  Given a system $M$ in a thermal bath we obtain a generalized detailed balance
relation for the ratio $r=\pi_\tau(K\to J)/\pi_\tau(J\to K)$ of the transition
probabilities $M:J\to K$ and $M:K\to J$ in time $\tau$. We assume an active
bath, containing solute molecules in metastable states. These molecules may
react with $M$ and the transition $J\to K$ occurs through different channels
$\alpha$ involving different reactions with the bath. We find that $r=\sum
p^\alpha r^\alpha$, where $p^\alpha$ is the probability that channel $\alpha$
occurs, and $r^\alpha$ depends on the amount of heat (more precisely enthalpy)
released to the bath in channel $\alpha$.
