Certain perfect gas is found to obey `PV^(n)` = constant during adiabatic process.
The volume expansion coefficient at temperature `T` is

Options

A. `(1-n)/(T)`
B. `(1)/(1-n)T`
C. `(n)/(T)`
D. `(1)/(nT)`

Explanation

To find the volume expansion coefficient (β) of a perfect gas that obeys the relation \( PV^n = \text{constant} \) during an adiabatic process, we can follow these steps:

### Step 1: Understand the Volume Expansion Coefficient
The volume expansion coefficient (β) is defined as:
\[ \beta = \frac{\Delta V}{V \Delta T} \]
where \( \Delta V \) is the change in volume, \( V \) is the initial volume, and \( \Delta T \) is the change in temperature.

### Step 2: Use the Given Relation
We know that the gas obeys the relation \( PV^n = \text{constant} \). For an ideal gas, we can also use the ideal gas equation:
\[ PV = nRT \]
From this, we can express pressure \( P \) in terms of volume \( V \) and temperature \( T \):
\[ P = \frac{nRT}{V} \]

### Step 3: Substitute Pressure into the Given Relation
Substituting \( P \) into the relation \( PV^n = \text{constant} \):
\[ \left(\frac{nRT}{V}\right)V^n = \text{constant} \]
This simplifies to:
\[ nRTV^{n-1} = \text{constant} \]

### Step 4: Differentiate with Respect to Temperature
Since \( nR \) is a constant, we can differentiate both sides with respect to temperature \( T \):
\[ \frac{d}{dT}(nRTV^{n-1}) = 0 \]
Using the product rule, we have:
\[ nR \left( V^{n-1} + (n-1)V^{n-2}\frac{dV}{dT} \right) = 0 \]
This implies:
\[ V^{n-1} + (n-1)V^{n-2}\frac{dV}{dT} = 0 \]

### Step 5: Solve for \(\frac{dV}{dT}\)
Rearranging the equation gives:
\[ (n-1)V^{n-2}\frac{dV}{dT} = -V^{n-1} \]
Dividing both sides by \( V^{n-2} \) (assuming \( V \neq 0 \)):
\[ \frac{dV}{dT} = -\frac{V}{n-1} \]

### Step 6: Substitute into the Volume Expansion Coefficient Formula
Now substituting \( \frac{dV}{dT} \) into the formula for the volume expansion coefficient:
\[ \beta = \frac{\Delta V}{V \Delta T} = \frac{dV}{dT} \cdot \frac{1}{V} \]
Substituting \( \frac{dV}{dT} \):
\[ \beta = \left(-\frac{V}{n-1}\right) \cdot \frac{1}{V} = -\frac{1}{n-1} \]

### Step 7: Final Expression for Volume Expansion Coefficient
Thus, the volume expansion coefficient at temperature \( T \) is:
\[ \beta = -\frac{1}{n-1} \]