 Solution

For identical atoms, use $\mu =m_A/2$ and the homonuclear rotational symmetry number $\sigma =2$. With a rate defined as reaction events per volume and time, this halves the geometric prefactor relative to distinguishable labelled pairs. Do not independently count both ordered pairs and then add another unrelated factor of two. If the rate is instead defined by disappearance of A, each event consumes two atoms, so $-d{n}_A/dt=2r$; that rate coefficient is twice the event-rate coefficient.