If the temperature is raised by 1 K from 300 K, the percentage change in the speed of sound in the gaseous mixture is `[R=8.314" J mol"^(-1)K^(-1)]`

Options

A. `0.167%`
B. `0.334%`
C. `1%`
D. `2%`

Answer

To solve the problem of finding the percentage change in the speed of sound in a gaseous mixture when the temperature is raised by 1 K from 300 K, we can follow these steps:

### Step 1: Understand the formula for the speed of sound
The speed of sound \( V_s \) in a gas is given by the formula:
\[ V_s = \sqrt{\frac{\gamma R T}{M}} \]
where:
- \( \gamma \) is the adiabatic index (ratio of specific heats),
- \( R \) is the universal gas constant (8.314 J mol\(^{-1}\)K\(^{-1}\)),
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas.

### Step 2: Determine the change in speed of sound with temperature
When the temperature increases by \( \Delta T = 1 \, K \), we can express the change in speed of sound as:
\[ \Delta V_s = V_s(T + \Delta T) - V_s(T) \]

### Step 3: Use the formula for percentage change
The percentage change in speed of sound can be expressed as:
\[ \frac{\Delta V_s}{V_s} \times 100\% \]

### Step 4: Use the approximation for small changes
For small changes in temperature, we can use the approximation:
\[ \frac{\Delta V_s}{V_s} \approx \frac{1}{2} \frac{\Delta T}{T} \]
This approximation comes from differentiating the speed of sound formula with respect to temperature.

### Step 5: Substitute the values
Given:
- \( \Delta T = 1 \, K \)
- \( T = 300 \, K \)

Substituting these values into the equation:
\[ \frac{\Delta V_s}{V_s} \approx \frac{1}{2} \cdot \frac{1}{300} \]

### Step 6: Calculate the percentage change
Now, we can calculate the percentage change:
\[ \frac{\Delta V_s}{V_s} \times 100\% = \left(\frac{1}{2} \cdot \frac{1}{300}\right) \times 100\% \]
\[ = \frac{50}{300} = \frac{1}{6} \approx 0.167\% \]

### Final Answer
The percentage change in the speed of sound when the temperature is raised by 1 K from 300 K is approximately \( 0.167\% \).

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