Calculating Heat: How to Apply Calculus for Accurate Results

[s7b]

I'm doing some sample problems to prepare for midterm and am stuck on this:

The question talked about many materials at low temperatures obeying Debye's Law C=A(t/θ)^3

it said that for a diamond θ is 1860K and asked to evaluate the specific heats at 20K and 100K.

For this I just used that formula given. The part I'm having trouble with is how much heat is required to heat one mole of diamond between 20K and 100K.

I know that
To heat the diamond from 20 to 21 K, you need:

0.0024 J/molK

from 21 to 22 K, you probably need a little more

0.0026 J/molK ( more or less)

and so on until you heat it from 99 to 100 K where you need:

0.301 J/molK

so you need to add up

0.0024 + 0.0026 + ... +..... + 0.301 to get to the final answer - it should probably look like:

(0.0024 + 0.301) / 2 x (100 - 20 K) = 12.136 J/mol

But this is not the correct answer. How can I apply calculus in order to obtain a more correct answer?

 

[gsal]

Well, I have not met that Debye dude, but I think that you need to simply integrate and evaluate the expression at the two limits

your integral would result in something like this: [A/(θ)^3] T^4/4

evaluate that expression at 100 and 20 and subtract such numbers.

does this help? maybe?

 

[s7b]

This makes more sense then my attempt. Thanks :)