Choose the correct alternative (a), (b), (c) or (d) for each of the questions given below :
For an adiabatic change of a perfect gas the relation between pressure and volume is :

Options

A. `PV^(gamma)=`constant
B. `P^(gamma)V=`constant
C. PV = constant
D. `PV^(gamma-1)=`constant

Answer

To solve the question regarding the relation between pressure and volume for an adiabatic change of a perfect gas, we can follow these steps:

### Step 1: Understand the Adiabatic Process
In an adiabatic process, there is no heat exchange with the surroundings. Therefore, the first law of thermodynamics simplifies to:
\[ \Delta U = W \]
where \( \Delta U \) is the change in internal energy and \( W \) is the work done on or by the gas.

### Step 2: Use the Ideal Gas Law
For a perfect gas, the ideal gas law is given by:
\[ PV = nRT \]
where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

### Step 3: Apply the Adiabatic Condition
For an adiabatic process involving a perfect gas, the following relationships hold:
1. \( PV^{\gamma} = \text{constant} \)
2. \( TV^{\gamma - 1} = \text{constant} \)

Here, \( \gamma \) (gamma) is the heat capacity ratio defined as:
\[ \gamma = \frac{C_p}{C_v} \]
where \( C_p \) is the heat capacity at constant pressure and \( C_v \) is the heat capacity at constant volume.

### Step 4: Identify the Correct Relation
From the relationships stated above, the correct relation for pressure and volume during an adiabatic change is:
\[ PV^{\gamma} = \text{constant} \]

### Step 5: Choose the Correct Option
Now, we can analyze the options provided in the question. The correct relation is represented by:
- Option (a): \( PV^{\gamma} = \text{constant} \) (This is correct)
- Option (b): Incorrect
- Option (c): Incorrect
- Option (d): Incorrect

Thus, the correct answer is option (a).