# BR thermochem...heat capacity and KE

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#### happyfellow

##### Full Member
10+ Year Member
I'm looking at BR thermochemistry question #74. Here's some background in case you don't have the book:

So two different metals are heated in a chamber and experience different changes in temperature. This is due to their different heat capacities. If two metals have different heat capacities what does this tell us about their kinetic energies when they are both at 50 degrees celsius? Apparently, the metal with the higher heat capacity has more kinetic energy? Could someone help explain this?

thanks,

happy

#### RogueUnicorn

##### rawr.
10+ Year Member
you know that a higher heat capacity means more energy taken in per degree of temperature raised. which means at a given temp, the metal with a higher capacity is holding more heat. the only way this heat transfer to the metal can be manifested is through kinetic energy at the atomic level, since obviously we're not forming bonds or changing nuclear structure. if i had paid attention in physical chem i would be able to give you a rigorous quantitative answer, but hopefully this somewhat hand-waving explanation makes some sense.

#### kentavr

##### Full Member
10+ Year Member
15+ Year Member
1. It is not very correct to talk about kinetic energy here. Let's say: internal energy. The heat capacity of the solid body is a complicated question and it is far beyond MCAT(phonon theory)
2. It is possible to analyse this problem with easy math, but with the time frame given, it is best to use bleargh explanation above.
3. However, I show how to do it anyway:
The liquid raise its temperature: so Q= C1 * M1 * (T-T1)
(C1, M1,T1 ) are heat capacity,Mass and initial temp for liquid)
The metal drop its temperature: Q= C2*M2*(T2-T)
Notice that Final temperature the same for liquid and metal, and amount of heat transfer equals Q=Q or
C1 * M1 * (T-T1) = C2*M2*(T2-T)
Let's substitute the numbers from the problem and get how T depend on C2.
We get( I skip arithmetics)
T = (50C2+100C1)/(C2+4C1)
So, if C2(metal heat capacity) and C1 >0 (it is a fact then nominator is always larger then the denomentor and the function grows if C2 grows. (You can check it with the first derivatives if you'd like)

That means that the larger the C2 the higher the T. As required. But I would not use this method on MCAT. (too time consuming).
Just say, that high heat capacitors can store more internal energy and transfer it to heat.
HTH