Question

In an adiabatic expansion of an ideal gas the product of pressure and volume

• A. Decreases
• B. Increases
• C. Remains constant
• D. At first increases and then decreases

Answer 1

The Correct Option is A

In an adiabatic process, no heat is exchanged with the surroundings. According to the first law of thermodynamics and for an ideal gas undergoing adiabatic expansion, the product of pressure and volume, \( P \cdot V \), decreases. This is because, as the gas expands, its pressure decreases and its volume increases, but the rate at which the volume increases is slower than the pressure decreases, leading to an overall decrease in the product of \( P \) and \( V \).

So, the correct answer is (A): Decreases

Answer 2

In an adiabatic expansion, the process occurs without heat exchange, and the first law of thermodynamics gives us the equation: \[ dQ = 0 \quad \Rightarrow \quad dU = -PdV \] For an ideal gas, the internal energy \( U \) depends only on temperature, so: \[ dU = n C_V dT \] where \( n \) is the number of moles and \( C_V \) is the molar specific heat capacity at constant volume. Combining these equations gives the relationship: \[ PdV = n C_V dT \] For an ideal gas undergoing adiabatic expansion, we also know the following relation between pressure and volume during the process: \[ P V^\gamma = \text{constant} \] where \( \gamma = \frac{C_P}{C_V} \) is the adiabatic index. This means that the product of pressure and volume raised to the power of \( \gamma \) remains constant during the adiabatic expansion. Therefore, since \( \gamma > 1 \), the product \( P \cdot V \) decreases during expansion.

Thus, the product of pressure and volume decreases in an adiabatic expansion of an ideal gas.