B3-4. Kinetic Theory and Internal Energy

1. Kinetic Theory of Gases

The kinetic theory connects the macroscopic properties of a gas (like pressure and temperature) to the microscopic motion of its constituent particles.

Assumptions of an Ideal Gas:

An ideal gas is a theoretical construct. For a real gas to act "ideally" (which happens best at high temperatures and low pressures), we assume:

  1. Particles have negligible volume compared to the total volume of the container.
  2. Collisions between particles and the container walls are perfectly elastic (no kinetic energy is lost).
  3. There are absolutely no intermolecular forces between particles (except during collisions).
  4. The duration of collisions is negligible compared to the time between collisions.

2. Average Kinetic & Internal Energy

Because we assume there are absolutely no intermolecular forces in an ideal gas, the particles have exactly zero potential energy. Therefore, the total Internal Energy ($U$) of an ideal gas is purely the sum of the kinetic energies of all its particles.

Average Kinetic Energy of a Single Molecule ($E_K$):

$$E_K = \dfrac{3}{2} k_B T = \dfrac{3}{2} \dfrac{R}{N_A} T$$

Total Internal Energy of the Gas ($U$):

To find the total internal energy, we simply multiply the average kinetic energy of a single molecule by the total number of molecules ($N$), or apply it in moles ($n$).

$$U = \dfrac{3}{2} n R T = \dfrac{3}{2} N k_B T$$
  • $U$: Total Internal Energy (J)
  • $E_K$: Average kinetic energy of one molecule (J)
  • $T$: Absolute temperature (K)

Example 1: Internal Energy of Helium

Problem: Calculate the temperature of $3.0$ g of helium gas kept at a constant volume in a cylinder, if its total internal energy is $600$ J. (Helium has a mass number of $4$).

Solution:

  • Step 1: Calculate the number of moles ($n$).
    The molar mass ($M$) of Helium is $4$ g mol⁻¹.
    $n = \dfrac{m}{M} = \dfrac{3.0}{4.0} = 0.75$ mol
  • Step 2: State the relevant Internal Energy equation.
    $$U = \dfrac{3}{2} n R T$$
  • Step 3: Rearrange for Temperature ($T$) and solve.
    $$T = \dfrac{2U}{3nR} = \dfrac{2 \times 600}{3 \times 0.75 \times 8.31} = \dfrac{1200}{18.6975} \approx \mathbf{64 \text{ K}}$$