By increasing temperature of a gas by `6^@ C` its pressure increases by 0.4%, at constant volume. Then initial temperature of gas is

Options

A. 1000 K
B. 1500 K
C. 2000 K
D. 750 K

Answer

To solve the problem of finding the initial temperature of the gas given the increase in temperature and pressure, we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Problem**:
- We are given that the temperature of a gas increases by \(6^\circ C\) and the pressure increases by \(0.4\%\) at constant volume. We need to find the initial temperature of the gas.

2. **Setting Up Initial and Final Conditions**:
- Let the initial pressure be \(P_0\) and the initial temperature be \(T\).
- After the increase, the final pressure \(P\) becomes:
\[ P = P_0 + 0.004 P_0 = 1.004 P_0 \]
- The final temperature after the increase is:
\[ T + 6 \]

3. **Using the Ideal Gas Law**:
- According to the ideal gas law, at constant volume, we have:
\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]
- Here, \(P_1 = P_0\), \(T_1 = T\), \(P_2 = 1.004 P_0\), and \(T_2 = T + 6\).

4. **Setting Up the Equation**:
- Plugging in the values, we get:
\[ \frac{P_0}{T} = \frac{1.004 P_0}{T + 6} \]
- We can cancel \(P_0\) from both sides (assuming \(P_0 \neq 0\)):
\[ \frac{1}{T} = \frac{1.004}{T + 6} \]

5. **Cross-Multiplying**:
- Cross-multiplying gives us:
\[ T + 6 = 1.004 T \]

6. **Rearranging the Equation**:
- Rearranging the equation leads to:
\[ 1.004 T - T = 6 \]
- Simplifying this gives:
\[ 0.004 T = 6 \]

7. **Solving for T**:
- Dividing both sides by \(0.004\):
\[ T = \frac{6}{0.004} = 1500 \text{ K} \]

8. **Conclusion**:
- The initial temperature of the gas is \(1500 \text{ K}\).

### Final Answer
The initial temperature of the gas is \(1500 \text{ K}\).