A gas is compressed adiabatically to half its volume. By what factor does the pressure of the gas increase?

Options

A. 2
B. 2.64
C. 1.4
D. 2.4

Explanation

To solve the problem of how much the pressure of a gas increases when it is compressed adiabatically to half its volume, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
In an adiabatic process, there is no heat exchange with the surroundings. The relationship between pressure (P) and volume (V) for an adiabatic process is given by the equation:
\[ PV^\gamma = \text{constant} \]
where \(\gamma\) (gamma) is the heat capacity ratio, defined as \(C_P/C_V\).

2. **Set Up the Initial and Final Conditions**:
Let the initial pressure be \(P_i\) and the initial volume be \(V_i\). When the gas is compressed to half its volume, the final volume \(V_f\) becomes:
\[ V_f = \frac{V_i}{2} \]

3. **Apply the Adiabatic Condition**:
Using the adiabatic condition, we can write:
\[ P_i V_i^\gamma = P_f V_f^\gamma \]

4. **Express Final Pressure in Terms of Initial Pressure**:
Rearranging the equation gives:
\[ P_f = P_i \left(\frac{V_i}{V_f}\right)^\gamma \]
Substituting \(V_f = \frac{V_i}{2}\):
\[ P_f = P_i \left(\frac{V_i}{\frac{V_i}{2}}\right)^\gamma = P_i \left(2\right)^\gamma \]

5. **Calculate the Pressure Increase Factor**:
Therefore, the factor by which the pressure increases is:
\[ \frac{P_f}{P_i} = 2^\gamma \]
If \(\gamma\) is given as 1.4, we calculate:
\[ 2^{1.4} \]

6. **Compute the Value**:
Using a calculator or logarithmic tables:
\[ 2^{1.4} \approx 2.639 \]
Thus, the final pressure \(P_f\) is approximately:
\[ P_f \approx 2.639 P_i \]

### Conclusion:
The pressure of the gas increases by a factor of approximately **2.639** when compressed adiabatically to half its volume.