A bicycle tyre is filled with air having pressure of `270kPa` at `27^(@)C`.The approximate pressure of the air in the tyre when the temperature increases

Options

A. `262kPa`
B. `270kPa`
C. `360kPa`
D. `278kPa`

Answer

To solve the problem of finding the approximate pressure of air in a bicycle tire when the temperature increases, we can use the ideal gas law, which states that for a gas at constant volume, the pressure is directly proportional to the temperature in Kelvin.

### Step-by-Step Solution:

1. **Identify Given Values**:
- Initial Pressure, \( P_1 = 270 \, \text{kPa} \)
- Initial Temperature, \( T_1 = 27^\circ C \)

2. **Convert Temperature to Kelvin**:
- To convert Celsius to Kelvin, use the formula:
\[ T(K) = T(°C) + 273 \]
- Therefore,
\[ T_1 = 27 + 273 = 300 \, K \]

3. **Assume Final Temperature**:
- Let's assume the final temperature \( T_2 \) is increased to \( 36^\circ C \).
- Convert \( T_2 \) to Kelvin:
\[ T_2 = 36 + 273 = 309 \, K \]

4. **Use the Direct Proportionality of Pressure and Temperature**:
- According to the ideal gas law for constant volume:
\[ \frac{P_1}{P_2} = \frac{T_1}{T_2} \]
- Rearranging gives:
\[ P_2 = P_1 \times \frac{T_2}{T_1} \]

5. **Substitute the Known Values**:
- Plug in the values:
\[ P_2 = 270 \, \text{kPa} \times \frac{309 \, K}{300 \, K} \]

6. **Calculate \( P_2 \)**:
- Performing the calculation:
\[ P_2 = 270 \times \frac{309}{300} = 270 \times 1.03 = 278.1 \, \text{kPa} \]

7. **Conclusion**:
- The approximate pressure of the air in the tire when the temperature increases to \( 36^\circ C \) is \( P_2 \approx 278.1 \, \text{kPa} \).

### Final Answer:
The approximate pressure of the air in the tire when the temperature increases is **278 kPa**.

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