Question

Assume a solid of $N$ atoms, each vibrating about its mean position. An oscillation in one dimension has average energy $\frac{1}{2}k_BT$. In 3 dimensions, the average energy is:

• A. $3k_BT$
• B. $3RT$
• C. $9RT$
• D. $R$

Hint:

Energy in solids due to vibrations is partitioned equally between potential and kinetic energy (Equipartition theorem).

Answer 1

The Correct Option is B

In 3 dimensions, each dimension contributes $\frac{1}{2}k_B T$ from kinetic energy and $\frac{1}{2}k_B T$ from potential energy, giving $k_B T$ per dimension. Thus:
\[ \text{Total Energy} = 3k_B T \]
Since $R = N_A k_B$, this becomes:
\[ E = 3RT \]