1. **State the problem:** (a)(i) Define internal energy. (a)(ii) State the kinetic theory assumption leading to zero potential energy between ideal gas molecules. (b) Use the equation $pV = \frac{1}{3} Nm\langle c^2 \rangle$ to show that the mean kinetic energy $\langle E_k \rangle$ of a molecule is $\frac{3}{2} kT$. 2. **Definitions and assumptions:** - Internal energy is the total energy contained within a system due to the kinetic and potential energies of its molecules. - The kinetic theory assumption is that ideal gas molecules have no intermolecular forces, so potential energy between molecules is zero. 3. **Derivation:** Given: $$pV = \frac{1}{3} Nm \langle c^2 \rangle$$ where: - $p$ = pressure - $V$ = volume - $N$ = number of molecules - $m$ = mass of one molecule - $\langle c^2 \rangle$ = mean square speed of molecules From the ideal gas law: $$pV = NkT$$ where $k$ is Boltzmann constant and $T$ is temperature. Equate the two expressions for $pV$: $$NkT = \frac{1}{3} Nm \langle c^2 \rangle$$ Divide both sides by $N$: $$kT = \frac{1}{3} m \langle c^2 \rangle$$ Multiply both sides by $\frac{3}{2}$: $$\frac{3}{2} kT = \frac{1}{2} m \langle c^2 \rangle$$ Note that the mean kinetic energy per molecule is: $$\langle E_k \rangle = \frac{1}{2} m \langle c^2 \rangle$$ Therefore: $$\langle E_k \rangle = \frac{3}{2} kT$$ 4. **Explanation:** - The internal energy includes kinetic energy of molecules. - The assumption of no intermolecular forces means potential energy is zero. - Using the kinetic theory relation and ideal gas law, we relate pressure, volume, and temperature to molecular speeds. - This leads to the expression for mean kinetic energy in terms of temperature and Boltzmann constant. **Final answer:** $$\boxed{\langle E_k \rangle = \frac{3}{2} kT}$$