When an ideal gas at pressure P, temperature T and volume V is isothermally compressed to V/n. its pressure becomes `P_i`. If the gas is compressed adiabatically to V/n, its pressure becomes `P_a`. The ratio `P_i//P_a` is: ( `l = C_P //C_V`)

Options

A. `1`
B. `n`
C. `n^("'l")`
D. `n^((1-l))`

Explanation

To solve the problem, we need to find the ratio \( \frac{P_i}{P_a} \) where \( P_i \) is the pressure after isothermal compression and \( P_a \) is the pressure after adiabatic compression.

### Step-by-step Solution:

1. **Isothermal Compression**:
- For an ideal gas undergoing isothermal compression, we can use the ideal gas law, which states:
\[ PV = nRT \]
- Initially, the gas has pressure \( P \), volume \( V \), and temperature \( T \). After isothermal compression to volume \( \frac{V}{n} \), the new pressure \( P_i \) can be calculated as:
\[ P_i \cdot \frac{V}{n} = P \cdot V \]
- Rearranging gives:
\[ P_i = nP \]
- This is our **Equation 1**.

2. **Adiabatic Compression**:
- For an adiabatic process, we use the relation:
\[ PV^\gamma = \text{constant} \]
- Here, \( \gamma \) is the ratio of specific heats, which is given as \( l = \frac{C_P}{C_V} \).
- The initial conditions are the same: pressure \( P \) and volume \( V \). After adiabatic compression to volume \( \frac{V}{n} \), the new pressure \( P_a \) can be expressed as:
\[ P \cdot V^\gamma = P_a \cdot \left(\frac{V}{n}\right)^\gamma \]
- Rearranging gives:
\[ P_a = P \cdot n^\gamma \]
- This is our **Equation 2**.

3. **Finding the Ratio**:
- Now, we need to find the ratio \( \frac{P_i}{P_a} \):
\[ \frac{P_i}{P_a} = \frac{nP}{n^\gamma P} \]
- Simplifying this, we have:
\[ \frac{P_i}{P_a} = \frac{n}{n^\gamma} = n^{1 - \gamma} \]
- Since \( \gamma = l \), we can write:
\[ \frac{P_i}{P_a} = n^{1 - l} \]

### Final Result:
The ratio \( \frac{P_i}{P_a} \) is:
\[ \frac{P_i}{P_a} = n^{1 - l} \]