Mixed Air Temperature Calculator

The formula

T(t) = T_env + (T₀ − T_env)e^(−kt); mixing weights by mc

Exponential, so it never quite arrives

Newton's law of cooling says the rate of heat loss is proportional to the temperature excess over the surroundings. That makes the excess decay exponentially — the same mathematics as radioactive decay or an RC circuit — so the object approaches ambient temperature asymptotically and never formally reaches it.

The practical consequence is that cooling is fastest at the start. A cup of coffee loses far more heat in its first minute than its tenth, which is the basis of the counter-intuitive advice to add cold milk late if you want coffee cool soonest: keeping it hot longer makes it shed heat faster to the room.

Mixing is a different calculation. The equilibrium temperature is the average weighted by mass times specific heat — the thermal mass — not by mass alone. A small amount of water can shift a larger amount of metal considerably, because water's specific heat is roughly ten times that of most metals.

The law is an approximation valid for modest temperature differences and convection-dominated cooling. At large differences radiation takes over, and since radiated power goes as the fourth power of absolute temperature, the decay is then much faster than exponential.