# Relationship between Specific heat capacity and Newton's law of cooling

I have been faced with trying to find a relationship between specific heat capacity and Newtons'law of cooling. Reading around, I have seen that a objects rate of cooling is based on surface area, the temperature of the surroundings etc.

To put this into context, I am aiming at taking 10 different metals with different specific heat capacities and heating them to 100 degrees Celsius. From there(as they cool), I aim to take the necessary temperature points and create the curve for the Newton cooling curve. I have not tested this yet in real life as of yet and would like to know if this even exists.

In short, is there a trend between metals of varying SHC's and their respective cooling curve(Or cooling constant K)?

Thanks, Suraj

As a side note, the metals will be cooled in air. I will be heating them in water and, using an IR sensor, measuring the temperature as they cool.

• Hi, welcome to physics stack exchange. What progress have you made towards answering this question, and what, if any, are the roadblocks you've encountered? – Al Nejati Sep 24 '18 at 21:59

The Newton's law ("of cooling") is actually a formula that describes the rate of heat exchange with the medium. It is the rate of loosing (or gaining) heat and not the rate of temperature change. This last one depends on the heat capacity of the body. The same heat transfer to the medium will produce a large drop in temperature for a body with low heat capacity and small drop for a body with high capacity. The rate of heat transfer is $$\frac{dQ}{dt}=-hA[T(t)-T_{medium}]$$ where h is the coefficient of heat transfer, A is the contact area and T(t) is the variable temperature of the cooling object. dQ is the heat transferred to the medium in the time interval dt. This is related to the heat capacity by $$dQ=m C dT$$ where m is the mass of the cooling body and C is the heat capacity (in J/kg). By combining the two equations you will get a differential equation whose solution is the temperature variation in time, T(t). The rate of cooling is $$\frac{dT}{dt}$$ and will depend on h,A, C, m and the temperature of the medium.