At which temperature the speed of sound in hydrogen will be same as that of speed of sound in oxygen at `100^@C`

Options

A. `-148^(@)C`
B. `-212.5^(@)C`
C. `317.5^(@)C`
D. `-249.7^(@)C`

Answer

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

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

### Step 2: Set up the equation for equal speeds
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 set up the equation:
\[ \sqrt{\frac{\gamma_{H_2} R T_{H_2}}{M_{H_2}}} = \sqrt{\frac{\gamma_{O_2} R T_{O_2}}{M_{O_2}}} \]

### Step 3: Simplify the equation
Since \( \gamma \) and \( R \) are constants, we can simplify this to:
\[ \frac{T_{H_2}}{M_{H_2}} = \frac{T_{O_2}}{M_{O_2}} \]

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

### Step 5: Find the molar masses
The molar mass of hydrogen (\( H_2 \)) is approximately 2 g/mol, and the molar mass of oxygen (\( O_2 \)) is approximately 32 g/mol.

### Step 6: Convert the temperature of oxygen to Kelvin
The temperature of oxygen given in the problem is 100°C. To convert this to Kelvin:
\[ T_{O_2} = 100 + 273 = 373 \, K \]

### Step 7: Substitute the values into the equation
Now we can substitute the values into our equation:
\[ T_{H_2} = 373 \cdot \frac{2}{32} \]

### Step 8: Calculate \( T_{H_2} \)
Calculating this gives:
\[ T_{H_2} = 373 \cdot \frac{1}{16} = \frac{373}{16} \approx 23.2 \, K \]

### Step 9: Convert the temperature back to Celsius
To convert from Kelvin to Celsius:
\[ T_{H_2} = 23.2 - 273 \approx -249.8 \, °C \]

### Final Answer
The temperature at which the speed of sound in hydrogen will be the same as that in oxygen at 100°C is approximately **-249.8°C**.

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