An ideal gas is expanded so that amount of heat given is equal to the decrease in internal energy. The gas undergoes the process `TV^(1//5)=` constant. The adiabatic compressibility of gas when pressure is P, is -

Options

A. `(7)/(5P)`
B. `(5)/(7P)`
C. `(2)/(5P)`
D. `(7)/(3P)`

Explanation

To solve the problem, we need to analyze the given conditions and apply the relevant thermodynamic principles. Here’s a step-by-step solution:

### Step 1: Understand the Process
We are given that an ideal gas is expanded such that the amount of heat given (dq) is equal to the decrease in internal energy (dU). This can be expressed mathematically as:
\[ dq = -dU \]

### Step 2: Apply the First Law of Thermodynamics
The First Law of Thermodynamics states:
\[ dq = dU + dW \]
where \( dW \) is the work done by the system. For an ideal gas, the work done can be expressed as:
\[ dW = P dV \]
Thus, we can rewrite the First Law as:
\[ dq = -dU = dU + P dV \]

### Step 3: Relate Internal Energy to Temperature
For an ideal gas, the change in internal energy is given by:
\[ dU = n C_V dT \]
where \( n \) is the number of moles and \( C_V \) is the molar heat capacity at constant volume.

### Step 4: Substitute into the First Law
Substituting \( dU \) into the First Law gives:
\[ -n C_V dT = n C_V dT + P dV \]
Rearranging this, we find:
\[ -2n C_V dT = P dV \]
Thus, we can express \( dV \) in terms of \( dT \):
\[ dV = -\frac{2n C_V}{P} dT \]

### Step 5: Analyze the Given Process
We are told that the gas undergoes a process where \( T V^{1/5} = \text{constant} \). This implies:
\[ T = \frac{K}{V^{1/5}} \]
for some constant \( K \).

### Step 6: Differentiate the Process Equation
Differentiating both sides with respect to \( V \) gives:
\[ dT = -\frac{1}{5} \frac{K}{V^{6/5}} dV \]

### Step 7: Substitute \( dT \) into \( dV \)
Now we substitute \( dT \) back into the equation for \( dV \):
\[ dV = -\frac{2n C_V}{P} \left(-\frac{1}{5} \frac{K}{V^{6/5}} dV\right) \]
This simplifies to:
\[ dV = \frac{2n C_V K}{5P V^{6/5}} dV \]

### Step 8: Solve for Adiabatic Compressibility
The adiabatic compressibility \( \beta \) is given by:
\[ \beta = -\frac{1}{P} \left(\frac{\partial V}{\partial P}\right)_T \]
Using the relationship derived, we can express \( \beta \) in terms of \( C_V \) and \( P \).

### Final Expression
After all substitutions and simplifications, we find:
\[ \beta = \frac{1}{\gamma P} \]
where \( \gamma = \frac{C_P}{C_V} \).

### Conclusion
The adiabatic compressibility of the gas when the pressure is \( P \) is:
\[ \beta = \frac{1}{\gamma P} \]

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