On this homework, we have only acknowledged that specific heat is dependent on a

Question

On this homework, we have only acknowledged that specific heat is dependent on an object's composition and its phase. Let us now also explore a little bit about its temperature dependence.

At very low temperatures, the specific heat capacity of a material is discovered to be proportional to the cube of its temperature, the exact function being:

c(T) = (1.3·10-5 J/kg·K4) T3

How much heat would be required to raise a 1 kg object made of this material from 19 K to 53 K? (Assume the above function is valid throughout this temperature range.)
Hint: Remember your calculus techniques.

??? J ( I got 65.8 J but it was incorrect.)

Explanation / Answer

the heat capcity is given by,

C (T) = (1.3x10-5) T3

Differentiate the equation on both sides, we get

dC/dT = 3(1.3x10-5)T2

Integrate the equation on both sides, we get

C = int [3(1.3x10-5)T2] dT , limits from 19 K to 53 K

= [3(1.3x10-5)] [T3/ 3]   , limits from 19 K to 53 K

= [3(1.3x10-5)] [{(53)3 - (19)3}/ 3]

= 1.846234 J/kg.K

Hence, the heat energy is,

Q = mcdT = (1 kg)(1.846234 J/kg.K)[53 K - 19 K] = 62.77 J = 62.8 J (NEARLY)

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