To calculate the average speed of a gaseous Chlorine molecule ($\text{Cl}_2$) at a temperature of $-155^\circ \text{C}$, we can use the formula derived from the kinetic theory of gases:

$v_{\text{avg}} = \sqrt{\frac{8RT}{\pi M}}$

Where:

• $v_{\text{avg}}$ is the average speed of the gas molecules.
• $R$ is the universal gas constant.
• $T$ is the absolute temperature in Kelvin.
• $M$ is the molar mass of the gas.

Step 1: Convert the temperature to Kelvin

The temperature in Celsius is given as $-155^\circ \text{C}$. To convert to Kelvin:

$T(K) = T(°C) + 273.15 = -155 + 273.15 = 118.15 \, \text{K}$

Step 2: Determine the molar mass of chlorine gas

The molar mass of $\text{Cl}_2$ is approximately:

$M_{\text{Cl}_2} = 2 \times 35.453 \, \text{g/mol} = 70.906 \, \text{g/mol}$

Step 3: Apply the formula

Now, we can substitute the known values into the formula. We'll convert the molar mass into kilograms per mole for consistency with SI units:

$M_{\text{Cl}_2} = 70.906 \, \text{g/mol} = 70.906 \times 10^{-3} \, \text{kg/mol}$

The gas constant is given as $R = 8.31 \times 10^3 \, \text{g} \cdot \text{m}^2 \cdot \text{s}^{-2} \cdot \text{mol}^{-1} \cdot \text{K}^{-1}$, which is consistent with our mass units.

Let me calculate the final value.The average speed of a chlorine gas molecule ($\text{Cl}_2$) at $-155^\circ \text{C}$ is approximately $5938 \, \text{m/s}$.

Would you like further details or have any questions?

Here are 5 related questions for exploration:

1. How does the average speed of gas molecules change with temperature?
2. How would the speed differ for a lighter gas, such as helium, at the same temperature?
3. What assumptions are made in the kinetic theory of gases for calculating speed?
4. How does the molar mass of a gas affect its average molecular speed?
5. What is the difference between root mean square speed and average speed?

Tip: Temperature and molar mass are the main factors affecting molecular speed; lighter molecules move faster at a given temperature.