Fundamental note of a closed pipe of lengt is l `f_(0)` at `0^(@)C` first overtone of te same pipe at `273^(@)C` is (assum there is no effect of temperature on te dimension of the pipe.)

Options

A. `3f_(0)`
B. `f_(0)`
C. `(3sqrt(2))f_(0)`
D. `3//sqrt(2)f_(0)`

Explanation

To solve the problem, we need to find the frequency of the first overtone of a closed pipe at a temperature of 273°C, given that the fundamental frequency at 0°C is \( f_0 \).

### Step-by-Step Solution:

1. **Understanding the Frequencies in a Closed Pipe:**
- The frequency of the \( n \)-th harmonic in a closed pipe is given by the formula:
\[ f_n = \frac{(2n-1) V}{4L} \]
- For the fundamental frequency (first harmonic, \( n=1 \)):
\[ f_0 = \frac{V}{4L} \]
- For the first overtone (second harmonic, \( n=2 \)):
\[ f_1 = \frac{3V}{4L} \]

2. **Effect of Temperature on Velocity of Sound:**
- The velocity of sound \( V \) in a gas is given by:
\[ V = \sqrt{\frac{\gamma RT}{M}} \]
- Here, \( \gamma \) is the adiabatic index, \( R \) is the universal gas constant, \( T \) is the absolute temperature in Kelvin, and \( M \) is the molar mass of the gas.
- The velocity of sound is proportional to the square root of the temperature:
\[ V \propto \sqrt{T} \]

3. **Calculating Frequencies at Different Temperatures:**
- At \( 0^\circ C \) (which is 273 K), the fundamental frequency is \( f_0 \).
- At \( 273^\circ C \) (which is 546 K), the first overtone frequency can be calculated as follows:
\[ f_1 = \frac{3V_2}{4L} \]
- Since \( V_2 \) at 546 K can be expressed in terms of \( V_1 \) at 273 K:
\[ V_2 = V_1 \sqrt{\frac{546}{273}} = V_1 \sqrt{2} \]

4. **Relating Frequencies:**
- The frequency at 546 K can now be expressed as:
\[ f_1 = \frac{3V_1 \sqrt{2}}{4L} \]
- We know that \( V_1 = 4L f_0 \), so substituting this into the equation gives:
\[ f_1 = 3 \cdot \frac{4L f_0 \sqrt{2}}{4L} = 3\sqrt{2} f_0 \]

5. **Final Result:**
- Therefore, the frequency of the first overtone at 273°C is:
\[ f_1 = 3\sqrt{2} f_0 \]

### Conclusion:
The first overtone of the closed pipe at \( 273^\circ C \) is \( 3\sqrt{2} f_0 \).