# Which will cool faster?

I just sat my first materials science exam today and one question just stuck out in my mind which I was unsure to answer, maybe someone here could give me an insight.

"1kg of Aluminium requires twice as much energy to raise its temperature than 1kg of steel. However, aluminium has a larger thermal conductivity than steel. If a hot coin is placed on both an alminium slab and steel slab of same mass, which coin will cool faster."

I initially answer the question by saying Aluminium but I wondered to myself why would they add the fact that steel requires less energy, so I changed my answer to steel.

My basic thought process was if the steel requires less energy to raise it temperature then the energy transferred from the coin to steel will happen twice as fast as the aluminium but because the thermal expansivity of aluminium is not twice that of steel (the question provided this data), the steel coin will cool faster.

More graphically, these numbers are purely fictious!

Coin with 200J amount of heat energy ---TRANSFER AT 100J/s---> Aluminium needs 200J to raise temperature. This means the aluminium will take 2 seconds to heat up and the coin 2 seconds to cool.

Coin with 200J of heat energy ---TRANSFER AT 80J/s (since it has lower conductivity)--->Steel only needs 100J to raise temperature. This means the steek will take 1.25 seconds to heat up and the coin 1.25 seconds to cool.

Is this analogy correct? And more importantly did I answer my exam question correct?

I have just looked at my post again and now I think I might have answered this wrongly by saying steel, because it is asking purely which coin will cool faster then I suspect since the heat transfer i.e. conductivity is the only parameter that matters here and not how much energy each surface needs because the different energies will result in one surface being warmer than the other.

This is the same flipping and flapping I done in the exam. Very Annoying.

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I'd go with aluminium (it is more commonly used as heatsink material thatn steel)

The only thing that seems important to this question is the thermal conductivity, not the energy temp requirements.

So the Alu will absorb the heat energy from the coin faster, but will take longer to get hotter (temperature wise). That could make sense.

I'd go with aluminium (it is more commonly used as heatsink material thatn steel)

The only thing that seems important to this question is the thermal conductivity, not the energy temp requirements.

So the Alu will absorb the heat energy from the coin faster, but will take longer to get hotter (temperature wise). That could make sense.

Yes this is what I think happens now, pity I changed my mind.

The correct answer is aluminum. Here is why;

Fourier's law states that the rate of conductive heat transfer is proportional to the transfer coefficient and the temperature gradient of the material(s). Or in other words;

q = k * $$\nabla$$T

Aluminum has a much higher conduction coefficient (k) value than steel, therefor the amount of heat transferred from the coin and the material around it will be greater given the same $$\nabla$$T.

Aluminum also has a much greater specific heat (Cv) than steel does. The first law of thermodynamics states;

$$\Delta$$Q = Mass * Cv * $$\Delta$$T

This means that for the same amount of heat transferred (Q) from the coin, the aluminum will have a lower temperature. This means that while the coin is cooling, the aluminum block will have a larger $$\nabla$$T than the steel, thereby having a higher rate of heat transfer as you can see from Fourier's law.