The density of an ideal gas is 0.03 g/`cm^3`, its pressure is `10^6` dy/`cm^2`. What is its R.M.S velocity (in cm/s)?

Options

A. `10^8`
B. `3 xx 20^4`
C. `10^6`
D. `10^4`

Answer

To find the root mean square (R.M.S) velocity of an ideal gas given its density and pressure, we can use the following steps:

### Step-by-Step Solution:

1. **Understand the Given Data**:
- Density (d) = 0.03 g/cm³
- Pressure (P) = \(10^6\) dy/cm²

2. **Use the Formula for R.M.S Velocity**:
The R.M.S velocity (v_rms) can be calculated using the formula:
\[ v_{rms} = \sqrt{\frac{3P}{d}} \]
where P is the pressure and d is the density.

3. **Substitute the Values**:
Substitute the given values of pressure and density into the formula:
\[ v_{rms} = \sqrt{\frac{3 \times 10^6 \, \text{dy/cm}^2}{0.03 \, \text{g/cm}^3}} \]

4. **Calculate the Expression Inside the Square Root**:
First, calculate \(3 \times 10^6\):
\[ 3 \times 10^6 = 3000000 \]
Now divide this by the density:
\[ \frac{3000000}{0.03} = 100000000 \]

5. **Take the Square Root**:
Now, take the square root of \(100000000\):
\[ v_{rms} = \sqrt{100000000} = 10000 \, \text{cm/s} \]

### Final Answer:
The R.M.S velocity of the gas is \(10^4\) cm/s.

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