At what temperature will the speed of sound in hydrogen be the same as in oxygen at `100^(@)C` ? Molar Masses of oxygen and hydrogen are in the ratio `16: 1` .

Explanation

To find the temperature at which the speed of sound in hydrogen equals the speed of sound in oxygen at 100°C, we can follow these steps:

### Step 1: Understand the formula for the speed of sound in gases
The speed of sound \( v \) in a gas is given by the formula:
\[ v = \sqrt{\frac{\gamma R T}{M}} \]
where:
- \( \gamma \) is the specific heat ratio (Cp/Cv),
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas.

### Step 2: Set up the equation for both gases
Since we want the speed of sound in hydrogen (\( v_{H_2} \)) to equal the speed of sound in oxygen (\( v_{O_2} \)), we can write:
\[ \sqrt{\frac{\gamma_{O_2} R T_{O_2}}{M_{O_2}}} = \sqrt{\frac{\gamma_{H_2} R T_{H_2}}{M_{H_2}}} \]

### Step 3: Cancel out common terms
Since both hydrogen and oxygen are diatomic gases, their specific heat ratios (\( \gamma \)) are equal. The universal gas constant \( R \) is also a constant, so we can cancel these terms:
\[ \frac{T_{O_2}}{M_{O_2}} = \frac{T_{H_2}}{M_{H_2}} \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ T_{H_2} = T_{O_2} \cdot \frac{M_{H_2}}{M_{O_2}} \]

### Step 5: Substitute known values
We know that:
- The temperature of oxygen \( T_{O_2} = 100°C = 373 K \) (since we convert Celsius to Kelvin by adding 273).
- The molar masses of oxygen and hydrogen are in the ratio \( M_{O_2} : M_{H_2} = 16 : 1 \). Thus, \( \frac{M_{H_2}}{M_{O_2}} = \frac{1}{16} \).

Substituting these values into the equation:
\[ T_{H_2} = 373 K \cdot \frac{1}{16} \]

### Step 6: Calculate \( T_{H_2} \)
Calculating gives:
\[ T_{H_2} = \frac{373}{16} \approx 23.31 K \]

### Step 7: Convert to Celsius
To convert back to Celsius:
\[ T_{H_2} = 23.31 K - 273 \approx -249.69°C \]

### Final Answer
The temperature at which the speed of sound in hydrogen equals that in oxygen at 100°C is approximately:
\[ -249.69°C \]

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