An ideal gas `(gamma = (5)/(3))` is adiabatically compressed from `640 cm^(3)` to `80cm^(3)`. If the initial pressure is `P` then find the final pressure?

Explanation

To find the final pressure of an ideal gas that is adiabatically compressed, we can use the relationship that holds for adiabatic processes:

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

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 heat capacity ratio (given as \( \frac{5}{3} \)).

### Step-by-Step Solution:

1. **Identify the given values:**
- Initial volume, \( V_1 = 640 \, \text{cm}^3 \)
- Final volume, \( V_2 = 80 \, \text{cm}^3 \)
- Initial pressure, \( P_1 = P \)
- Heat capacity ratio, \( \gamma = \frac{5}{3} \)

2. **Set up the equation for adiabatic process:**
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]

3. **Substitute the known values into the equation:**
\[ P \cdot (640)^{\frac{5}{3}} = P_2 \cdot (80)^{\frac{5}{3}} \]

4. **Rearranging the equation to solve for \( P_2 \):**
\[ P_2 = P \cdot \frac{(640)^{\frac{5}{3}}}{(80)^{\frac{5}{3}}} \]

5. **Simplify the fraction:**
\[ P_2 = P \cdot \left( \frac{640}{80} \right)^{\frac{5}{3}} \]
\[ \frac{640}{80} = 8 \]
Therefore,
\[ P_2 = P \cdot 8^{\frac{5}{3}} \]

6. **Calculate \( 8^{\frac{5}{3}} \):**
\[ 8^{\frac{5}{3}} = (2^3)^{\frac{5}{3}} = 2^5 = 32 \]

7. **Final expression for \( P_2 \):**
\[ P_2 = 32P \]

### Final Answer:
The final pressure after adiabatic compression is \( 32P \).