At the room temperature the velocity of sound in `O_(2)` gas is v. Then in mixture of `H_(2)` and `O_(2)` gas the speed of sound at same temperature :-

Options

A. will be less than v
B. wil be more than v
C. will be equal to v
D. nothing can be said

Explanation

To determine the speed of sound in a mixture of hydrogen (H₂) and oxygen (O₂) gas at room temperature, we can follow these steps:

### Step 1: Understand the formula for the speed of sound in a gas
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: Identify the parameters for O₂
At room temperature, the speed of sound in oxygen (O₂) is given as \( v \). The molar mass of O₂ is approximately 32 g/mol (16 g/mol for each oxygen atom).

### Step 3: Analyze the mixture of H₂ and O₂
When we create a mixture of hydrogen (H₂) and oxygen (O₂), we need to calculate the effective molar mass of the mixture. The molar mass of H₂ is approximately 2 g/mol (1 g/mol for each hydrogen atom).

### Step 4: Calculate the average molar mass of the mixture
Assuming we have equal moles of H₂ and O₂, the average molar mass \( M_{mix} \) can be calculated as:
\[ M_{mix} = \frac{M_{H_2} + M_{O_2}}{2} = \frac{2 + 32}{2} = \frac{34}{2} = 17 \text{ g/mol} \]

### Step 5: Compare the molar masses
Now we compare the molar mass of the mixture (17 g/mol) with the molar mass of O₂ (32 g/mol). Since 17 g/mol < 32 g/mol, the molar mass of the mixture is less than that of pure O₂.

### Step 6: Determine the effect on the speed of sound
Since the speed of sound is inversely proportional to the square root of the molar mass (as seen in the formula), a decrease in molar mass leads to an increase in the speed of sound. Therefore, we conclude that the speed of sound in the mixture of H₂ and O₂ will be greater than that in pure O₂.

### Conclusion
The speed of sound in the mixture of H₂ and O₂ at the same temperature will be greater than \( v \), the speed of sound in pure O₂.

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