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. `330sqrt(2) m//s`
D. `330//sqrt(2) m//s`

Answer

To solve the problem of finding the new velocity of sound 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 an ideal gas is given by the formula:
\[ V = \sqrt{\frac{\gamma R T}{M}} \]
where:
- \( V \) = velocity of sound
- \( \gamma \) = adiabatic index (constant for a given gas)
- \( R \) = universal gas constant (constant)
- \( T \) = absolute temperature
- \( M \) = molar mass of the gas (constant)

### Step 2: Analyze the changes in temperature and pressure
In the problem, we are told that:
- The temperature is doubled: \( T' = 2T \)
- The pressure is halved: \( P' = \frac{P}{2} \)

However, the velocity of sound is primarily dependent on the temperature. The pressure does not directly affect the velocity of sound in an ideal gas under these conditions.

### Step 3: Substitute the new temperature into the velocity formula
Since the pressure does not affect the velocity, we can express the new velocity (\( V' \)) as:
\[ V' = \sqrt{\frac{\gamma R (2T)}{M}} \]

### Step 4: Relate the new velocity to the original velocity
We can express \( V' \) in terms of the original velocity (\( V \)):
\[ V' = \sqrt{2} \cdot \sqrt{\frac{\gamma R T}{M}} \]
This means:
\[ V' = \sqrt{2} \cdot V \]

### Step 5: Calculate the new velocity
Given that the original velocity \( V \) at NTP is 330 m/s:
\[ V' = \sqrt{2} \cdot 330 \]

### Step 6: Compute the numerical value
Calculating \( \sqrt{2} \) (approximately 1.414):
\[ V' \approx 1.414 \cdot 330 \]
\[ V' \approx 466.62 \, \text{m/s} \]

Thus, the new velocity of sound when the temperature is doubled and pressure is halved is approximately:
\[ V' \approx 330 \sqrt{2} \, \text{m/s} \]

### Final Answer
The answer is \( 330 \sqrt{2} \, \text{m/s} \).

---