If`lambda=2.5` and volume is equal to `(1)/(8)` times to the initial volume then perssure p' is equal to (initial pressure = p)

Options

A. `p'=p`
B. `p'=2p`
C. `p'=pxx(2)^(15//2)`
D. `p'=7p`

Answer

To solve the problem, we need to find the new pressure \( P' \) when the volume is reduced to \( \frac{1}{8} \) of the initial volume and the process is adiabatic with \( \lambda = 2.5 \).

### Step-by-Step Solution:

1. **Understand the Adiabatic Process:**
In an adiabatic process, the relationship between pressure and volume is given by:
\[ P V^\lambda = \text{constant} \]
where \( \lambda \) is the adiabatic index.

2. **Define Initial and Final Conditions:**
Let:
- Initial pressure = \( P \)
- Initial volume = \( V_1 \)
- Final volume = \( V' = \frac{1}{8} V_1 \)
- Final pressure = \( P' \)

3. **Apply the Adiabatic Condition:**
According to the adiabatic condition:
\[ P V_1^\lambda = P' V'^\lambda \]
Substituting \( V' = \frac{1}{8} V_1 \):
\[ P V_1^\lambda = P' \left(\frac{1}{8} V_1\right)^\lambda \]

4. **Simplify the Equation:**
This can be rewritten as:
\[ P V_1^\lambda = P' \left(\frac{1}{8^\lambda} V_1^\lambda\right) \]
Rearranging gives:
\[ P' = P \cdot 8^\lambda \]

5. **Substitute the Value of \( \lambda \):**
Given \( \lambda = 2.5 \):
\[ P' = P \cdot 8^{2.5} \]

6. **Calculate \( 8^{2.5} \):**
We can express \( 8 \) as \( 2^3 \):
\[ 8^{2.5} = (2^3)^{2.5} = 2^{3 \cdot 2.5} = 2^{7.5} = 2^{15/2} \]

7. **Final Expression for \( P' \):**
Therefore, we have:
\[ P' = P \cdot 2^{15/2} \]

### Conclusion:
The final pressure \( P' \) is given by:
\[ P' = P \cdot 2^{15/2} \]