Let speed of sound waves in hydrogen gas at room temperature is `v_(0)`. What will be the speed of sound waves in a room which contains an equimolar mixture of hydrogen and 'He' at same temperature :-

Options

A. `sqrt((5)/(7))v_(0)`
B. `sqrt((7)/(5))v_(0)`
C. `sqrt((2)/(5))v_(0)`
D. None

Answer

To find the speed of sound waves in an equimolar mixture of hydrogen and helium at room temperature, we can follow these steps:

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

### Step 2: Calculate speed of sound in hydrogen
For hydrogen (\( H_2 \)):
- It is a diatomic gas, thus \( \gamma_H = \frac{C_p}{C_v} = \frac{7}{5} \).
- The molar mass \( M_H \) of hydrogen is \( 2 \, \text{g/mol} \).

Using the formula:
\[ v_0 = \sqrt{\frac{\gamma_H RT}{M_H}} = \sqrt{\frac{\frac{7}{5}RT}{2}} \]

### Step 3: Calculate speed of sound in helium
For helium (\( He \)):
- It is a monoatomic gas, thus \( \gamma_{He} = \frac{C_p}{C_v} = \frac{5}{3} \).
- The molar mass \( M_{He} \) of helium is \( 4 \, \text{g/mol} \).

Using the formula:
\[ v_{He} = \sqrt{\frac{\gamma_{He} RT}{M_{He}}} = \sqrt{\frac{\frac{5}{3}RT}{4}} \]

### Step 4: Calculate the effective \( \gamma \) and \( M \) for the mixture
Since we have an equimolar mixture of hydrogen and helium, we can find the effective \( \gamma \) and \( M \) for the mixture.

1. **Effective \( C_p \) and \( C_v \)**:
- For hydrogen: \( C_{p,H} = \frac{7R}{2} \) and \( C_{v,H} = \frac{5R}{2} \)
- For helium: \( C_{p,He} = \frac{5R}{2} \) and \( C_{v,He} = \frac{3R}{2} \)

Using the formula for \( \gamma \) of the mixture:
\[ \gamma_{mix} = \frac{N_1C_{p,H} + N_2C_{p,He}}{N_1C_{v,H} + N_2C_{v,He}} \]
Since \( N_1 = N_2 = 1 \):
\[ \gamma_{mix} = \frac{\frac{7R}{2} + \frac{5R}{2}}{\frac{5R}{2} + \frac{3R}{2}} = \frac{12R/2}{8R/2} = \frac{12}{8} = \frac{3}{2} \]

2. **Effective molar mass \( M_{mix} \)**:
\[ M_{mix} = \frac{M_H + M_{He}}{2} = \frac{2 + 4}{2} = 3 \, \text{g/mol} \]

### Step 5: Calculate the speed of sound in the mixture
Using the effective values:
\[ v_{mix} = \sqrt{\frac{\gamma_{mix} RT}{M_{mix}}} = \sqrt{\frac{\frac{3}{2}RT}{3}} = \sqrt{\frac{RT}{2}} \]

### Step 6: Relate \( v_{mix} \) to \( v_0 \)
Now, we can relate \( v_{mix} \) to \( v_0 \):
\[ v_{mix} = \sqrt{\frac{RT}{2}} = \sqrt{\frac{7}{10} \cdot \frac{RT}{2}} = \sqrt{\frac{7}{10}} \cdot v_0 \]

### Final Result
Thus, the speed of sound in the equimolar mixture of hydrogen and helium is:
\[ v_{mix} = \sqrt{\frac{5}{7}} v_0 \]