The density of carbon dioxide gas at `0^(@)C` and at pressure `1.0 xx 10^(5) Nm^(-2) is 1.98 kg m^(-3)`. Find the rms velocity of its molecules at `0^(@)C` and also at `30^(@)C`, assuming pressure to be constant.

Explanation

To find the root mean square (rms) velocity of carbon dioxide gas at two different temperatures (0°C and 30°C), we can follow these steps:

### Step 1: Understand the formula for rms velocity
The root mean square velocity (Vrms) of gas molecules can be calculated using the formula:
\[ V_{rms} = \sqrt{\frac{3P}{\rho}} \]
where:
- \( P \) is the pressure of the gas,
- \( \rho \) is the density of the gas.

### Step 2: Calculate Vrms at 0°C
Given:
- Pressure \( P = 1.0 \times 10^5 \, \text{N/m}^2 \)
- Density \( \rho = 1.98 \, \text{kg/m}^3 \)

Substituting these values into the formula:
\[ V_{rms} = \sqrt{\frac{3 \times (1.0 \times 10^5)}{1.98}} \]

Calculating the numerator:
\[ 3 \times (1.0 \times 10^5) = 3.0 \times 10^5 \]

Now, substituting into the formula:
\[ V_{rms} = \sqrt{\frac{3.0 \times 10^5}{1.98}} \]

Calculating the division:
\[ \frac{3.0 \times 10^5}{1.98} \approx 151515.15 \]

Now, taking the square root:
\[ V_{rms} \approx \sqrt{151515.15} \approx 389.2 \, \text{m/s} \]

### Step 3: Calculate Vrms at 30°C
To find the rms velocity at 30°C, we can use the relationship that the rms velocity is proportional to the square root of the temperature:
\[ \frac{V_{rms}(T_2)}{V_{rms}(T_1)} = \sqrt{\frac{T_2}{T_1}} \]

Where:
- \( T_1 = 0°C = 273 \, K \)
- \( T_2 = 30°C = 303 \, K \)

Substituting the known values:
\[ \frac{V_{rms}(30°C)}{389.2} = \sqrt{\frac{303}{273}} \]

Calculating the right side:
\[ \sqrt{\frac{303}{273}} \approx \sqrt{1.109} \approx 1.05 \]

Now, substituting back to find \( V_{rms}(30°C) \):
\[ V_{rms}(30°C) = 389.2 \times 1.05 \approx 409.66 \, \text{m/s} \approx 410 \, \text{m/s} \]

### Final Answer
The root mean square velocity of carbon dioxide gas at 0°C is approximately **389.2 m/s**, and at 30°C it is approximately **410 m/s**.

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