The ratio of speed of sound in monomatomic gas to that in water vapours at any temperature is. (when molecular weight of gas is `40 gm//mol` and for water vapours is `18 gm//mol`)

Options

A. `0.75`
B. `0.73`
C. `0.68`
D. None of these

Explanation

To find the ratio of the speed of sound in a monatomic gas to that in water vapour, we can use the formula for the speed of sound in a gas, which is given by:

\[ v = \sqrt{\frac{\gamma P}{\rho}} \]

Where:
- \( v \) is the speed of sound,
- \( \gamma \) is the adiabatic index (ratio of specific heats),
- \( P \) is the pressure,
- \( \rho \) is the density of the gas.

### Step 1: Identify the values of \( \gamma \) for both gases

For a monatomic gas, \( \gamma \) is typically \( \frac{5}{3} \).
For water vapour, which is a diatomic gas, \( \gamma \) is approximately \( \frac{4}{3} \).

### Step 2: Write the formula for the speed of sound in both gases

Let \( v_{mono} \) be the speed of sound in the monatomic gas and \( v_{water} \) be the speed of sound in water vapour.

\[ v_{mono} = \sqrt{\frac{\gamma_{mono} P_{mono}}{\rho_{mono}}} \]
\[ v_{water} = \sqrt{\frac{\gamma_{water} P_{water}}{\rho_{water}}} \]

### Step 3: Find the ratio of the speeds of sound

We can find the ratio of the speeds of sound in the monatomic gas and water vapour:

\[ \frac{v_{mono}}{v_{water}} = \frac{\sqrt{\frac{\gamma_{mono} P_{mono}}{\rho_{mono}}}}{\sqrt{\frac{\gamma_{water} P_{water}}{\rho_{water}}}} = \sqrt{\frac{\gamma_{mono} P_{mono} \rho_{water}}{\gamma_{water} P_{water} \rho_{mono}}} \]

### Step 4: Use the ideal gas law to express pressure and density

Using the ideal gas law \( P = \frac{nRT}{V} \) and \( \rho = \frac{m}{V} \), we can express the densities and pressures in terms of molecular weights.

For a gas, the density can be expressed as:

\[ \rho = \frac{M}{RT} \]

Where \( M \) is the molar mass.

### Step 5: Substitute the values of molecular weights

Given:
- Molecular weight of monatomic gas \( M_{mono} = 40 \, \text{g/mol} \)
- Molecular weight of water vapour \( M_{water} = 18 \, \text{g/mol} \)

Substituting these values into the ratio:

\[ \frac{v_{mono}}{v_{water}} = \sqrt{\frac{\frac{5}{3} \cdot P_{mono} \cdot \frac{18}{RT}}{\frac{4}{3} \cdot P_{water} \cdot \frac{40}{RT}}} \]

### Step 6: Simplify the expression

Assuming pressures are equal for both gases at the same temperature, we can simplify:

\[ \frac{v_{mono}}{v_{water}} = \sqrt{\frac{5 \cdot 18}{4 \cdot 40}} = \sqrt{\frac{90}{160}} = \sqrt{\frac{9}{16}} = \frac{3}{4} \]

### Final Result

The ratio of the speed of sound in monatomic gas to that in water vapour is:

\[ \frac{v_{mono}}{v_{water}} = \frac{3}{4} \]