For an isothermal process:
A. dQ = dW
B. dQ = dU
C. dW = dU
D. dQ = dU + dW

We need to identify the correct relation for heat (dQdQ) and work (dWdW) in an isothermal process.
In thermodynamics, an isothermal process is a process that occurs at a constant temperature. For an ideal gas undergoing an isothermal process, the internal energy (UU) remains constant because the temperature is constant.
The first law of thermodynamics states:
dQ=dU+dWdQ = dU + dW
Where:
• dQdQ is the heat added to the system,
• dUdU is the change in internal energy,
• dWdW is the work done by the system.
For an isothermal process:
• Since the temperature is constant, the internal energy UU of an ideal gas does not change, i.e., dU=0dU = 0.
• Therefore, the heat added to the system dQdQ is entirely used for doing work dWdW.
Thus, for an isothermal process:
dQ=dWdQ = dW
Analysis of the Options:
• A. dQ=dWdQ = dW: This is correct. For an isothermal process, the heat added is equal to the work done.
• B. dQ=dUdQ = dU: Incorrect. In an isothermal process, dU=0dU = 0, so dQ≠dUdQ \neq dU.
• C. dW=dUdW = dU: Incorrect. The work done is not equal to the change in internal energy in an isothermal process. In fact, dU=0dU = 0 for an ideal gas.
• D. dQ=dU+dWdQ = dU + dW: This is the general first law of thermodynamics. It applies to all processes, not specifically to isothermal processes.
The correct option is: A) dQ=dWdQ = dW.