As the temperature is raised from $$20^\circ$$C to $$40^\circ$$C, the average kinetic energy of neon atoms changes by a factor of which of the following?

According to the Kinetic Molecular Theory of Gases, the average kinetic energy ($$\text{KE}_{\text{avg}}$$) of an ideal gas molecule is directly proportional to its absolute temperature ($$T$$) in Kelvin:

$$\text{KE}_{\text{avg}} = \frac{3}{2}k_B T$$

Where $$k_B$$ is the Boltzmann constant. Because the identity of the gas (Neon) does not affect this relationship, the ratio of the changing kinetic energies is simply equal to the ratio of their absolute temperatures:

$$\frac{\text{KE}_2}{\text{KE}_1} = \frac{T_2}{T_1}$$

• Step 1: Convert temperatures from Celsius to Kelvin

  The absolute scale requires adding $$273$$ (or more precisely, $$273.15$$) to the Celsius value. Using standard textbook rounding:

  $$T_1 = 20^\circ\text{C} + 273 = 293 \text{ K}$$

  $$T_2 = 40^\circ\text{C} + 273 = 313 \text{ K}$$




• Step 2: Find the ratio factor

  Substitute the absolute temperatures into our proportionality ratio equation:

  $$\text{Factor} = \frac{T_2}{T_1} = \frac{313}{293}$$

Conclusion:

The average kinetic energy shifts by a direct absolute temperature factor scale of $$\frac{313}{293}$$.

Answer: Option C — $$\frac{313}{293}$$