The equipartition theorem is valid only if kT is very much greater than the separation between energy levels of the mode of motion, Δε. Translational and rotational energy levels are sufficiently close together that this is true for most molecules at room temperature. However, the separation between vibrational energy levels is much greater, such that the equipartition theorem may be used to calculate the total contribution to the internal energy from all modes of motion only at high temperatures. The exact expression for the average contribution to the energy from vibration isgiven by Δε{1 - e-Δε/kT}. (a) For chlorine, Cl2, the separation between vibrational energy levels is 6.70 kJ mol-1. Estimate the temperature at which the equipartition theorem becomes valid. (b) An exponential function may be expanded as ex = 1 + x + (1)/(2)x2 + ... if x < 1. Show that in th is case, the exact expression reduces to the result obtained from the equipartition theorem.

The equipartition theorem is valid only if kT is very much greater than the separation between energy levels of the mode of motion, Δε. Translational and rotational energy levels are sufficiently close together that this is true for most molecules at room temperature. However, the separation between vibrational energy levels is much greater, such that the equipartition theorem may be used to calculate the total contribution to the internal energy from all modes of motion only at high temperatures. The exact expression for the average contribution to the energy from vibration isgiven by Δε{1 - e-Δε/kT}. (a) For chlorine, Cl2, the separation between vibrational energy levels is 6.70 kJ mol-1. Estimate the temperature at which the equipartition theorem becomes valid. (b) An exponential function may be expanded as ex = 1 + x + (1)/(2)x2 + ... if x < 1. Show that in th is case, the exact expression reduces to the result obtained from the equipartition theorem.