The pressure and volume of a given mass of gas at a given temperature are P and V respectively. Keeping the temperature constant, the pressure is increased by 10% and the system is allowed to achieve a steady-state, then the pressure is decreased by 10% what cann be said about the final volume?

Options

A. less than V
B. more than V
C. equal to V
D. less than V for diatomic and more than V for monoatomic

Answer

To solve the problem, we need to analyze the changes in pressure and volume of the gas while keeping the temperature constant. We will use the ideal gas law, which states that for a given mass of gas at constant temperature, the product of pressure (P) and volume (V) remains constant.

### Step-by-Step Solution:

1. **Initial Conditions**:
Let the initial pressure be \( P_1 \) and the initial volume be \( V_1 \). According to the ideal gas law, we have:
\[ P_1 V_1 = nRT \]
where \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature.

2. **Increase in Pressure**:
The pressure is increased by 10%. Therefore, the new pressure \( P_2 \) becomes:
\[ P_2 = 1.1 P_1 \]
Since the temperature is constant, we can apply the ideal gas law again:
\[ P_1 V_1 = P_2 V_2 \]
Substituting \( P_2 \):
\[ P_1 V_1 = (1.1 P_1) V_2 \]
Dividing both sides by \( P_1 \) (assuming \( P_1 \neq 0 \)):
\[ V_1 = 1.1 V_2 \]
Rearranging gives us:
\[ V_2 = \frac{V_1}{1.1} \]

3. **Decrease in Pressure**:
Now, the pressure is decreased by 10% from \( P_2 \). The new pressure \( P_3 \) is:
\[ P_3 = P_2 - 0.1 P_2 = 0.9 P_2 = 0.9 (1.1 P_1) = 0.99 P_1 \]

4. **Final Volume Calculation**:
Again applying the ideal gas law:
\[ P_2 V_2 = P_3 V_3 \]
Substituting \( P_2 \) and \( P_3 \):
\[ (1.1 P_1) \left(\frac{V_1}{1.1}\right) = (0.99 P_1) V_3 \]
Simplifying the left side:
\[ P_1 V_1 = 0.99 P_1 V_3 \]
Dividing both sides by \( P_1 \) (assuming \( P_1 \neq 0 \)):
\[ V_1 = 0.99 V_3 \]
Rearranging gives us:
\[ V_3 = \frac{V_1}{0.99} \]

5. **Conclusion**:
Since \( V_3 \) is greater than \( V_1 \) (because \( 0.99 < 1 \)), we conclude that the final volume \( V_3 \) is slightly larger than the initial volume \( V_1 \).

### Final Answer:
The final volume \( V_3 \) is greater than the initial volume \( V_1 \).