A gas at the temperature 250 K is contained in a closed vessel. If the gas is heated through 1K, then the percentage increase in its pressure will be

Options

A. `0.4%`
B. `0.2%`
C. `0.1%`
D. `0.8%`

Answer

To solve the problem, we need to determine the percentage increase in pressure when the temperature of a gas in a closed vessel is increased by 1 K from an initial temperature of 250 K. We'll use the ideal gas law and the relationship between pressure and temperature.

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- Initial Temperature (T) = 250 K
- Change in Temperature (ΔT) = 1 K

2. **Calculate the Final Temperature:**
- Final Temperature (T') = T + ΔT = 250 K + 1 K = 251 K

3. **Understand the Relationship Between Pressure and Temperature:**
- According to the ideal gas law, for a fixed amount of gas in a closed vessel (where volume and number of moles are constant), pressure (P) is directly proportional to temperature (T). This can be expressed as:
\[ P \propto T \]
- Therefore, the change in pressure (ΔP) can be related to the change in temperature (ΔT) as follows:
\[ \frac{\Delta P}{P} = \frac{\Delta T}{T} \]

4. **Calculate the Change in Temperature Ratio:**
- We can substitute the values into the equation:
\[ \Delta T = T' - T = 251 K - 250 K = 1 K \]
- Now, substituting into the ratio:
\[ \frac{\Delta T}{T} = \frac{1 K}{250 K} = \frac{1}{250} \]

5. **Calculate the Percentage Change in Pressure:**
- To find the percentage change in pressure, we multiply the ratio by 100:
\[ \text{Percentage Change in Pressure} = \left(\frac{1}{250}\right) \times 100 = 0.4\% \]

6. **Conclusion:**
- The percentage increase in pressure when the gas is heated by 1 K is **0.4%**.

### Final Answer:
The percentage increase in pressure is **0.4%**.

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