Calculate the average speed of gaseous Chlorine (C*l_{2}) molecule at a very cold temperature of - 155 deg * C Use R=8.31*10^ 3 g.m^ 2 s^ -2 mol^ -1 K^ -1

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.

Calculate the average speed of gaseous Chlorine (Cl2) molecule at a very cold temperature of -155°C. Use R = 8.31 × 10^3 g.m^2 s^-2 mol^-1 K^-1.

Learn how to calculate the average speed of a gaseous Chlorine (Cl2) molecule at -155°C using the kinetic theory of gases. This problem involves molecular speed calculation, thermodynamics, and the use of the gas constant (R = 8.31 × 10^3 g.m^2 s^-2 mol^-1 K^-1). Suitable for advanced high school or college-level students.

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