \(\Delta U = 0\).

  2. Adiabatic compression from \((T2,V2)\) back to \((T1,V1)\). The heat term \(q = 0\); the work term \(w\) is negative, and the resulting \(\Delta U\) is again zero because the final state is identical to the initial one.

Both routes lead to the same change in internal energy, confirming that \(U\) is a state function.

7. Practical Implications and Examples

  1. Ideal‑gas internal energy: For any ideal gas (vibrations frozen),

    \[

    U = nC_VT = \frac{f}{2}nRT.

    \]

    Example: 1 mol of a linear diatomic gas at 300 K (vibration frozen) → \(U = \tfrac{5}{2}R(300) \approx 3.11\times10^3\;\text{J}\).

  2. Isothermal expansion of an ideal gas: Because \(U\) depends only on \(T\), \(\Delta U = 0\) for an isothermal process even though heat is transferred and work is done.

  3. Phase change (melting of ice): Temperature remains constant while a large amount of heat is supplied. The energy is used to increase intermolecular potential energy (break the crystal lattice), so \(\Delta U\) is large despite \(\Delta T = 0\).

  4. Heat capacity of solids (Dulong–Petit law): Each atom in a solid behaves like a three‑dimensional harmonic oscillator. Each vibrational mode contributes \(\tfrac{1}{2}RT\) kinetic + \(\tfrac{1}{2}RT\) potential, giving

    \[

    U \approx 3nRT,\qquad C_V \approx 3nR.

    \]

8. Suggested Diagram

Illustration of a collection of molecules: random arrows show translational motion, curved arrows around each sphere depict rotation, springs between neighbouring spheres represent intermolecular potential energy, and a small oscillating sphere on a spring shows vibrational motion.

9. Summary Checklist

  • Internal energy \(U\) is a state function – only the initial and final states matter.

  • \(U\) equals the sum of microscopic kinetic (translational, rotational, vibrational) and potential energies of all molecules.

  • For an ideal gas (vibrations frozen) \(U = \dfrac{f}{2}\,nRT\) and \(\Delta U = CV\Delta T\) with \(CV = \dfrac{f}{2}nR\).

  • Real substances have significant potential‑energy terms, especially during phase changes and in condensed phases.

  • The first law (Cambridge convention) is \(\displaystyle \Delta U = q + w\).

  • Heat capacities are linked to internal energy:

    \[

    CV = \left(\frac{\partial U}{\partial T}\right)V,\qquad Cp = \left(\frac{\partial H}{\partial T}\right)p.

    \]