Hydrogen (H2) gas can exist in two forms. If the proton spins are in an exchange symmetric triplet (S=1) state, it is known as orthohydrogen. If the proton spins are in an exchange antisymmetric singlet (S=0) state, it is known as para-hydrogen. The symmetry of the total wave function must be antisymmetric overall, so that the rotational part of the wave function must be antisymmetric for ortho-hydrogen (so that the angular momentum quantum number J is 1,3,5,…) or symmetric for para-hydrogen (so that J=0,2,4,…). The proton separation in H2 is 7.4×10−11 m. Estimate the spacing in kelvin between the ground state and first excited state in para-hydrogen.
Show that the ratio f of ortho-hydrogen to para-hydrogen is given by
f=3∑J=0,2,4,…(2J+1)e−J(J+1)ℏ2/2IkBT∑J=1,3,5,…(2J+1)e−J(J+1)ℏ2/2IkBT,(29.39)
and find f at 50 K.