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# Temperature difference. watch

1. If you have 2 cups of coffee in a room temperature. One at 50 degrees C and the other at 30 degrees C. The one with the biggest temperature difference (the 50 degrees C) one will change temperature the fastest, am I right?

However, if both coffees are at 50 degrees for example, but one of them is 50ml and the other 100ml. Which one will cool first and why?
2. If you have 2 cups of coffee in a room temperature. One at 50 degrees C and the other at 30 degrees C. The one with the biggest temperature difference (the 50 degrees C) one will change temperature the fastest, am I right?
Newton's law of cooling states that

where Ta is the ambient temperature and T is the temperature of the object. So yes you are right, as the cooling rate is proportional to the temp difference between the bodies involved.

However, if both coffees are at 50 degrees for example, but one of them is 50ml and the other 100ml. Which one will cool first and why?
The 50ml will cool first as, assuming they are in the same shaped and sized cup, they have (almost) identical surface profiles (and hence cool at the same rate by convection and radiation) but the 100ml one has twice the energy to lose (due to it having twice the volume).
3. The conclusion is correct, but it may be worth mentioning a couple of things regarding cooling laws.
Newton's Law actually states that the rate of loss of heat (not temperature) is proportional to the temperature difference.
Of course, dQ=mC.dT so for a constant mass, m, of the same liquid with sp. heat capacity C
dQ/dt= k(dT)
dT/dt=(k/mC).dT
k is a cooling constant.
This of course gives the result mentioned by s_l_n_h above.
It also shows why decreasing the mass of coffee, assuming all else remains the same, increases the rate of fall of the temperature.
Secondly, Newton's Law of Cooling only applies when there is forced convection. The law is not quite true for still air when heat loss is due to natural convection.
In this case you would use the "five fourths" law.
Rate of loss of heat = constant times temperature difference to power 5/4
This would still give the same result as far as the fall in temperature is concerned.

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Updated: February 1, 2010
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