Question

For an ideal gas undergoing an adiabatic process, which quantity remains constant?

• A. Temperature
• B. Pressure
• C. \( PV^{\gamma} \)
• D. Volume

Hint:

For a {monatomic} ideal gas, \( \gamma = 5/3 \approx 1.67 \).
For a {diatomic} ideal gas, \( \gamma = 7/5 = 1.4 \).
The adiabatic curve on a P-V diagram is steeper than the isothermal curve.

Answer 1

The Correct Option is C

Step 1: Recall what an adiabatic process means. 
An adiabatic process is a thermodynamic change in which the system does not exchange heat with its surroundings.
Mathematically, this condition is written as $dQ = 0$.

Step 2: Identify the governing relation for an ideal gas.
For an ideal gas undergoing a reversible adiabatic change, pressure and volume are related through Poisson’s equation:

\[ P V^{\gamma} = \text{constant} \]

Here, $\gamma$ is the adiabatic index, defined as the ratio of specific heats:

\[ \gamma = \frac{C_p}{C_v} \]

Step 3: Explain the physical behavior during the process.
When a gas expands adiabatically, it performs work using its internal energy, which leads to a decrease in temperature.
Conversely, during adiabatic compression, work is done on the gas, causing its temperature to rise.
As a result, pressure, volume, and temperature all change during the process.

Despite these changes, certain combinations of variables remain unchanged.
For an adiabatic process, the invariant relationships include:

• $P V^{\gamma} = \text{constant}$
• $T V^{\gamma-1} = \text{constant}$
• $P^{1-\gamma} T^{\gamma} = \text{constant}$

Step 4: Final conclusion.
The defining constant quantity for an adiabatic process involving an ideal gas is:

\[ \boxed{P V^{\gamma}} \]