How to calculate thermal equilibrium temperature of three different substances?

Question

I am trying to calculate the thermal equilibrium temperature of three different substances (with three difference specific heats). The substances are:

Substance 1 - mass: 0.6 kg, specific heat: 4170 J/kg°C, initial temperature: 90°C

Substance 2 - mass: 0.2 kg, specific heat: 840 J/kg°C, initial temperature: 20°C

Substance 3 - mass: 0.1 kg, specific heat: 110 J/kg°C, initial temperature: 10°C

I have already calculated the equilibrium temperature of Substances 1 and 2, which is approximately 85.61°C. What I am trying to figure out is, when Substance 1 and 2 are 80°C, what will be the equilibrium temperature when Substance 3 is added? I am confused about if I need to add, multiply, or find the average of specific heats for Substances 1 and 2. I would like to know what is the formula for calculating equilibrium temperature of 3 or more substances (with different specific heats).

Answer 1

Think about it: the specific heat is, in principle, something characteristic of the substance. If you add substances 1 and 2, you will get a resulting mixture, let's call it 1/2. Do you think that it's specific heat will be more like 1's or 2's?

I think it is intuitive to say that it will be closer to the specific heat of substance 1, since it is predominant in your mixture (you added a larger mass of substance 1 to the mixture, compared with substance 2).

You can work this out from here. Hope it helped.