The velocity of sound in air at NTP is 330 m/s. What will be its value when temperature is doubled and pressure is halved ?

Options

A. 330 m/s
B. 165 m/s
C. `330 sqrt2` m/s
D. `330//sqrt2` m/s

Answer

To solve the problem of finding the velocity of sound in air when the temperature is doubled and the pressure is halved, we can follow these steps:

### Step 1: Understand the relationship between velocity of sound, temperature, and pressure
The velocity of sound in a gas is given by the formula:
\[ v = \sqrt{\frac{\gamma \cdot R \cdot T}{M}} \]
where:
- \( v \) = velocity of sound
- \( \gamma \) = adiabatic index (ratio of specific heats)
- \( R \) = universal gas constant
- \( T \) = absolute temperature
- \( M \) = molar mass of the gas

For air, the velocity of sound is primarily dependent on the temperature and is independent of pressure.

### Step 2: Identify the initial conditions
At Normal Temperature and Pressure (NTP), the velocity of sound in air is given as:
\[ v_0 = 330 \, \text{m/s} \]

### Step 3: Determine the new temperature and pressure conditions
- The temperature is doubled: \( T' = 2T \)
- The pressure is halved: \( P' = \frac{P}{2} \)

### Step 4: Calculate the new velocity of sound
Since the velocity of sound is dependent on the square root of the temperature, we can express the new velocity \( v' \) as:
\[ v' = v_0 \cdot \sqrt{\frac{T'}{T}} \]
Substituting \( T' = 2T \):
\[ v' = v_0 \cdot \sqrt{\frac{2T}{T}} = v_0 \cdot \sqrt{2} \]

### Step 5: Substitute the initial velocity
Now substituting the initial velocity \( v_0 = 330 \, \text{m/s} \):
\[ v' = 330 \cdot \sqrt{2} \]

### Step 6: Calculate the final value
Using the approximate value of \( \sqrt{2} \approx 1.414 \):
\[ v' \approx 330 \cdot 1.414 \approx 466.62 \, \text{m/s} \]

### Conclusion
The new velocity of sound in air when the temperature is doubled and the pressure is halved is approximately:
\[ v' \approx 466.62 \, \text{m/s} \]