Velocity of hydrogen at NTP is V. The velocity of sound in a mixture of hydrogen and oxygen in the ratio of `4:1` at NTP is

Options

A. `(1)/(5)V`
B. `(1)/(4)V`
C. `(1)/(3)V`
D. `(1)/(2)V`

Explanation

To find the velocity of sound in a mixture of hydrogen and oxygen in the ratio of 4:1 at NTP, we can follow these steps:

### Step 1: Understand the Mixture Composition
Given the ratio of hydrogen to oxygen is 4:1, we can assume:
- Volume of Hydrogen (H₂) = 4V
- Volume of Oxygen (O₂) = V

### Step 2: Calculate Molar Masses
- Molar mass of Hydrogen (H₂) = 2 g/mol (1 g/mol per H atom)
- Molar mass of Oxygen (O₂) = 32 g/mol (16 g/mol per O atom, and O₂ has 2 atoms)

### Step 3: Calculate the Densities
Using the formula for density (ρ):
- Density of Hydrogen (ρ₁) = Molar mass of H₂ / Volume = 2 g/mol / V
- Density of Oxygen (ρ₂) = Molar mass of O₂ / Volume = 32 g/mol / V

### Step 4: Calculate the Density of the Mixture
The total mass of the mixture can be calculated as:
- Mass of Hydrogen = 4V * ρ₁ = 4V * (2 g/mol / V) = 8 g
- Mass of Oxygen = V * ρ₂ = V * (32 g/mol / V) = 32 g
- Total mass of the mixture = 8 g + 32 g = 40 g

The total volume of the mixture is:
- Total Volume = 4V + V = 5V

Now, the density of the mixture (ρ_mixture) is:
\[ \rho_{mixture} = \frac{\text{Total mass}}{\text{Total volume}} = \frac{40 g}{5V} = 8 g/V \]

### Step 5: Use the Formula for Velocity of Sound
The velocity of sound (v) in a medium is given by:
\[ v = \sqrt{\frac{\gamma P}{\rho}} \]
Where:
- γ (gamma) is the adiabatic index (ratio of specific heats), which is constant for both hydrogen and oxygen.
- P is the pressure, which is also constant at NTP.

### Step 6: Relate the Velocities
The velocity of sound in hydrogen (v₁) is given as V. Therefore, we can write:
\[ \frac{v_{mixture}}{v_{H_2}} = \sqrt{\frac{\rho_{H_2}}{\rho_{mixture}}} \]
Substituting the densities:
\[ \frac{v_{mixture}}{V} = \sqrt{\frac{2 g/V}{8 g/V}} \]
This simplifies to:
\[ \frac{v_{mixture}}{V} = \sqrt{\frac{2}{8}} = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

### Step 7: Calculate the Velocity of Sound in the Mixture
Thus, the velocity of sound in the mixture is:
\[ v_{mixture} = V \cdot \frac{1}{2} = \frac{V}{2} \]

### Final Answer
The velocity of sound in the mixture of hydrogen and oxygen at NTP is \( \frac{V}{2} \).

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