A possible ion–molecule reaction mechanism for the synthesis of ammonia in interstellar gas clouds i — Physical Chemistry Chemistry Question
Interstellar chemistry
A possible ion–molecule reaction mechanism for the synthesis of ammonia in interstellar gas clouds is shown below
N+ + H2 → NH+ + H k1
NH+ + H2 → NH2 + + H k2
NH2+ + H2 → NH3+ + H k3
NH3+ + H2 → NH4+ + H k4
NH4+ + e– → NH3 + H k5
NH4+ + e– → NH2 + 2H k6
Use the steady state approximation to derive equations for the concentrations of the intermediates NH+, NH2+, NH3+ and NH4+ in terms of the reactant concentrations [N+], [H2] and . Treat the electrons as you would any other reactant.
Model Answer
We can apply the SSA to NH+, NH2+, NH3+ and NH4+.
d[NH+]/dt = 0 = k1[N+][H2] – k2[NH+][H2] ⇒ [NH+] = (k1/k2)[N+]
d[NH2+]/dt = 0 = k2[NH+][H2] – k3[NH2+][H2] ⇒ [NH2+] = (k2/k3)[NH+] = (k2/k3)(k1/k2)[N+] = (k1/k3)[N+]
d[NH3+]/dt = 0 = k3[NH2+][H2] – k4[NH3+][H2] ⇒ [NH3+] = (k3/k4)[NH2+] = (k1/k4)[N+]
d[NH4+]/dt = 0 = k4[NH3+][H2] – k5[NH4+] – k6[NH4+]
[NH4+] = (k4[NH3+][H2]) / ((k5 + k6)) = (k1[N+][H2]) / ((k5 + k6))
Show that the overall rate of production of NH3 is given by d[NH3]/dt = k2nd[N+][H2] where k2nd is the second order rate constant for the reaction. Give an expression for k2nd in terms of the rate constants for the elementary steps, k1 to k6.
Model Answer
d[NH3]/dt = k5[NH4+] = k5(k1[N+][H2])/((k5 + k6)) * = (k1k5)/(k5+k6) [N+][H2]
where k2nd = (k1k5)/(k5+k6)
What is the origin of the activation energy in a chemical reaction?
Model Answer
Chemical reactions involve the making and breaking of bonds. The activation energy is related to the energy required to break the initial bond or provide a sufficient rearrangement of the reactant geometries to initiate reaction.
The rates of many ion-molecule reactions show virtually no dependence on temperature. What does this imply about their activation energy?
Model Answer
The temperature dependence of a rate constant k is described by the Arrhenius equation. k(T) = A exp(−Ea / RT)
Where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T the temperature. Virtually no temperature dependence therefore indicates that the activation energy is very close to zero.
What relevance does this have to reactions occurring in the interstellar medium?
Model Answer
Temperatures in the interstellar medium are extremely low. Only reactions with very low activation energies can occur.