In general cases of second-order reactions in solution, a species R of molecules with natural lifetime lambda(0) reacts with another species S with an intrinsic rate constant k(r) that varies with the mutual distance r, following mutual diffusion in the field of an intermolecular potential U(r). An average rate constant k(2) is defined from the mean lifetime lambda (given by the mean first-passage time) of R as the proportionality coefficient of lambda(-1)-lambda(0)(-1) to the concentration of S when concentrations of R and S are both sufficiently low. It is proved rigorously that k(2) has a general formula of 1/(k(TST)(-1)+k(D)(-1)), where k(TST) represents the transition-state-theory rate constant determined by assuming the thermal-equilibrium distribution of r in the potential U(r), and is independent of the mutual diffusion constant D, while the D dependence of k(2) is isolated in k(D)(>0) which decreases as D decreases. Since k(2) nearly equals k(TST) for a large D but approaches k(D) as D decreases, k(D) is called the diffusion limited rate constant. So far it has only been known that this formula is applicable when molecules react coming into contact, but this formula is shown to hold in the general cases. Traditionally, the mean lifetime has been calculated by numerically solving Kolmogorov’s backward equation, which is a second-order differential equation. A new method is given to calculate k(2) and k(D) for arbitrary lambda(0), D, k(r), and U(r) by solving Fredholm’s integral equation of the second kind. This equation can numerically be solved with much higher precision than the backward equation, since in the digitization approximation of a continuous variable, a double differential in the latter is approximated by a double difference among slightly different small terms, while an integral in the former is approximated by a sum of them.