A long tube contains air pressure of `1 atm` and a temperature of `59^(@) C`. The tube is open at one end and closed at the other by a movable piston . A tuning fork near the open end is vibrating with a frequency of `500 Hz`. Resonance is produced when the piston is at distances `16 cm , 49.2 cm and 82.4 cm` from open end. Molar mass of air is `28.8 g//mol`.
The speed of sound in air at `59^(@) C` is

Options

A. `332 m//s`
B. `342 m//s`
C. `352 m//s`
D. `362 m//s`

Explanation

To find the speed of sound in air at 59°C, we can use the information about the resonance produced in a tube that is open at one end and closed at the other. Here are the steps to solve the problem:

### Step 1: Understand the resonance conditions
In a tube that is open at one end and closed at the other, the resonant lengths correspond to odd multiples of quarter wavelengths (λ/4). The distances at which resonance occurs are given as:
- \( L_1 = 16 \, \text{cm} \)
- \( L_2 = 49.2 \, \text{cm} \)
- \( L_3 = 82.4 \, \text{cm} \)

### Step 2: Calculate the difference in lengths
The difference between the lengths at which resonance occurs can be calculated:
\[ L_2 - L_1 = 49.2 \, \text{cm} - 16 \, \text{cm} = 33.2 \, \text{cm} \]
This difference corresponds to half a wavelength (λ/2) because the tube supports a fundamental mode and its harmonics.

### Step 3: Find the wavelength
Since the difference \( L_2 - L_1 \) corresponds to half a wavelength:
\[ \frac{\lambda}{2} = 33.2 \, \text{cm} \implies \lambda = 66.4 \, \text{cm} \]
Convert this to meters:
\[ \lambda = 0.664 \, \text{m} \]

### Step 4: Use the frequency to find the speed of sound
The speed of sound \( v \) can be calculated using the formula:
\[ v = f \cdot \lambda \]
where \( f \) is the frequency of the tuning fork, given as \( 500 \, \text{Hz} \). Thus:
\[ v = 500 \, \text{Hz} \cdot 0.664 \, \text{m} = 332 \, \text{m/s} \]

### Final Answer
The speed of sound in air at 59°C is approximately \( 332 \, \text{m/s} \).

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