To determine the average speed ($\overline{v}$) of nitrogen gas ($N_2$) at a given temperature, we can use the following formula derived from the kinetic theory of gases:

$\overline{v} = \sqrt{\frac{8RT}{\pi M}}$

Where:

• $R$ is the universal gas constant ($8.314$ J/mol·K),
• $T$ is the temperature in Kelvin ($445$ K),
• $M$ is the molar mass of $N_2$ (which is $28.02 \, \text{g/mol} = 0.02802 \, \text{kg/mol}$).

Let's calculate:

$\overline{v} = \sqrt{\frac{8 \times 8.314 \times 445}{\pi \times 0.02802}}$

Now, I'll compute the value.The average speed of $N_2$ molecules at 445 K is approximately $579.9$ m/s.

Would you like more details or have any questions about the calculation?

Here are 5 related questions you might find interesting:

1. How does the average speed of $N_2$ change if the temperature increases to 500 K?
2. What would be the average speed of oxygen gas ($O_2$) at 445 K?
3. How does the molar mass of a gas affect its average speed at a given temperature?
4. What is the root mean square speed of $N_2$ at 445 K?
5. How would you determine the average kinetic energy of an $N_2$ molecule at 445 K?

Tip: Remember that lighter gas molecules move faster at the same temperature compared to heavier ones.