Question

For a thermodynamic system undergoing an adiabatic process, what is the relationship between pressure \(P\) and volume \(V\)?

• A. \(PV = \text{constant}\)
• B. \(P/V = \text{constant}\)
• C. \(PV^\gamma = \text{constant}\)
• D. \(P^2V = \text{constant}\)

Hint:

Adiabatic process relations: \[ PV^\gamma = \text{constant}, \quad TV^{\gamma-1} = \text{constant} \]

Answer 1

The Correct Option is C

Concept: An adiabatic process is a thermodynamic process in which no heat is exchanged between the system and its surroundings. \[ Q = 0 \] For an ideal gas undergoing an adiabatic process, the relation between pressure and volume is given by \[ PV^\gamma = \text{constant} \] where

• \(P\) = pressure
• \(V\) = volume
• \(\gamma = \frac{C_p}{C_v}\) = adiabatic index

Step 1: Use the first law of thermodynamics.} \[ dQ = dU + dW \] For adiabatic process \[ dQ = 0 \] \[ dU = -dW \]
Step 2: Derive the pressure–volume relation.} Using ideal gas relations and thermodynamic identities, the final relation becomes \[ PV^\gamma = \text{constant} \]
Step 3: Interpretation.} During adiabatic expansion, volume increases and pressure decreases more rapidly compared to an isothermal process. Thus, the correct relation is \[ PV^\gamma = \text{constant} \]