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. Let's break this down step by step.

### Step 1: Isothermal Compression
In an isothermal process, the temperature remains constant. The relationship between pressure and volume for an ideal gas can be expressed using the equation:
\[ PV = \text{constant} \]
Initially, we have:
- Pressure = \( P \)
- Volume = \( V \)

After isothermal compression to volume \( \frac{V}{n} \), the new pressure \( P_i \) can be calculated as follows:
\[ P \cdot V = P_i \cdot \frac{V}{n} \]
Rearranging gives:
\[ P_i = nP \]

### Step 2: Adiabatic Compression
In an adiabatic process, there is no heat exchange with the surroundings. The relationship between pressure and volume can be described by the equation:
\[ PV^\gamma = \text{constant} \]
where \( \gamma = \frac{C_P}{C_V} = l \).

Initially, we have:
- Pressure = \( P \)
- Volume = \( V \)

After adiabatic compression to volume \( \frac{V}{n} \), the new pressure \( P_a \) can be calculated as follows:
\[ P \cdot V^\gamma = P_a \cdot \left(\frac{V}{n}\right)^\gamma \]
Rearranging gives:
\[ P_a = P \cdot \frac{V^\gamma}{\left(\frac{V}{n}\right)^\gamma} = P \cdot \frac{V^\gamma}{\frac{V^\gamma}{n^\gamma}} = P \cdot n^\gamma \]

### Step 3: Finding the Ratio \( \frac{P_i}{P_a} \)
Now we can find the ratio \( \frac{P_i}{P_a} \):
\[ \frac{P_i}{P_a} = \frac{nP}{n^\gamma P} \]
The \( P \) cancels out:
\[ \frac{P_i}{P_a} = \frac{n}{n^\gamma} = n^{1 - \gamma} \]
Since \( \gamma = l \), we can rewrite this as:
\[ \frac{P_i}{P_a} = n^{1 - l} \]

### Final Answer
Thus, the ratio \( \frac{P_i}{P_a} \) is:
\[ \frac{P_i}{P_a} = n^{1 - l} \]