The ratio of velocity of sound in hydrogen and oxygen at STP is

Options

A. (a)`16:1`
B. (b)`8:1`
C. (c)`4:1`
D. (d)`2:1`

Answer

To find the ratio of the velocity of sound in hydrogen (H₂) and oxygen (O₂) at standard temperature and pressure (STP), we can use the formula for the velocity of sound in a gas, which is given by:

\[ v = \sqrt{\frac{\gamma RT}{M}} \]

where:
- \( v \) is the velocity of sound,
- \( \gamma \) 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.

### Step 1: Write the equations for the velocities of sound in hydrogen and oxygen.

For hydrogen (H₂):
\[ v_{H_2} = \sqrt{\frac{\gamma RT}{M_{H_2}}} \]

For oxygen (O₂):
\[ v_{O_2} = \sqrt{\frac{\gamma RT}{M_{O_2}}} \]

### Step 2: Set up the ratio of the velocities.

To find the ratio of the velocities of sound in hydrogen to that in oxygen, we can write:
\[ \frac{v_{H_2}}{v_{O_2}} = \frac{\sqrt{\frac{\gamma RT}{M_{H_2}}}}{\sqrt{\frac{\gamma RT}{M_{O_2}}}} \]

### Step 3: Simplify the ratio.

Since \(\gamma\), \(R\), and \(T\) are the same for both gases, they will cancel out:
\[ \frac{v_{H_2}}{v_{O_2}} = \sqrt{\frac{M_{O_2}}{M_{H_2}}} \]

### Step 4: Substitute the molar masses.

The molar mass of hydrogen (H₂) is approximately 2 g/mol, and the molar mass of oxygen (O₂) is approximately 32 g/mol. Thus:
\[ \frac{v_{H_2}}{v_{O_2}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4 \]

### Step 5: Write the final ratio.

Therefore, the ratio of the velocity of sound in hydrogen to that in oxygen is:
\[ \frac{v_{H_2}}{v_{O_2}} = 4:1 \]

### Conclusion

The ratio of the velocity of sound in hydrogen and oxygen at STP is \(4:1\).

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