Smoluchowski equation with a sink term is widely used as a model of a rate process in a slowly relaxing environment. Two approximate solutions for the rate constant obtained for a steeply growing sink are tested numerically using an exponential sink. Both analytical solutions are in a good agreement with the numerical results over a wide range of the problem parameters (environment relaxation rate). They show how the rate constant Gamma decreases when the viscosity eta of the environment increases. If the dependence is approximated by the fractional power law, Gamma proportional to eta(-alpha), the exponent alpha is always less than unity and depends on eta. It tends to zero for fast relaxation of the environment (small eta) and increases when the relaxation slows down (eta grows).