Statement: Tds = du pdV

Explanation:
To understand why option C is the correct answer, let's break down the given statement and analyze each term separately.

Tds:
Tds represents the change in entropy of the system. Entropy is a measure of the system's disorder or randomness. For a reversible process, the change in entropy can be calculated using the equation Tds = du + pdV, where du is the change in internal energy and pdV is the work done by the gas.

du:
du represents the change in internal energy of the system. Internal energy is the sum of the kinetic and potential energies of the gas molecules. For a perfect gas, the internal energy depends only on the temperature of the gas.

pdV:
pdV represents the work done by the gas during the process. It is the product of pressure (p) and change in volume (dV). For a perfect gas, the work done can be calculated using the equation pdV = nRT, where n is the number of moles of the gas, R is the gas constant, and T is the temperature.

Irreversible Process:
An irreversible process is a process that cannot be reversed to its original state by any means. In an irreversible process, there may be energy losses due to friction, heat transfer to the surroundings, or other irreversibilities.

Explanation of the Correct Answer:
In an irreversible process, the system does not follow a well-defined path and there may be energy losses. However, the change in internal energy (du) and the change in entropy (ds) of the system can still be calculated using the initial and final states of the system.

The given statement Tds = du + pdV holds true for a reversible process, where there are no energy losses. In an irreversible process, the term pdV represents the work done by the gas, but it may not be equal to the exact work done due to energy losses.

However, the change in internal energy (du) and the change in entropy (ds) can still be calculated using the initial and final states of the system. Therefore, the statement Tds = du + pdV is also applicable to an irreversible process.

Conclusion:
The correct statement applicable to a perfect gas that is also true for an irreversible process is:
Tds = du + pdV (Option C)