Find the temperatures at which methane and ethane will have the same rms speed as carbon dioxide at `400^(@)C`. Also calculate the mean velocity and most probable velocity of methane molecules at `400^(@)C`.

Explanation

To solve the problem, we need to find the temperatures at which methane (CH₄) and ethane (C₂H₆) will have the same root mean square (RMS) speed as carbon dioxide (CO₂) at 400°C. We will also calculate the mean velocity and most probable velocity of methane molecules at this temperature.

### Step 1: Convert the temperature of CO₂ from Celsius to Kelvin
The temperature of CO₂ is given as 400°C. To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
\[
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