Sample Quiz #2 Solutions

1. K = z(prod) / z(react) exp(-dE_o/kT)
   
   dE_o = E_o(products) - E_o(reactants) but we're ignoring that term.
   
   z(prod) = z(Cl) * z(O₂) which counts all available product states.
   
   z(react) = z(ClO) * z(O) ditto for reactants
   
   Since both are products (actually as many such terms as there are stoichiometric moles in the generic reaction...here all stoichiometric coefficients are 1) in the multiplicative sense,
   
   z _TRANS (prod) = z _TRANS (Cl) * z _TRANS (O₂)
   
   z _TRANS (react) = z _TRANS (ClO) * z _TRANS (O)
   
   and K _TRANS = z _TRANS (prod) / z _TRANS (react)
   
   Since there as many numerator as denominator terms, all the common factors in z _TRANS will cancel out, leaving only:
   
   K _TRANS = ( m_Cl * m_O2 / m_ClO * m_O ) ^3/2 = 1.62
   
   
   
2. z = z _TRANS * z _INTERNAL always (since TRANS is always separable)
   
   Thus, since E = kT ² ( d ln[z] / dT ) _V , that ln[z] gives E _TRANS always
   
   But z _INTERNAL is NOT z _EL * z _ROT * z _VIB since different EL's contribute!
   
   Instead z ^INTERNAL = z ₀^INTERNAL + z ₁^INTERNAL. Why "+" and not "*"?
   
   Because z is defined as the "sum over states." It conveniently separates into multiplicative terms only in the case that every cross-term (each EL with every VIB, say) is represented in the sum, and that comes from a product of sums! But not a SUM of sums! In the present case, for example, there's no state which combines e ₀^EL with e ₁^VIB! So that cross-term's NOT PRESENT. No separation.
   
   So we're stuck with z = z^TRANS * ( z₀^INTERNAL + z₁^INTERNAL)
   
   and there's no way of factoring out, say, z ^VIB! Logically, we wouldn't know which electronic state to take it from. So there's no E ^VIB possible even though E ^TRANS is separable.
   
   The best we can do is extract a E ^INTERNAL which will have a mixture of electronic states 0 and 1.