In an adiabatic process on a gas with `(gamma = 1.4)` th pressure is increased by 0.5% . The volume decreases by about

Options

A. `0.36%`
B. `0.5%`
C. `0.7%`
D. `1%`

Answer

To solve the problem of finding the decrease in volume during an adiabatic process where the pressure of a gas is increased by 0.5% and the adiabatic exponent (gamma) is 1.4, we can follow these steps:

### Step-by-Step Solution:

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

2. **Take the Natural Logarithm**:
Taking the natural logarithm of both sides gives:
\[ \ln P + \gamma \ln V = \ln \text{constant} \]

3. **Differentiate the Equation**:
Differentiating both sides with respect to time (or any variable) gives:
\[ \frac{dP}{P} + \gamma \frac{dV}{V} = 0 \]

4. **Rearrange the Equation**:
Rearranging the equation, we can express the change in volume in terms of the change in pressure:
\[ \frac{dV}{V} = -\frac{1}{\gamma} \frac{dP}{P} \]

5. **Substitute the Given Values**:
We know that the percentage change in pressure is given as 0.5%. Thus:
\[ \frac{dP}{P} = 0.5\% = \frac{0.5}{100} = 0.005 \]
Substituting this value into the equation gives:
\[ \frac{dV}{V} = -\frac{1}{1.4} \times 0.005 \]

6. **Calculate the Change in Volume**:
Performing the calculation:
\[ \frac{dV}{V} = -\frac{0.005}{1.4} \approx -0.00357 \]
To express this as a percentage:
\[ \text{Percentage change in volume} = -0.00357 \times 100 \approx -0.357\% \]

7. **Conclusion**:
The volume decreases by approximately 0.36%.

### Final Answer:
The volume decreases by about **0.36%**.