Based on the idea that selective neutrality is the limit when the selective disadvantage becomes indefinitely small, a model of neutral (and nearly neutral) mutations is proposed that assumes that the selection coefficient (s') against the mutant at various sites within a cistron (gene) follows a Gamma distribution; f(s') = alpha(beta)e(-alphas')s'(beta-1)/Gamma(beta), in which alpha = beta/ s' and s' is the mean selection coefficient against the mutants ( s' > 0; 1 [unk] beta > 0). The mutation rate for alleles whose selection coefficients s' lie in the range between 0 and 1/(2N(e)), in which N(e) is the effective population size, is termed the effectively neutral mutation rate (denoted by v(e)). Using the model of "infinite sites" in population genetics, formulas are derived giving the average heterozygosity ( h(e)) and evolutionary rate per generation (k(g)) in terms of mutant substitutions. It is shown that, with parameter values such as beta = 0.5 and s' = 0.001, the average heterozygosity increases much more slowly as N(e) increases, compared with the case in which a constant fraction of mutations are neutral. Furthermore, the rate of evolution per year (k(1)) becomes constant among various organisms, if the generation span (g) in years is inversely proportional to radicalN(e) among them and if the mutation rate per generation is constant. Also, it is shown that we have roughly k(g) = v(e). The situation becomes quite different if slightly advantageous mutations occur at a constant rate independent of environmental conditions. In this case, the evolutionary rate can become enormously higher in a species with a very large population size than in a species with a small population size, contrary to the observed pattern of evolution at the molecular level.