What is the approximate thermal energy in kJ/mol of gas molecules at -20 °C?

To estimate the thermal energy (\( E \)) of gas molecules at a given temperature, we use the equation:

\[
E = \frac{3}{2} k_B T
\]

where:
- \( k_B \) is the Boltzmann constant, \( 1.38 \times 10^{-23} \, \text{J/K} \),
- \( T \) is the temperature in Kelvin.

Since the problem asks for thermal energy in \( \text{kJ/mol} \), we can adjust our calculation to include Avogadro's number \( N_A = 6.022 \times 10^{23} \, \text{mol}^{-1} \) to scale it to per mole:

\[
E_{\text{mol}} = \frac{3}{2} k_B T \times N_A
\]

### Step 1: Convert Temperature to Kelvin
Temperature in Celsius is \(-20^\circ \text{C}\):
\[
T = -20 + 273.15 = 253.15 \, \text{K}
\]

### Step 2: Substitute Values
\[
E_{\text{mol}} = \frac{3}{2} \times (1.38 \times 10^{-23}) \times 253.15 \times (6.022 \times 10^{23})
\]

### Step 3: Simplify Calculation
Let's go through the arithmetic:

1. \( k_B T = 1.38 \times 10^{-23} \times 253.15 = 3.49 \times 10^{-21} \, \text{J} \)
2. Multiply by \( \frac{3}{2} \): \( \frac{3}{2} \times 3.49 \times 10^{-21} = 5.24 \times 10^{-21} \, \text{J} \)
3. Convert to per mole by multiplying by \( N_A \): \( 5.24 \times 10^{-21} \times 6.022 \times 10^{23} = 3.16 \, \text{kJ/mol} \)

### Answer
The approximate thermal energy of gas molecules at \(-20^\circ \text{C}\) is **3.16 kJ/mol**.

Would you like further details or have any questions on this?

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### Related Questions
1. How does thermal energy change with temperature for an ideal gas?
2. What is the thermal energy of gas molecules at 25 °C in kJ/mol?
3. How does the factor \(\frac{3}{2}k_B T\) arise in kinetic theory?
4. What assumptions are made in calculating thermal energy of an ideal gas?
5. How would thermal energy per mole differ for a diatomic gas?

#### Tip
For quick thermal energy estimates, remember that every 10 K increase roughly adds 0.12 kJ/mol for ideal gases.

Discover how to calculate the thermal energy of gas molecules at -20 °C. Using principles from the kinetic theory and equipartition theorem, this step-by-step solution explains the formula E = (3/2) k_B T and scales the result to kJ/mol using Avogadro's number. Ideal for advanced high school or undergraduate physics students.

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