Equilibrium with respect to matter flow

1. The problem statement, all variables and given/known data

A two component gaseous system has a fundamental equation of the form

S=AU^1/3V^1/3N^1/3 + (BN₁N₂)/N

N=N₁+N₂

where A and B are positive constants. V is volume , S is entropy, U is internal energy, N is mole number. A closed cylinder of total volume 2V₀ is separated into two equal subvolumes by a rigid diathermal partition permeable only to the first component. One mole of the first component at a temperature T_l is introduced in the left subvolume and a mixture of 1/2 mole of each component is introduced in the right subvolume at a temperature T_r.



Find the equilibrium T^e and the mole numbers in each subvolume.





2. Relevant equations

At equilibrium the chemical potentials of species 1 on right and left chambers are equal and also temperatures are equal.





3. The attempt at a solution



The main issue is that whenever I compute the chemical potential of species 1 in the left chamber or the right chamber and equate them



(∂S/∂N₁)_left=(∂S/∂N₁)_right



I end up getting 0 = 0. I expected to obtain an expression that helps in evaluate the amount of species (1) in the left and right chamber. Am I missing something? Help is appreciated.