THERMAL EQUILIBRIUM

If you place a hot cup of coffee and a cold glass of milk on the table and leave the room for several hours, when you return the coffee and the milk are both at room temperature. We say that the coffee, the milk, and the air in the room are in thermal equilibrium. A faster way to reach thermal equilibrium, at least between the coffee and the milk, is to pour the milk into the coffee and stir.
On an atomic scale, what does it mean to say that the molecules of the coffee and those of the milk are in thermal equilibrium? We obtain a hint from our discussion of Brownian motion. The cigarette smoke represents a well stirred mixture of smoke particles and air molecules. These should therefore be in thermal equilibrium, just as the well stirred molecules in the coffee and milk. In the case of Brownian motion, the smoke particles and the air molecules had the same average thermal kinetic energy. We expect that the same may be true for the molecules in the mixture of coffee and milk.
It is an almost general rule that when two objects are in thermal equilibrium, the molecules that make up these objects have the same average thermal kinetic energy. When we first placed a hot cup of coffee and a cold cup of milk on the table, the molecules in the coffee had a greater average kinetic energy, and the molecules in the milk a lesser average kinetic energy, than the molecules in the air. But after a few hours, the coffee molecules slowed down and the milk molecules speeded up until all three sets of molecules, coffee, milk, and air attained the same average kinetic energy.
The process of reaching thermal equilibrium is usually a result of random collisions between molecules. If a fast molecule collides with a slow one, chances are that the slow one will speed up and the fast one will slow down. It requires a detailed analysis, which we will not attempt, to show that if the collisions are random, they tend to equalize the kinetic energy of the particles.