I read an article about that in order to determine K for their equilibrium at different temperatures they needed to know the concentrations of all...

I read an article about that in order to determine K for their equilibrium at different temperatures they needed to know the concentrations of all their reactants and products. They didn't explain mathematically how they determined the concentrations of each species at the different temperatures. I am wondering if there was any math involved, if so did they assume their determinations were temperature independent and why. If so, isn't this a problem and did they compensate?