Recall the relationships between the coefficients of thermal expansion.

The coefficient of linear expansion is $\alpha$.
The coefficient of area expansion is $\beta$.
The coefficient of volume expansion is $\gamma$.

These coefficients are related as follows:

• Option A: $\alpha = 2\beta$ Substitute $\beta = 2\alpha$: $\alpha = 2(2\alpha) \implies \alpha = 4\alpha$. This is incorrect.

• Option B: $\gamma = \frac{2\beta}{3}$ Substitute $\beta = 2\alpha$ and $\gamma = 3\alpha$: $3\alpha = \frac{2(2\alpha)}{3}$ $3\alpha = \frac{4\alpha}{3}$. This is incorrect. Alternatively, from $\beta = 2\alpha$ and $\gamma = 3\alpha$, we can derive $\gamma = \frac{3}{2}\beta$. So, $\frac{2\beta}{3}$ is incorrect.

• Option C: $\beta = \frac{3\gamma}{2}$ Substitute $\beta = 2\alpha$ and $\gamma = 3\alpha$: $2\alpha = \frac{3(3\alpha)}{2}$ $2\alpha = \frac{9\alpha}{2}$. This is incorrect. Alternatively, from $\beta = 2\alpha$ and $\gamma = 3\alpha$, we can derive $\beta = \frac{2}{3}\gamma$. So, $\frac{3\gamma}{2}$ is incorrect.

Since none of the first three options are correct, the correct choice is "None of the other options".

The correct option is $\boxed{\text{None of the other options}}$.

: Adiabatic Expansion of an Ideal Monatomic Gas

An adiabatic process means no heat exchange with the surroundings ($Q = 0$).
An expansion means the volume of the gas increases, and the gas does work on its surroundings ($W > 0$).

The First Law of Thermodynamics states $\Delta U = Q - W$.
Since $Q = 0$ for an adiabatic process, the equation becomes $\Delta U = -W$.
Since it's an expansion, $W > 0$. Therefore, $\Delta U = -(\text{positive value})$, which means $\Delta U < 0$.
A decrease in internal energy ($\Delta U < 0$) for an ideal gas implies a decrease in its temperature ($T$).

• I. The work done is zero. In an expansion (not free expansion), the gas does work against external pressure, so $W > 0$. This statement is incorrect.

• II. The average speed of the atoms increases. Since the temperature decreases, the average kinetic energy of the atoms decreases. The average speed of the atoms is directly related to their average kinetic energy, so the average speed must decrease. This statement is incorrect.

• III. The average speed of the atoms remains constant. As explained above, the temperature decreases, so the average speed of the atoms decreases. This statement is incorrect.

• IV. The average speed of the atoms decrease. As the temperature of the gas decreases during an adiabatic expansion, the average kinetic energy of its atoms decreases, which in turn means their average speed decreases. This statement is correct.

Only statement IV is correct.

The correct option is $\boxed{\text{IV only}}$.