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: Understanding the Concept:
An adiabatic process is a thermodynamic process in which there is no exchange of heat \( (dQ = 0) \) between the system and its surroundings.
Step 2: Key Formula or Approach:
For a reversible adiabatic process involving an ideal gas, the relationship between pressure \( (P) \) and volume \( (V) \) follows the Poisson's law:
\[ PV^{\gamma} = \text{constant} \]
Where \( \gamma \) (gamma) is the adiabatic index, defined as the ratio of specific heats \( C_p / C_v \).
Step 3: Detailed Explanation:
During an adiabatic expansion, the gas does work at the expense of its internal energy, causing the temperature to drop.
Similarly, during adiabatic compression, the temperature rises.
Thus, \( T \), \( P \), and \( V \) all change during the process.
However, the specific combination \( PV^{\gamma} \) remains invariant throughout the process for an ideal gas.
Other constant relations include:
- \( TV^{\gamma-1} = \text{constant} \)
- \( P^{1-\gamma} T^{\gamma} = \text{constant} \)
Step 4: Final Answer:
In an adiabatic process, the quantity \( PV^{\gamma} \) remains constant.