If the density of air at NTP is `1.293kg//m^(3)` and `gamma=1.41`, then the velocity of sound in air at NTP is :

Options

A. 102.3 m/s
B. 252.3 m/s
C. 332.3 m/s
D. 432.3 m/s

Answer

To find the velocity of sound in air at Normal Temperature and Pressure (NTP), we can use the formula for the velocity of sound in a gas:

\[ V = \sqrt{\frac{\gamma P}{\rho}} \]

where:
- \( V \) is the velocity of sound,
- \( \gamma \) is the adiabatic index (ratio of specific heats),
- \( P \) is the pressure,
- \( \rho \) is the density of the gas.

### Step 1: Identify the given values
- Density of air, \( \rho = 1.293 \, \text{kg/m}^3 \)
- Adiabatic index, \( \gamma = 1.41 \)
- Pressure at NTP, \( P = 1 \, \text{atm} = 1.01 \times 10^5 \, \text{Pa} \)

### Step 2: Substitute the values into the formula
Now, we can substitute the values into the formula for the velocity of sound:

\[ V = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{1.41 \times (1.01 \times 10^5)}{1.293}} \]

### Step 3: Calculate the numerator
Calculate \( \gamma P \):

\[ \gamma P = 1.41 \times 1.01 \times 10^5 = 1.42441 \times 10^5 \, \text{Pa} \]

### Step 4: Calculate the entire expression under the square root
Now substitute this back into the equation:

\[ V = \sqrt{\frac{1.42441 \times 10^5}{1.293}} \]

### Step 5: Perform the division
Calculate the division:

\[ \frac{1.42441 \times 10^5}{1.293} \approx 110.17 \times 10^3 \approx 110170.2 \]

### Step 6: Take the square root
Now take the square root:

\[ V \approx \sqrt{110170.2} \approx 331.8 \, \text{m/s} \]

### Step 7: Round the answer
Rounding to one decimal place gives us:

\[ V \approx 332.3 \, \text{m/s} \]

### Conclusion
Thus, the velocity of sound in air at NTP is approximately **332.3 m/s**.

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