At room temperature `(27^(@)C)` the velocity of sound in air is 330 m/s. the increase in velocity of sound when temperature is increased by `1^(@)C` is

Options

A. 2 m/s
B. 1 m/s
C. 10 m/s
D. 0.55 m/s

Answer

To solve the problem of finding the increase in the velocity of sound in air when the temperature is increased by \(1^\circ C\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between velocity of sound and temperature**:
The velocity of sound in air is given by the formula:
\[ V = \sqrt{\frac{\gamma RT}{M}} \]
where:
- \(V\) is the velocity of sound,
- \(\gamma\) is the adiabatic index (approximately 1.4 for air),
- \(R\) is the universal gas constant,
- \(T\) is the absolute temperature in Kelvin,
- \(M\) is the molar mass of air.

2. **Convert the given temperature to Kelvin**:
The room temperature is given as \(27^\circ C\). To convert this to Kelvin:
\[ T_1 = 27 + 273 = 300 \, K \]

3. **Calculate the new temperature after increasing by \(1^\circ C\)**:
The new temperature \(T_2\) after an increase of \(1^\circ C\) will be:
\[ T_2 = 27 + 1 + 273 = 301 \, K \]

4. **Use the formula for velocity at both temperatures**:
The velocity of sound at \(T_1\) (300 K) is given as \(V_1 = 330 \, m/s\).
We can express the velocities at both temperatures:
\[ V_1 = \sqrt{\frac{\gamma R T_1}{M}} \quad \text{and} \quad V_2 = \sqrt{\frac{\gamma R T_2}{M}} \]

5. **Find the ratio of velocities**:
Since \(\gamma\), \(R\), and \(M\) are constants, we can find the ratio of the velocities:
\[ \frac{V_2}{V_1} = \sqrt{\frac{T_2}{T_1}} = \sqrt{\frac{301}{300}} \]

6. **Calculate \(V_2\)**:
Now substituting \(V_1 = 330 \, m/s\):
\[ V_2 = V_1 \cdot \sqrt{\frac{301}{300}} = 330 \cdot \sqrt{\frac{301}{300}} \]
Approximating \(\sqrt{\frac{301}{300}} \approx 1.005\):
\[ V_2 \approx 330 \cdot 1.005 \approx 331.65 \, m/s \]

7. **Calculate the increase in velocity**:
The increase in velocity \(\Delta V\) is:
\[ \Delta V = V_2 - V_1 = 331.65 - 330 = 1.65 \, m/s \]

### Final Answer:
The increase in the velocity of sound when the temperature is increased by \(1^\circ C\) is approximately \(1.65 \, m/s\).

---