For a gas `gamma = 5//3` and 800 c.c, of this gas is suddenly compressed to 100 c.c. If the intial pressure is P, then final pressure will be

Options

A. P/32
B. `(24//5)P`
C. 0.83333333333333
D. 32 P

Explanation

To solve the problem, we will use the principles of an adiabatic process for an ideal gas. The relationship for an adiabatic process is given by:

\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]

where:
- \( P_1 \) is the initial pressure,
- \( V_1 \) is the initial volume,
- \( P_2 \) is the final pressure,
- \( V_2 \) is the final volume,
- \( \gamma \) is the adiabatic index (given as \( \frac{5}{3} \)).

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Initial volume \( V_1 = 800 \, \text{cc} \)
- Final volume \( V_2 = 100 \, \text{cc} \)
- Initial pressure \( P_1 = P \)
- Adiabatic index \( \gamma = \frac{5}{3} \)

2. **Write the Adiabatic Condition:**
- According to the adiabatic process, we can write:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]

3. **Rearranging the Equation for Final Pressure:**
- We need to find \( P_2 \):
\[ P_2 = P_1 \left( \frac{V_1}{V_2} \right)^\gamma \]

4. **Substituting the Known Values:**
- Substitute \( P_1 = P \), \( V_1 = 800 \, \text{cc} \), \( V_2 = 100 \, \text{cc} \), and \( \gamma = \frac{5}{3} \):
\[ P_2 = P \left( \frac{800}{100} \right)^{\frac{5}{3}} \]

5. **Calculating the Volume Ratio:**
- Calculate the volume ratio:
\[ \frac{800}{100} = 8 \]

6. **Applying the Volume Ratio to the Pressure Equation:**
- Now substitute this back into the equation for \( P_2 \):
\[ P_2 = P \cdot 8^{\frac{5}{3}} \]

7. **Calculating \( 8^{\frac{5}{3}} \):**
- First, calculate \( 8^{\frac{1}{3}} = 2 \) (since \( 2^3 = 8 \)).
- Then, \( 8^{\frac{5}{3}} = (2^5) = 32 \).

8. **Final Expression for \( P_2 \):**
- Substitute this value back into the equation:
\[ P_2 = P \cdot 32 \]

9. **Conclusion:**
- Therefore, the final pressure \( P_2 \) is:
\[ P_2 = 32P \]

### Final Answer:
The final pressure will be \( 32P \).