300 cc of a gas is compressed to 150 cc at the atmospheric pressure of `10^6` dyne/`cm^2`. If the change is sudden, what is final pressure ?
[Given `gamma=1.4`]

Answer

To solve the problem, we will use the principles of the adiabatic process, where the relationship between pressure and volume for an ideal gas can be expressed as:

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

Where:
- \( P_1 \) = Initial pressure
- \( V_1 \) = Initial volume
- \( P_2 \) = Final pressure
- \( V_2 \) = Final volume
- \( \gamma \) = Heat capacity ratio (given as 1.4)

### Step-by-Step Solution:

1. **Identify Given Values**:
- Initial volume, \( V_1 = 300 \, \text{cc} \)
- Final volume, \( V_2 = 150 \, \text{cc} \)
- Initial pressure, \( P_1 = 10^6 \, \text{dyne/cm}^2 \)
- \( \gamma = 1.4 \)

2. **Write the Adiabatic Relation**:
Using the adiabatic condition:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]

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

4. **Substituting the Values**:
- Substitute \( P_1, V_1, V_2, \) and \( \gamma \):
\[ P_2 = 10^6 \left( \frac{300}{150} \right)^{1.4} \]

5. **Calculating the Volume Ratio**:
- Calculate \( \frac{300}{150} = 2 \)

6. **Calculating \( P_2 \)**:
\[ P_2 = 10^6 \cdot 2^{1.4} \]

7. **Finding \( 2^{1.4} \)**:
- Using a calculator or logarithmic tables, we find:
\[ 2^{1.4} \approx 2.639 \]

8. **Final Calculation**:
\[ P_2 = 10^6 \cdot 2.639 \]
\[ P_2 \approx 2.639 \times 10^6 \, \text{dyne/cm}^2 \]

### Final Answer:
The final pressure \( P_2 \) is approximately \( 2.639 \times 10^6 \, \text{dyne/cm}^2 \).

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