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

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

Let us analyze the thermodynamic behavior of an ideal gas undergoing an irreversible expansion under isothermal conditions step-by-step:

1. Internal Energy Change ($∆U$) of an Ideal Gas:
For an ideal gas, the internal energy ($U$) is a function of temperature only. Since the process is carried out under isothermal conditions, the temperature remains constant ($T=\text{constant}$ or $∆T=0$).
Therefore, the change in internal energy is:
$∆U=0$

2. Total Entropy Change ($∆S^{\text{total}}$):
According to the second law of thermodynamics, for any spontaneous, irreversible process, the total entropy of the universe (system + surroundings) must increase:
$∆S^{\text{total}}=∆S^{\text{system}}+∆S^{\text{surroundings}}>0$
Since the expansion of the gas is irreversible, the total entropy change cannot be zero. Thus:
$∆S^{\text{total}}\ne 0$

Combining these two results, we get:
$∆U=0$ and $∆S^{\text{total}}\ne 0$