Equipartition Theory of Energy

1. z y x Equipartition Theory of Energy The Equipartition Theory of Energydistributes an ideal gas molecule’s internal energy more or less equally among a molecule’s independent translational, rotational, and vibrational motions or degrees of freedom. At room temperature molecules are generally in their lowest electronic state or ground electronic state and the energy of the electronic ground state can be assigned a value of zero and subsequently ignored in what follows. Consider a monatomic species such as He. A He atom can undergo only translational motion and hence can only store translational energy: since, 1) if we consider He a point mass, there is no mass off of the center of mass and it cannot store rotational energy and 2) monatomic He is not bound to other atoms against which it could vibrate and therefore cannot store vibrational energy. This translational motion can be resolved into translational motion along each of the three independent cartesian directions: 12.1

2. According to the equipartition theory of energy, each energy term that involves the square of a coordinate, contributes 1/2 kT to the total energy of each molecule or 1/2 RT to the total energy of a mole of these molecules. Each cartesian direction or translational degree of freedom in He contributes a term to total energy involving the square of a coordinate, e.g., 1/2 mvx2, and therefore each of these degrees of freedom contributes 1/2 RT to the total molar internal energy: Etrans = Ex,trans + Ey,trans + Ez,trans = 1/2 RT + 1/2 RT + 1/2 RT = 3/2 RT Since He can only translate, this is also the total internal energy in He: Etotal = Etrans = 3/2 RT The constant volume molar heat capacity is just the partial derivative of the total molar internal energy and for He would be: CV = (E/T)V = [ (3/2 RT) / T ]V = 3/2 R = 3/2 (1.987 cal / mole K) = 2.980 cal / mole K The experimental value for He is 2.981 cal / mole K and we can see that the equipartition theory of energy works very well for He! Because the equipartition theory is a classical mechanical theory and ignores the quantitzation of vibrational energies, the theory unfortunately does not work well for molecules, accept for some heavy diatomics such as I2 (g). 12.2

3. What is CP for Rn? Calculate the enthalpy change in Joules when 2.00 moles of an ideal monatomic gas are heated at constant pressure from 300 to 600 K. 12.3