Oxygen is `16` times heavier than hydrogen. At `NTP` equal volumn of hydrogen and oxygen are mixed. The ratio of speed of sound in the mixture to that in hydrogen is

Options

A. `sqrt(8)`
B. `sqrt((1)/(8))`
C. `sqrt((2)/(17))`
D. `sqrt((32)/(17))`

Explanation

To solve the problem of finding the ratio of the speed of sound in a mixture of hydrogen and oxygen to that in hydrogen, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that oxygen is 16 times heavier than hydrogen. At Normal Temperature and Pressure (NTP), equal volumes of hydrogen and oxygen are mixed. We need to find the ratio of the speed of sound in the mixture to that in hydrogen.

2. **Formula for Speed of Sound**:
The speed of sound \( v \) in a gas is given by the formula:
\[ v = \sqrt{\frac{\gamma RT}{M}} \]
where:
- \( \gamma \) 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.

3. **Proportionality of Speed of Sound**:
From the formula, we can see that the speed of sound is inversely proportional to the square root of the molar mass:
\[ v \propto \frac{1}{\sqrt{M}} \]

4. **Calculating Molar Mass of the Mixture**:
Let’s denote the molar mass of hydrogen \( M_H = 2 \) g/mol and the molar mass of oxygen \( M_O = 32 \) g/mol. Since equal volumes of the gases are mixed at NTP, the number of moles of hydrogen and oxygen will be the same.

Let the number of moles of each gas be \( n \). The molar mass of the mixture \( M_{mixture} \) can be calculated as:
\[ M_{mixture} = \frac{n \cdot M_O + n \cdot M_H}{n + n} = \frac{n \cdot 32 + n \cdot 2}{2n} = \frac{32 + 2}{2} = \frac{34}{2} = 17 \text{ g/mol} \]

5. **Finding the Ratio of Speeds**:
Now, we can find the ratio of the speed of sound in the mixture to that in hydrogen:
\[ \frac{v_{mixture}}{v_H} = \frac{M_H}{M_{mixture}} = \frac{2}{17} \]

6. **Final Result**:
Thus, the ratio of the speed of sound in the mixture to that in hydrogen is:
\[ \frac{v_{mixture}}{v_H} = \frac{2}{17} \]