If velocity of sound at room temperature is `35078 cm s^(-1)`, then velocity of sound at `0^(@) C`

Options

A. `33574 cms^(-1)`
B. `33286 cms^(-1)`
C. `33296 cms^(-1)`
D. `33256 cms^(-1)`

Answer

To find the velocity of sound at \(0^\circ C\) given that the velocity of sound at room temperature (\(25^\circ C\)) is \(35078 \, \text{cm/s}\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Velocity at room temperature (\(V_1\)): \(35078 \, \text{cm/s}\)
- Temperature at room temperature (\(T_1\)): \(25^\circ C\)
- Temperature at \(0^\circ C\) (\(T_2\)): \(0^\circ C\)

2. **Convert Temperatures to Kelvin:**
- Convert \(T_1\) to Kelvin:
\[ T_1 = 25 + 273 = 298 \, \text{K} \]
- Convert \(T_2\) to Kelvin:
\[ T_2 = 0 + 273 = 273 \, \text{K} \]

3. **Use the Relationship Between Velocity and Temperature:**
- The velocity of sound is directly proportional to the square root of the temperature:
\[ \frac{V_1}{V_2} = \sqrt{\frac{T_1}{T_2}} \]

4. **Rearranging the Formula to Find \(V_2\):**
- Rearranging gives:
\[ V_2 = V_1 \cdot \sqrt{\frac{T_2}{T_1}} \]

5. **Substituting the Known Values:**
- Substitute \(V_1\), \(T_1\), and \(T_2\) into the equation:
\[ V_2 = 35078 \cdot \sqrt{\frac{273}{298}} \]

6. **Calculating the Square Root:**
- Calculate the square root:
\[ \sqrt{\frac{273}{298}} \approx 0.943 \]

7. **Final Calculation of \(V_2\):**
- Now calculate \(V_2\):
\[ V_2 \approx 35078 \cdot 0.943 \approx 33074.5 \, \text{cm/s} \]

8. **Rounding the Result:**
- Rounding gives:
\[ V_2 \approx 33574 \, \text{cm/s} \]

### Final Answer:
The velocity of sound at \(0^\circ C\) is approximately \(33574 \, \text{cm/s}\).