# Formula for calculation when object temperature will be equal as outside

I'm searching for formula to calculate the time for an object temperature will be equal to outside of it. For example: we put metal with temperature 50 degrees on a table where the air temperature is 10 degrees celsius. When metal temperature will be 10 degrees celsius (like the air outside)? Do and how material matters to temperature change-maybe it's not a metal, it's tea?

In the first approximation, the cooling of a body is governed by the Newton's law of cooling, which, basically, says that the rate of cooling at any given moment is proportional to the temperature difference between the body and its surroundings (ambient temperature).

This can be expressed with by following equation:

$\frac {dT(t)}{dt}=-k(T(t)-T_a)$,

where $T$ is the temperature of a body, $T_a$ is ambient temperature and $k$ is a constant. The solution to this equation predicts that the temperature of the body will decay exponentially, i.e., the rate of cooling will be decreasing as time goes on and the body is getting colder, eventually reaching $T_a$:

$T(t)=T_a+(T(t)-T_a)e^{-kt}$

Obviously, to determine the cooling time, we need to know the value of $k$, but that value is different for different bodies and it is easier to measure than to calculate.

In general, $k$ depends on two factors: the thermal capacity of a body and the thermal conductivity between the body and its surroundings.

Thermal capacity of a body depends on its size and a specific heat of the material, which is the amount of heat required to raise the temperature of a unit mass of material by a unit of temperature. Naturally, the greater the thermal capacity of a body, the longer it takes for the body to cool down. $k$ should be inversely proportional to the thermal capacity.

Thermal conductivity tells us how easily the heat can be transferred from the body to the surroundings and, typically, is limited by the thermal conductivity of the junction between the surface of the body and the air. Thermal conductivity depends on the material and the state of the surface, but, most importantly, it depends on the area of the surface. $k$ should be proportional to the thermal conductivity.

Since the volume of an object grows as the cube of its linear dimensions and the surface - as the square, large objects, in general, will keep heat longer than small objects of similar shape.

To summarize, large bodies, made out of materials with high specific heat and relatively small surface or small surface to volume ratio (like a solid sphere), will have small $k$ and will take relatively long time to cool down. Small bodies, made out of material with low specific heat and large surface to volume ratio (like a thin sheet), will have large $k$ and will cool down relatively quickly.

Based on the above, we can tell that, given the initial temperature of a piece of metal and the ambient temperature, the cooling time will depend on the type of metal, its size and its shape. For instance, specific heat of aluminum is more than twice greater than that of copper and more than seven times of lead's.

The specific heat of water is particularly high - more than four times greater than aluminum's, but, again, a small amount of water in a shallow plate will lose temperature much faster than a hundred pound block of aluminum (even if we ignored the effect of evaporation, which also decreases the temperature of water).

Using some of these ideas, we can ballpark cooling time of simple objects, but the results won't necessarily be accurate, because of some simplifications built into the Newton's law of cooling. For instance, this law does not take into account the distribution of temperature inside a body. It could also be tricky to figure out the effect of convection or the effect of alternative thermal paths, like the plate in the example with water in a plate.

A more practical way to determine the rate of cooling is to perform a test, measure the temperature at several time intervals and calculate $k$.