For irreversible expansion of an ideal gas under isothermal condition, the correct option is :

The correct option is ∆U=0, ∆Sᵗᵒᵗᵃˡ≠0.

Let's break down the physical principles behind this result step-by-step:

1. Isothermal Condition and Internal Energy Change (ΔU):
For an ideal gas, the internal energy ($U$) depends solely on its absolute temperature.
An isothermal process is defined as one that occurs at a constant temperature, meaning:

$\Delta T=0$

Because the temperature does not change, there is no change in the internal energy of the ideal gas. Therefore, we have:

$\Delta U=0$

2. Irreversible Process and Total Entropy Change (ΔS_total):
According to the Second Law of Thermodynamics, the total entropy change of the universe (system + surroundings) for any spontaneous or irreversible process must always be greater than zero:

$\Delta S_{\text{total}}=\Delta S_{\text{system}}+\Delta S_{\text{surroundings}}>0$

Since the total entropy change is strictly positive for an irreversible process, it cannot be equal to zero:

$\Delta S_{\text{total}}\ne 0$

Combining these two conclusions, for the irreversible isothermal expansion of an ideal gas:

$\Delta U=0$
and

$\Delta S_{\text{total}}\ne 0$