Noura Dawass

5.2. M ETHODS 5 97 g ωω ( r ) = n P i = 1 n P j = 1( j 6 = i ) N i N j g i j ( r ) + n P i = 1 N 2 i g ii ( r ) µ n P i = 1 N i ¶ 2 = n P i = 1 n P j = 1( j 6 = i ) N i N j + n P i = 1 N 2 i N i − 1 N i µ n P i = 1 N i ¶ 2 = n P i = 1 n P j = 1 N i N j − n P i = 1 N 2 i + n P i = 1 N i ( N i − 1) µ n P i = 1 N i ¶ 2 = n P i = 1 N i n P j = 1 N j − n P i = 1 N 2 i + n P i = 1 N 2 i − n P i = 1 N i µ n P i = 1 N i ¶ 2 = µ n P i = 1 N i ¶ 2 − n P i = 1 N i µ n P i = 1 N i ¶ 2 = n P i = 1 N i − 1 n P i = 1 N i (5.23) Similarly, for g ωα ( r ) g ωα ( r ) = n P i = 1 N i g i α ( r ) n P i = 1 N i = n P i = 1 N i n P i = 1 N i = 1 (5.24) Hence, for an ideal gas, Eqs. (5.21) and (5.22) provide the correct answer.