Chapter 12: 31P (page 509)

Suppose that the entropy of a certain substance (not anEinstein solid) is given by $S = a \sqrt{E}$ , 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, $C = \frac{a^{2} T}{2}$

Step by step solution

Identification of given data

The entropy of substance, $S = a \sqrt{E}$

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

Relation is used to solve the problem.

Definition of the temperature.

$\frac{1}{T} = \frac{\partial S}{\partial E}$

Where T is temperature, $\partial S$ is the change in entropy, and $\partial E$ is the change in energy.

Definition of specific heat capacity.

$C = \frac{\partial E}{\partial T}$

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

Finding the specific heat capacity as a function of time

Given in the question,

$S = a \sqrt{E}$

From the definition of temperature we know,

$\frac{1}{T} = \frac{\partial S}{\partial E}$

Substituting the value of S

$\frac{1}{T} = \frac{\partial (a \sqrt{E})}{\partial E} \frac{1}{T} = \frac{\partial (a \sqrt{E})}{\partial E} \frac{1}{T} = \frac{\partial (a \sqrt{E})}{\partial E} \frac{1}{T} = \frac{\partial (a \sqrt{E})}{\partial E} \frac{1}{T} = \frac{a \partial (\sqrt{E})}{\partial E} \frac{1}{T} = a (\frac{1}{2} E^{(1 / 2) - 1}) \frac{1}{T} = \frac{a}{2 \sqrt{E}}$

Further simplification will give,

$\sqrt{E} = \frac{aT}{2} E = \frac{a^{2} T^{2}}{4}$

The energy in terms of temperature can be given as, $E = \frac{a^{2} T^{2}}{4}$

Now from the definition of heat capacity, we know,

$C = \frac{\partial E}{\partial T}$

Now substituting the value of E.

$C = \frac{\partial (a^{2} T^{2} / 4)}{\partial T} C = \frac{a^{2}}{4} \frac{\partial (T^{2})}{\partial T} C = \frac{a^{2}}{4} (2 T) C = \frac{a^{2} T}{2}$

The specific heat capacity is, $C = \frac{a^{2} T}{2}$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Short Answer

Step by step solution

Identification of given data

Relation is used to solve the problem.

Finding the specific heat capacity as a function of time

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fields in Physics

Medical Physics

Magnetism and Electromagnetic Induction

Waves Physics

Kinematics Physics

Physics of Motion

Study anywhere. Anytime. Across all devices.

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

Step by step solution

Identification of given data

Relation is used to solve the problem.

Finding the specific heat capacity as a function of time

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fields in Physics

Medical Physics

Magnetism and Electromagnetic Induction

Waves Physics

Kinematics Physics

Physics of Motion

Study anywhere. Anytime. Across all devices.

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