Calculate the velocity of sound in air at NTP. The density of air at NTP is `1.29g//L`. Assume air to be diatomic with `gamma=1.4`. Also calculate the velocity of sound in air at `27^@C`.

Options

A. `347.6m//s`
B. `695.2m//s`
C. `331.6m//s`
D. none of these

Explanation

To calculate the velocity of sound in air at Normal Temperature and Pressure (NTP) and at a temperature of 27°C, we will follow these steps:

### Step 1: Understand the Given Data
- Density of air at NTP, \( \rho = 1.29 \, \text{g/L} = 1.29 \, \text{kg/m}^3 \) (conversion from grams per liter to kilograms per cubic meter).
- Pressure at NTP, \( P = 1.01 \times 10^5 \, \text{Pa} \).
- The value of \( \gamma \) (adiabatic constant for diatomic gases) is given as \( \gamma = 1.4 \).

### Step 2: Use the Formula for Velocity of Sound
The formula for the velocity of sound \( v \) in a medium is given by:
\[ v = \sqrt{\frac{\gamma P}{\rho}} \]

### Step 3: Substitute the Values
Substituting the known values into the formula:
\[ v = \sqrt{\frac{1.4 \times 1.01 \times 10^5}{1.29}} \]

### Step 4: Calculate the Velocity
Now we will calculate the value inside the square root:
1. Calculate \( 1.4 \times 1.01 \times 10^5 \):
\[ 1.4 \times 1.01 = 1.414 \quad \text{and} \quad 1.414 \times 10^5 = 141400 \]
2. Now divide by \( 1.29 \):
\[ \frac{141400}{1.29} \approx 109500 \]
3. Finally, take the square root:
\[ v \approx \sqrt{109500} \approx 331.6 \, \text{m/s} \]

### Step 5: Calculate the Velocity of Sound at 27°C
To find the velocity of sound at 27°C, we will use the relationship between the velocities at two different temperatures:
\[ \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} \]
Where:
- \( T_1 = 273 \, \text{K} \) (temperature at NTP)
- \( T_2 = 273 + 27 = 300 \, \text{K} \)

### Step 6: Substitute the Values
Rearranging the formula gives:
\[ v_2 = v_1 \times \sqrt{\frac{T_2}{T_1}} \]
Substituting the values:
\[ v_2 = 331.6 \times \sqrt{\frac{300}{273}} \]

### Step 7: Calculate the New Velocity
1. Calculate the ratio:
\[ \frac{300}{273} \approx 1.0985 \]
2. Take the square root:
\[ \sqrt{1.0985} \approx 1.049 \]
3. Multiply by \( v_1 \):
\[ v_2 \approx 331.6 \times 1.049 \approx 347.6 \, \text{m/s} \]

### Final Answers
- The velocity of sound in air at NTP is approximately **331.6 m/s**.
- The velocity of sound in air at 27°C is approximately **347.6 m/s**.