Entropy of a two-species classical ideal gas ($Z=Z_1^{N_A+N_B}/(N_A!N_B!)$)

Two physically distinguishable noninteracting species with fixed particle numbers and the same single-particle partition function have the displayed canonical partition function. If $Z_1\propto V{T}^{3/2}$, the leading large-particle-number entropy is $S=k_B{\sum }_{i=A,B}{N}_i[\ln (Z_1/N_i)+5/2]$. With $N=N_A+N_B$ and $x_i=N_i/N$, this equals the one-species expression at total number $N$ plus $-N{k}_B{\sum }_i{x}_i\ln x_i$. Merely attaching labels to otherwise identical indistinguishable atoms does not justify this mixing term.