The Dissociation Equation Versus the Saha Equation

(This treatment applies to diatomic but not polyatomic molecules.)

The dissociation equation can be written

N(A)N(B)/N(AB)  º K'(AB) = (g_Ag_B/g_AB)[(2πMkT)^3/2/h³](h²/8π²I)(1 - e^-s)exp(-D/kT), or

P(A)P(B)/P(AB)  º K(AB) = K'(AB)kT = [(2πM)^3/2(kT)^5/2/h³](g_Ag_B/g_AB)(h²/8π²I)(1 - e^-s)exp(-D/kT)

[K'(AB), or K(AB),  is sometimes called the "dissociation constant" even though it is a function of temperature.]  We compare this last expression above with the Saha Equation which can be written

P_q+1P_e/P_q  = [(2πm_e)^3/2(kT)^5/2/h³][2u_q+1(T)/u_q(T)]exp(-I_q /kT).

The expressions are similar, as we would expect since in one case two atoms, A and B, can combine to form molecule AB, and in the other case an ionized atom and an electron can combine to form an atom of one degree lower ionization.  However there are a number of differences in these situations and therefore differences and additional factors in the dissociation equation:

     (1) M is the reduced mass, M = M_AM_B/(M_A + M_B).  (Actually, in the Saha equation, m_e should be the reduced mass
          m_eM(A_q+1)/(m_e + M(A_q+1) » m_e.)

     (2) The ionization energy I_q+1 has been replaced by the dissociation energy of the molecule, D, i.e., energy D is required to dissociate the
          molecule from its ground state.

     (3) 2u_q+1(T)/u_q(T), the product of the electron and the (q+1)^th ionization state partition functions, divided by the q^th state partition function, has
          been replaced by g_Ag_B/g_AB, the product of the statistical weights of the ground states of atoms A and B, divided by the statistical weight of the
          ground state of molecule AB.  (A more rigorous treatment shows the appropriate factor is really u₀(A,T)u₀(B,T)/u₀(AB,T), but at the low
          temperatures at which molecules can exist, u₀(A,T) » g_A, etc.

     (4) A diatomic molecule has available two degrees of freedom not possessed by atoms - vibration along the axis joining A and B and rotation
          about an axis perpendicular to that axis.  Hence two factors occur which are not present in the Saha equation:

• h²/8π²I  is related to the rotational partition function, where I is the the moment of inertial of molecule AB,
   

• (1 - e^-s) is related to the vibrational partition function, where s = hν/kT, and
  ν is the fundamental vibrational frequency of molecule AB.

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  Dissociation constants for some diatomic molecules can be well approximated by a formula of form log K » A - B/T + ½ log T, where T is in kelvins and K is in dyn cm^-2.  For example, this approximation applies well to both the molecule TiO and also to ZrO, where the appropriate constants are: 

  Molecule	A	B
  TiO	10.69	35100
  ZrO	10.86	39600