A possible alternative mechanism to that shown in Problem 24 for the monochlorination of ethane would involve

Question:

A possible alternative mechanism to that shown in Problem 24 for the monochlorination of ethane would involve the following propagation steps:
CH3--H + ∙Cl -→ CH3--Cl + ∙H
∙H + Cl--Cl -→ H--Cl + ∙Cl
How do you know that the reaction does not take place by this mechanism?