A gas follows a process `TV^(n-1)=const ant`, where `T=` absolute temperature of the gas and `V=` volume of the gas. The bulk modulus of the gas in the process is given by

Options

A. (a) `(n-1)p`
B. (b) `p//(n-1)`
C. (c) `np`
D. (d) `p//n`

Explanation

To find the bulk modulus of the gas that follows the process \( TV^{(n-1)} = \text{constant} \), we will proceed step by step.

### Step 1: Understand the definition of bulk modulus
The bulk modulus \( B \) is defined as:
\[ B = -V \frac{dp}{dV} \]
where \( p \) is the pressure and \( V \) is the volume of the gas.

### Step 2: Use the given process equation
The process is given by:
\[ TV^{(n-1)} = \text{constant} \]
We can express this in terms of pressure using the ideal gas law, which states:
\[ PV = nRT \quad \Rightarrow \quad T = \frac{PV}{nR} \]

### Step 3: Substitute \( T \) in the process equation
Substituting \( T \) into the process equation:
\[ \left(\frac{PV}{nR}\right)V^{(n-1)} = \text{constant} \]
This simplifies to:
\[ \frac{PV^{n}}{nR} = \text{constant} \]
Thus, we can say:
\[ PV^{n} = \text{constant} \cdot nR \]

### Step 4: Differentiate both sides
Now, we differentiate both sides with respect to \( V \):
\[ P \cdot nV^{(n-1)} \frac{dV}{dV} + V^{n} \frac{dp}{dV} = 0 \]
This simplifies to:
\[ nPV^{(n-1)} + V^{n} \frac{dp}{dV} = 0 \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ V^{n} \frac{dp}{dV} = -nPV^{(n-1)} \]
Dividing both sides by \( V^{n} \):
\[ \frac{dp}{dV} = -\frac{nP}{V} \]

### Step 6: Substitute into the bulk modulus formula
Now substituting \( \frac{dp}{dV} \) into the bulk modulus formula:
\[ B = -V \left(-\frac{nP}{V}\right) = nP \]

### Conclusion
Thus, the bulk modulus of the gas in the process is:
\[ B = nP \]