Find the ratio of velocity of sound in Helium to that in nitrogen at same temperature. The molecular weights of helium and nitrogen are 4 and 28.

Answer

To find the ratio of the velocity of sound in helium to that in nitrogen at the same temperature, we can use the formula for the speed of sound in a gas, which is given by:

\[ V = \sqrt{\frac{\gamma R T}{M}} \]

where:
- \( V \) is the velocity of sound,
- \( \gamma \) is the adiabatic index (ratio of specific heats),
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

### Step 1: Write the formula for the speed of sound in helium and nitrogen

For helium:
\[ V_{He} = \sqrt{\frac{\gamma_{He} R T}{M_{He}}} \]

For nitrogen:
\[ V_{N2} = \sqrt{\frac{\gamma_{N2} R T}{M_{N2}}} \]

### Step 2: Find the ratio of the velocities

Now, we can find the ratio of the velocities:
\[ \frac{V_{He}}{V_{N2}} = \frac{\sqrt{\frac{\gamma_{He} R T}{M_{He}}}}{\sqrt{\frac{\gamma_{N2} R T}{M_{N2}}}} \]

### Step 3: Simplify the ratio

This can be simplified as:
\[ \frac{V_{He}}{V_{N2}} = \sqrt{\frac{\gamma_{He}}{M_{He}} \cdot \frac{M_{N2}}{\gamma_{N2}}} \]

### Step 4: Substitute the values

Given:
- Molecular weight of helium, \( M_{He} = 4 \)
- Molecular weight of nitrogen, \( M_{N2} = 28 \)
- \( \gamma_{He} = \frac{5}{3} \)
- \( \gamma_{N2} = \frac{7}{5} \)

Substituting these values into the equation:
\[ \frac{V_{He}}{V_{N2}} = \sqrt{\frac{\frac{5}{3}}{4} \cdot \frac{28}{\frac{7}{5}}} \]

### Step 5: Calculate the expression

Calculating the expression inside the square root:
\[ \frac{V_{He}}{V_{N2}} = \sqrt{\frac{5 \cdot 28 \cdot 5}{3 \cdot 4 \cdot 7}} \]

Calculating the numerator:
\[ 5 \cdot 28 \cdot 5 = 700 \]

Calculating the denominator:
\[ 3 \cdot 4 \cdot 7 = 84 \]

So we have:
\[ \frac{V_{He}}{V_{N2}} = \sqrt{\frac{700}{84}} \]

### Step 6: Simplify further

Now simplify \( \frac{700}{84} \):
\[ \frac{700}{84} = \frac{25}{3} \]

Thus:
\[ \frac{V_{He}}{V_{N2}} = \sqrt{\frac{25}{3}} = \frac{5}{\sqrt{3}} \]

### Step 7: Calculate the numerical value

Calculating \( \frac{5}{\sqrt{3}} \):
\[ \frac{5}{\sqrt{3}} \approx 2.89 \]

### Final Answer

Thus, the ratio of the velocity of sound in helium to that in nitrogen at the same temperature is approximately:
\[ \frac{V_{He}}{V_{N2}} \approx 2.89 \]