For an isothermal free expansion of an ideal gas into vacuum, which one of the following set of values is correct?

1. Work Done ($$w=0$$)

Free expansion means the gas expands into a vacuum. Because the external pressure ($$P_{\mathrm{ext}}$$) is zero, no opposing force exists.

Using the work formula:

$$w=-P_{\mathrm{ext}}\cdot\Delta V$$

$$w=0\cdot\Delta V=0$$

2. Change in Internal Energy ($$\Delta U=0$$)

The process is explicitly stated to be isothermal. For an ideal gas, internal energy depends strictly on temperature.

Since temperature remains constant ($$\Delta T=0$$), the internal energy does not change:

$$\Delta U=n\cdot C_v\cdot\Delta T=0$$

3. Heat Exchanged ($$q=0$$)

According to the First Law of Thermodynamics, the total energy change relates to heat and work:

$$\Delta U=q+w$$

Substituting the known values ($$\Delta U=0$$ and $$w=0$$):

$$0=q+0$$

Correct option: (D)