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Chapter 12: 31P (page 509)

Suppose that the entropy of a certain substance (not anEinstein solid) is given by S=aE, where ais a constant. Whatis the specific heat capacity Cas a function of the temperature T?

Short Answer

Expert verified

The specific heat capacity as a function of temperature can be given as,C=a2T2

Step by step solution

01

Identification of given data

The entropy of substance,S=aE

Where a is constant, E is energy and T is temperature

02

Relation is used to solve the problem.

  • Definition of the temperature.

1T=∂S∂E

Where T is temperature,∂S is the change in entropy, and∂E is the change in energy.

  • Definition of specific heat capacity.

C=∂E∂T

Where Cis specific heat capacity,∂T is the temperature change, and∂E is the change in energy.

03

Finding the specific heat capacity as a function of time

Given in the question,

S=aE

From the definition of temperature we know,

1T=∂S∂E

Substituting the value of S

1T=∂aE∂E1T=∂aE∂E1T=∂aE∂E1T=∂aE∂E1T=a∂E∂E1T=a12E1/2-11T=a2E

Further simplification will give,

E=aT2E=a2T24

The energy in terms of temperature can be given as, E=a2T24

Now from the definition of heat capacity, we know,

C=∂E∂T

Now substituting the value of E.

C=∂a2T2/4∂TC=a24∂T2∂TC=a242TC=a2T2

The specific heat capacity is, C=a2T2.

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