The `R.M.S`. Speed of the molecules of a gas of density `kg m^(-3)` and pressure `1.2xx10^(5)Nm^(-2)` is:

Options

A. `120 ms^(-1)`
B. `300 ms^(-1)`
C. `600 ms^(-1)`
D. `900 ms^(-1)`

Explanation

To find the R.M.S (Root Mean Square) speed of the molecules of a gas given its density and pressure, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for R.M.S Speed**:
The R.M.S speed (v_rms) of gas molecules can be calculated using the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the gas constant,
- \( T \) is the temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg/mol.

2. **Use the Ideal Gas Law**:
The ideal gas law states:
\[ PV = nRT \]
where:
- \( P \) is the pressure,
- \( V \) is the volume,
- \( n \) is the number of moles.

3. **Relate Moles to Mass and Density**:
The number of moles \( n \) can be expressed as:
\[ n = \frac{m}{M} \]
where \( m \) is the mass of the gas. Substituting this into the ideal gas law gives:
\[ PV = \frac{m}{M}RT \]

4. **Rearranging the Equation**:
Rearranging the ideal gas law, we can express \( \frac{RT}{M} \) in terms of pressure and density:
\[ P = \frac{m}{V} \cdot \frac{RT}{M} \]
Here, \( \frac{m}{V} \) is the density \( \rho \) of the gas. Thus, we can write:
\[ P = \frac{\rho RT}{M} \]
Rearranging gives:
\[ \frac{RT}{M} = \frac{P}{\rho} \]

5. **Substituting Back into R.M.S Speed Formula**:
Now substituting \( \frac{RT}{M} \) into the R.M.S speed formula:
\[ v_{rms} = \sqrt{3 \cdot \frac{P}{\rho}} \]

6. **Plugging in the Values**:
Given:
- Pressure \( P = 1.2 \times 10^5 \, \text{N/m}^2 \)
- Density \( \rho = 4 \, \text{kg/m}^3 \)

Substitute these values into the equation:
\[ v_{rms} = \sqrt{3 \cdot \frac{1.2 \times 10^5}{4}} \]

7. **Calculating the Result**:
First, calculate \( \frac{1.2 \times 10^5}{4} \):
\[ \frac{1.2 \times 10^5}{4} = 3.0 \times 10^4 \]
Now multiply by 3:
\[ 3 \cdot 3.0 \times 10^4 = 9.0 \times 10^4 \]
Finally, take the square root:
\[ v_{rms} = \sqrt{9.0 \times 10^4} = 300 \, \text{m/s} \]

### Final Answer:
The R.M.S speed of the molecules of the gas is \( 300 \, \text{m/s} \).

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