Shows two vessel A and B with rigid walls containing ideal gases. The pressure, temperature and the volume are `P_(A),T_(A),V in the vessel A and P_(B),T_(B),V in the vessel B. The vessels are mow connected through a small tube. Show that pressure p and the temperature T satisfy
`(P)/(T)=(1)/(2)((P_(A))/(T_(A))+(P_(B))/(T_(B))`
when equilibrium is archieved .

Explanation

Let the final pressure, volume and
temperature after connection are
` (P'_A) : partial pressure of A`
` (P'_B) : partial pressure of B`
` Now, ((P'_A)xx (2V)/T) = ((P_A)xx V/ (T_A))`
` or, (P'_A)/T = ((P_A)/(2T_A))………….(i)`
` Similarly ((P'_B)/T) = (P_B/2T_B)........(ii)`
Adding (i) and(ii)
` ((P'_A)/ T)+((P'_B)/T) = ((P_A)/(2T_A)) + ((P_B)/(2T_B))`
` = 1/2 [ (P_A)/(T_A) + (P_B)(T_B)]`
` rArr (P/T) = 1/2 [(P_A)/(T_A) +(P_B)/(T_B)]`
` [:. (P'_A)+(P'_B) = P]`