Three substances are added to a mug to make coffee – coffee is at \[{{65}^{0}}C\], milk is at \[{{65}^{0}}C\] and sugar, which is in equilibrium with coffee. Then –
A) Heat will be transferred from sugar to coffee to reach equilibrium.
B) The sugar is in equilibrium with the milk.
C) Heat will be transferred from the coffee to the sugar to reach equilibrium.
D) None of these conclusions are valid.

Hint: We need to understand the conditions of equilibrium in which the three substances are present at the time of mixing for the coffee. We can use the Zeroth law of thermodynamics to come to the conclusion about the heating processes involved.

Complete step-by-step answer:
We know from the Zeroth law of thermodynamics that if a system A is in equilibrium with a system B and a system C, then the system B and the system C will be in equilibrium with each other.
In the situation given to us, three ingredients for coffee are mixed together. We are given that the sugar and coffee are already in equilibrium. Also, we are given that the coffee and milk are in the same temperature in the same conditions, so the coffee and milk are also in equilibrium.
From the above arguments, we can understand that the three ingredients are in equilibrium with each other.
Also, when the sugar is added to coffee there will be a transfer of heat from coffee to the sugar to maintain thermal equilibrium.
From these, we can understand that the sugar is in equilibrium with the milk and the heat will be transferred from the coffee to the sugar upon mixing.

So, the correct answer is “Option B and C”.

Note: The equilibrium in the Zeroth law of thermodynamics takes into consideration all the physical aspects which can cause a change in temperature such as the entropy and pressure conditions. This is why the sugar and coffee are in equilibrium.