The velocity of sound in air at `27^(@)C` is 330 m/s. the velocity of sound when the temperature of the gas is raised to `227^(@)C` is

Options

A. 330 m/s
B. 660 m/s
C. 426 m/s
D. 213 m/s

Answer

To find the velocity of sound in air at a higher temperature, we can use the relationship between the velocity of sound and the absolute temperature. The formula we will use is:

\[ \frac{V_1}{V_2} = \sqrt{\frac{T_1}{T_2}} \]

Where:
- \(V_1\) = initial velocity of sound
- \(V_2\) = final velocity of sound
- \(T_1\) = initial temperature in Kelvin
- \(T_2\) = final temperature in Kelvin

### Step 1: Convert the temperatures from Celsius to Kelvin
- The initial temperature \(T_1\) at \(27^\circ C\):
\[ T_1 = 27 + 273 = 300 \, K \]
- The final temperature \(T_2\) at \(227^\circ C\):
\[ T_2 = 227 + 273 = 500 \, K \]

### Step 2: Write down the known values
- \(V_1 = 330 \, m/s\)
- \(T_1 = 300 \, K\)
- \(T_2 = 500 \, K\)

### Step 3: Use the formula to find \(V_2\)
Rearranging the formula to solve for \(V_2\):
\[ V_2 = V_1 \cdot \sqrt{\frac{T_2}{T_1}} \]

### Step 4: Substitute the known values into the equation
\[ V_2 = 330 \cdot \sqrt{\frac{500}{300}} \]

### Step 5: Calculate the square root
First, calculate \(\frac{500}{300}\):
\[ \frac{500}{300} = \frac{5}{3} \approx 1.6667 \]
Now, calculate the square root:
\[ \sqrt{1.6667} \approx 1.29099 \]

### Step 6: Calculate \(V_2\)
Now substitute back into the equation:
\[ V_2 = 330 \cdot 1.29099 \approx 426.327 \, m/s \]

### Step 7: Round the answer
Thus, the final velocity of sound at \(227^\circ C\) is approximately:
\[ V_2 \approx 426 \, m/s \]

### Conclusion
The velocity of sound when the temperature of the gas is raised to \(227^\circ C\) is approximately \(426 \, m/s\).

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