# Is thermal equilibrium possible by the Law of Cooling?

## Main Question or Discussion Point

Law of cooling states that the rate of loss of heat by a body is directly proportional to the excess temperature of the body above that of its surroundings. The equation of determining temperature in time T(t)=Ta+(T0-Ta)*e^kt (Ta-ambient temperature, T0-initial temperature, k-heat transfer constant) is a standart example of exponential decay BUT I don't understand the fact that in this equation ambient temperature in a graph is a horizontal asymptote, and If im not wrong, then the graph goes to infinity never reaching this point. That means the object which is cooling down never actually reaches temperature of the surroundings therefore thermal equilibrium is not reached (also remembering that ambient temperature is all the way a constant). How is that possible?

Thank you!

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I think that you can reasonably assume that the solution has reached the steady state when the temperature is 0.99 the ambient temperature

Nugatory
Mentor
ambient temperature in a graph is a horizontal asymptote, and If im not wrong, then the graph goes to infinity never reaching this point. That means the object which is cooling down never actually reaches temperature of the surroundings therefore thermal equilibrium is not reached (also remembering that ambient temperature is all the way a constant). How is that possible?
You'll see this happen in many problems: a bouncing ball slowing down and coming to rest, a damped harmonic oscillator, equalizing pressure between two volumes of gas, pretty much any situation in which a difference (in your question, the temperature difference between object and ambient) forces an action (in your question, a heat flow) that tends to reduce the difference.

What's going on in all of these problems is that the perfect mathematical curve with its asymptote is a description of an ideal system that exactly obeys the math. Real-world systems do not quite conform to this ideal; they're made up of atoms so if you study them at a fine enough scale their behavior is "lumpy", not a perfect mathematical smooth curve. When the lumpiness gets to be of about the same scale as the distance between the mathematical curve and the asymptote, we've reached steady state.