# Is thermal equilibrium possible by the Law of Cooling?

• marexz
In summary, the Law of Cooling states that the rate of heat loss by a body is directly proportional to its excess temperature compared to its surroundings. The equation for determining temperature in time is an example of exponential decay, but it doesn't seem to reach thermal equilibrium because the graph never reaches the horizontal asymptote of ambient temperature. This is because real-world systems are not perfect and have lumpy behavior that prevents them from exactly following the ideal mathematical curve. Therefore, even though the temperature may approach the ambient temperature, it may never actually reach it.
marexz
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 I am 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!

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

marexz said:
ambient temperature in a graph is a horizontal asymptote, and If I am 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.

1 person

## 1. What is thermal equilibrium?

Thermal equilibrium is a state in which two objects are at the same temperature and there is no heat transfer between them.

## 2. What is the Law of Cooling?

The Law of Cooling is a physical law that states that the rate of heat loss of an object is directly proportional to the temperature difference between the object and its surroundings.

## 3. How does the Law of Cooling apply to thermal equilibrium?

By following the Law of Cooling, an object will eventually reach thermal equilibrium with its surroundings as the rate of heat loss decreases and the temperature difference becomes smaller.

## 4. Is thermal equilibrium possible by the Law of Cooling in all situations?

No, thermal equilibrium is not always possible by the Law of Cooling, as there may be other factors such as external heat sources or insulation that can affect the rate of heat loss and prevent thermal equilibrium from being reached.

## 5. How can the Law of Cooling be used in practical applications?

The Law of Cooling can be used to determine the time it takes for an object to reach a certain temperature, which can be useful in cooking, refrigeration, and other temperature control processes.

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