The temperature at which the speed of sound in air becomes double of its value at `27^(@)C` is

Options

A. `54^(@)C`
B. `327^(@)C`
C. `927^(@)C`
D. `-123^(@)`C

Answer

To find the temperature at which the speed of sound in air becomes double its value at 27°C, we can follow these steps:

### Step 1: Understand the relationship between speed of sound and temperature
The speed of sound in air is given by the formula:
\[ V = \sqrt{\frac{\gamma RT}{M}} \]
where:
- \( V \) is the speed of sound,
- \( \gamma \) is the adiabatic index (constant for air),
- \( R \) is the universal gas constant (constant),
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of air (constant).

From this formula, we can see that the speed of sound is directly proportional to the square root of the absolute temperature \( T \).

### Step 2: Set up the relationship for the two temperatures
Let:
- \( V_1 \) be the speed of sound at \( T_1 = 27°C \),
- \( V_2 \) be the speed of sound at \( T_2 \) (the temperature we want to find).

Since we want \( V_2 \) to be double \( V_1 \):
\[ V_2 = 2V_1 \]

### Step 3: Write the proportionality equation
Using the relationship derived from the speed of sound:
\[ \frac{V_1}{V_2} = \sqrt{\frac{T_1}{T_2}} \]

Substituting \( V_2 = 2V_1 \):
\[ \frac{V_1}{2V_1} = \sqrt{\frac{T_1}{T_2}} \]
This simplifies to:
\[ \frac{1}{2} = \sqrt{\frac{T_1}{T_2}} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \left(\frac{1}{2}\right)^2 = \frac{T_1}{T_2} \]
\[ \frac{1}{4} = \frac{T_1}{T_2} \]

### Step 5: Rearrange to find \( T_2 \)
Rearranging gives:
\[ T_2 = 4T_1 \]

### Step 6: Substitute \( T_1 \) in Kelvin
Convert \( T_1 \) from Celsius to Kelvin:
\[ T_1 = 27°C + 273 = 300 K \]

Now substituting:
\[ T_2 = 4 \times 300 K = 1200 K \]

### Step 7: Convert \( T_2 \) back to Celsius
Convert \( T_2 \) back to Celsius:
\[ T_2 = 1200 K - 273 = 927°C \]

### Final Answer
The temperature at which the speed of sound in air becomes double its value at 27°C is:
\[ \boxed{927°C} \]