The density of `CO_(2)` gas at `0^(@)C` and at a pressure of `1.0 xx 10^(5) Nm^(-2)` is 1.98 `kgm^(-3)`. Calculate the root mean square velocity of its molecules at `0^(@)C` and `30^(@)C`. Assume the pressure to remain constant.

Explanation

To calculate the root mean square (RMS) velocity of CO₂ gas at two different temperatures (0°C and 30°C), we can follow these steps:

### Step 1: Understand the formula for RMS velocity
The RMS velocity \( v_{rms} \) of a gas can be calculated using the formula:
\[ v_{rms} = \sqrt{\frac{3p}{\rho}} \]
where:
- \( p \) = pressure of the gas (in Pascals)
- \( \rho \) = density of the gas (in kg/m³)

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

Substituting the 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 back:
\[ v_{rms} = \sqrt{\frac{3.0 \times 10^5}{1.98}} \approx \sqrt{151515.15} \approx 389.2 \, \text{m/s} \]

### Step 3: Calculate \( v_{rms} \) at 30°C
The RMS velocity is also related to temperature. The relationship is:
\[ \frac{v_{rms}(T_1)}{v_{rms}(T_2)} = \sqrt{\frac{T_1}{T_2}} \]
where \( T \) is the absolute temperature in Kelvin.

Convert temperatures from Celsius to Kelvin:
- \( T_1 = 0°C = 273 \, K \)
- \( T_2 = 30°C = 303 \, K \)

Using the relationship:
\[ \frac{v_{rms}(0°C)}{v_{rms}(30°C)} = \sqrt{\frac{273}{303}} \]

Let \( v_{rms}(30°C) \) be \( x \):
\[ \frac{389.2}{x} = \sqrt{\frac{273}{303}} \]

Cross-multiplying gives:
\[ 389.2 = x \cdot \sqrt{\frac{273}{303}} \]

Now, solve for \( x \):
\[ x = \frac{389.2}{\sqrt{\frac{273}{303}}} \]

Calculating \( \sqrt{\frac{273}{303}} \):
\[ \sqrt{\frac{273}{303}} \approx 0.956 \]

Now substituting back:
\[ x \approx \frac{389.2}{0.956} \approx 407.5 \, \text{m/s} \]

Thus, rounding gives:
\[ v_{rms}(30°C) \approx 410 \, \text{m/s} \]

### Final Results:
- \( v_{rms}(0°C) \approx 389.2 \, \text{m/s} \)
- \( v_{rms}(30°C) \approx 410 \, \text{m/s} \)

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