The ratio of R.M.S. velocity of air molecules at S. T. P. and velocity of sound in air at S. T. Pis about (y = 1.41 for air)

Options

A. 1.53
B. 1.46
C. 1
D. 1.48

Answer

To find the ratio of the R.M.S. velocity of air molecules at Standard Temperature and Pressure (S.T.P.) to the velocity of sound in air at S.T.P., we can follow these steps:

### Step 1: Write the formula for R.M.S. velocity
The R.M.S. velocity (\(v_{rms}\)) of gas molecules is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \(R\) is the universal gas constant,
- \(T\) is the absolute temperature,
- \(M\) is the molar mass of the gas.

### Step 2: Write the formula for the velocity of sound in a gas
The velocity of sound (\(v_{sound}\)) in a gas is given by the formula:
\[ v_{sound} = \sqrt{\frac{\gamma RT}{M}} \]
where:
- \(\gamma\) (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 3: Find the ratio of R.M.S. velocity to the velocity of sound
To find the ratio of \(v_{rms}\) to \(v_{sound}\), we can write:
\[ \text{Ratio} = \frac{v_{rms}}{v_{sound}} = \frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{\gamma RT}{M}}} \]

### Step 4: Simplify the ratio
Notice that \(R\), \(T\), and \(M\) are common in both expressions:
\[ \text{Ratio} = \frac{\sqrt{3RT}}{\sqrt{\gamma RT}} = \sqrt{\frac{3}{\gamma}} \]

### Step 5: Substitute the value of \(\gamma\)
Given that \(\gamma = 1.41\) for air, we can substitute this value:
\[ \text{Ratio} = \sqrt{\frac{3}{1.41}} \]

### Step 6: Calculate the numerical value
Calculating this gives:
\[ \text{Ratio} \approx \sqrt{2.126} \approx 1.46 \]

### Conclusion
Thus, the ratio of the R.M.S. velocity of air molecules at S.T.P. to the velocity of sound in air at S.T.P. is approximately \(1.46\).

---