What will be the work done by 3 moles of an ideal gas when it expands spontaneously in a vacuum?

The correct option is "zero".

To understand why the work done is zero, we need to analyze the thermodynamic definition of work during gas expansion.

When an ideal gas expands against an external pressure, the work done by or on the system is mathematically expressed as:

$$W=-P_{\text{ext}}\Delta V$$

In this formula:

• $W$ represents the work done.

• $P_{\text{ext}}$ represents the external pressure opposing the expansion.

• $\Delta V$ represents the change in volume of the gas.

In this problem, the gas expands spontaneously into a vacuum. A vacuum is a space entirely devoid of matter, which means there is no external gas or atmosphere to exert any opposing force or pressure. Therefore, the external pressure is exactly zero:

$$P_{\text{ext}}=0$$

Substituting this value into the work equation, we obtain:

$$W=-\left(0\right)\times \Delta V=0$$

This process is known as free expansion. Because there is no resisting external force to overcome, the gas does not perform any work during the expansion, regardless of the number of moles of gas present (3 moles in this case). Thus, the work done by the gas is zero.