Question

The condition \( dw = dq \) holds good in the following process:

• A. Adiabatic process
• B. Isothermal process
• C. Isochoric process
• D. Isobaric process

Hint:

In an isothermal process, the internal energy remains unchanged, meaning all heat supplied to the system is converted into work.

Answer 1

The Correct Option is B

Step 1: Understanding the Given Condition The first law of thermodynamics states: \[ dQ = dU + dW \] where: - \( dQ \) is the heat supplied, - \( dU \) is the change in internal energy, - \( dW \) is the work done by the system. Step 2: Applying the Isothermal Condition In an isothermal process, the temperature remains constant, meaning: \[ dU = 0 \] Thus, the first law reduces to: \[ dQ = dW \] which is exactly the given condition \( dw = dq \). Step 3: Why Other Options Are Incorrect - Adiabatic process: \( dQ = 0 \), meaning no heat exchange occurs.
- Isochoric process: \( dW = 0 \), as volume remains constant, meaning no work is done.
- Isobaric process: Heat is added, but part of it increases internal energy, so \( dQ \neq dW \).
Thus, the correct answer is Isothermal process.

Answer 2

In thermodynamics, the first law states:
\[ dU = dq - dw \] where:
- \( dU \) is the change in internal energy,
- \( dq \) is the heat added to the system,
- \( dw \) is the work done by the system.

The condition \( dw = dq \) implies:
\[ dU = dq - dw = 0 \] meaning the internal energy \( U \) remains constant during the process.

For an ideal gas, the internal energy depends only on temperature, so:
- If \( dU = 0 \), the temperature remains constant.

Therefore, the process where \( dw = dq \) holds is an
\[ \boxed{\text{Isothermal process}} \] because in an isothermal process, the temperature (and thus internal energy) is constant, so the heat added to the system is entirely used for doing work.