The r.m.s. velocity of the molecules of a gas at S.T.P. is `485.6 ms^(-1)`. Calculate the density of the gas.

Answer

To calculate the density of the gas given the RMS velocity, we can use the formula from kinetic theory of gases that relates pressure, density, and RMS velocity:

### Step-by-Step Solution:

1. **Understand the Relationship**: The formula relating pressure (P), density (ρ), and RMS velocity (V) is given by:
\[ P = \frac{1}{3} \rho V^2 \]

2. **Rearrange the Formula for Density**: We can rearrange this formula to solve for density (ρ):
\[ \rho = \frac{3P}{V^2} \]

3. **Identify the Values**:
- The RMS velocity \( V \) is given as \( 485.6 \, \text{m/s} \).
- At standard temperature and pressure (STP), the pressure \( P \) is \( 1 \, \text{atm} \), which can be converted to Pascals:
\[ P = 1.0132 \times 10^5 \, \text{Pa} \]

4. **Substitute the Values into the Density Formula**:
\[ \rho = \frac{3 \times (1.0132 \times 10^5)}{(485.6)^2} \]

5. **Calculate the Denominator**: First, calculate \( (485.6)^2 \):
\[ (485.6)^2 = 235,859.36 \, \text{m}^2/\text{s}^2 \]

6. **Calculate the Numerator**: Now calculate \( 3 \times (1.0132 \times 10^5) \):
\[ 3 \times (1.0132 \times 10^5) = 3.0396 \times 10^5 \, \text{Pa} \]

7. **Calculate Density**: Now substitute the values into the density formula:
\[ \rho = \frac{3.0396 \times 10^5}{235859.36} \]

8. **Final Calculation**:
\[ \rho \approx 1.29 \, \text{kg/m}^3 \]

### Final Answer:
The density of the gas is approximately \( 1.29 \, \text{kg/m}^3 \).