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

Published by Patrick Mutisya · 14 days ago

Cambridge A-Level Physics 9702 – Internal Energy

Internal Energy

Learning Objective

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.

Key Concepts

  • State function: Internal energy \$U\$ depends only on the current state (e.g., temperature, volume, pressure) and not on the path taken to reach that state.

  • Microscopic origin: \$U\$ is the total energy of all molecules, comprising translational, rotational, vibrational kinetic energies and intermolecular potential energies.

  • Random distribution: The molecular motions are random; the macroscopic \$U\$ is the statistical average of these microscopic contributions.

Mathematical Expression

The internal energy of a system containing \$N\$ molecules can be written as

\$U = \sum{i=1}^{N}\left(Ki^{\text{trans}}+Ki^{\text{rot}}+Ki^{\text{vib}}+V_i\right)\$

where \$Ki^{\text{trans}}\$, \$Ki^{\text{rot}}\$, \$Ki^{\text{vib}}\$ are the translational, rotational and vibrational kinetic energies of molecule \$i\$, and \$Vi\$ is its intermolecular potential energy.

Contributions in Different Types of Gas

Gas TypeDegrees of FreedomKinetic Energy ContributionPotential Energy Contribution
Monatomic ideal gas3 translational\$\frac{3}{2}Nk_{\mathrm B}T\$Negligible (ideal gas assumption)
Linear diatomic ideal gas (no vibration)3 translational + 2 rotational\$\frac{5}{2}Nk_{\mathrm B}T\$Negligible (ideal gas assumption)
Non‑linear polyatomic ideal gas (no vibration)3 translational + 3 rotational\$\frac{6}{2}Nk{\mathrm B}T = 3Nk{\mathrm B}T\$Negligible (ideal gas assumption)
Real gas / condensed phaseAll above + vibrational + intermolecularDepends on temperature (vibrational modes activated above characteristic temperatures)Significant; accounts for cohesion, phase changes, etc.

Internal Energy and the First Law of Thermodynamics

The first law relates the change in internal energy \$\Delta U\$ to heat \$Q\$ added to the system and work \$W\$ done by the system:

\$\Delta U = Q - W\$

Because \$U\$ is a state function, \$\Delta U\$ depends only on the initial and final states, not on the specific process.

Practical Implications

  1. For an ideal gas, \$U\$ is a function of temperature alone: \$U = f(T)\$. Hence, at constant temperature, \$\Delta U = 0\$ even if \$Q \neq 0\$ (e.g., isothermal expansion).

  2. During phase changes, large amounts of energy are transferred as \$Q\$ while \$T\$ remains constant; the energy goes into changing the potential energy (breaking or forming intermolecular bonds).

  3. In solids and liquids, vibrational kinetic energy and intermolecular potential energy are comparable; both must be considered when evaluating \$U\$.

Suggested Diagram

Suggested diagram: Schematic showing a collection of molecules with arrows indicating random translational motion, rotating arrows for rotational motion, and springs representing intermolecular potential energy.

Summary Checklist

  • Internal energy \$U\$ is a state function.

  • It is the sum of microscopic kinetic (translational, rotational, vibrational) and potential energies.

  • For ideal gases, \$U\$ depends only on temperature.

  • Real substances have significant potential energy contributions, especially during phase changes.

  • Use \$\Delta U = Q - W\$ to relate macroscopic heat and work to changes in internal energy.