An ideal gs at pressure P is adiabatically compressed so that its density becomes n times the initial vlaue The final pressure of the gas will be `(gamma=(C_(P))/(C_(V)))`

Options

A. `ngammaP`
B. `(n-gamma)P`
C. `n(gamma-1)P`
D. `n(1-gamma)P`

Explanation

To solve the problem of finding the final pressure of an ideal gas that is adiabatically compressed such that its density becomes n times the initial value, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**: For an adiabatic process involving an ideal gas, the relationship between pressure (P) and volume (V) is given by the equation:
\[ PV^\gamma = \text{constant} \]
where \(\gamma = \frac{C_P}{C_V}\) is the specific heat ratio.

2. **Define Initial Conditions**: Let \(P_1\) be the initial pressure and \(V_1\) be the initial volume of the gas. Therefore, we can write:
\[ P_1 V_1^\gamma = \text{constant} \]

3. **Express Density**: The density of a gas is defined as:
\[ \rho = \frac{m}{V} \]
where \(m\) is the mass of the gas. If we denote the initial density as \(\rho_1\), then the initial volume can be expressed as:
\[ V_1 = \frac{m}{\rho_1} \]

4. **Final Density**: According to the problem, the final density \(\rho_2\) becomes \(n\) times the initial density:
\[ \rho_2 = n \rho_1 \]
The final volume \(V_2\) can then be expressed as:
\[ V_2 = \frac{m}{\rho_2} = \frac{m}{n \rho_1} = \frac{V_1}{n} \]

5. **Apply the Adiabatic Condition**: Now, we can apply the adiabatic condition to the final state:
\[ P_2 V_2^\gamma = P_1 V_1^\gamma \]
Substituting \(V_2 = \frac{V_1}{n}\) into the equation gives:
\[ P_2 \left(\frac{V_1}{n}\right)^\gamma = P_1 V_1^\gamma \]

6. **Rearranging the Equation**: This can be rearranged to find \(P_2\):
\[ P_2 = P_1 \left(n^{-\gamma}\right) V_1^{\gamma} \cdot V_1^{\gamma} \]
Simplifying this gives:
\[ P_2 = P_1 n^{\gamma} \]

7. **Final Expression**: Therefore, the final pressure of the gas after adiabatic compression is:
\[ P_2 = n^{\gamma} P_1 \]

### Final Answer:
The final pressure of the gas after adiabatic compression is:
\[ P_2 = n^{\gamma} P \]
where \(P\) is the initial pressure of the gas.