Chapter 12: 29P (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 energy Eas a function of the temperature T?

Short Answer

Expert verified

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

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 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.

Finding the energy E as a function of time T.

Given in the question,

$S = a \sqrt{E}$

From the definition of the temperature we know,

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

Substituting the value of S

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

Further simplification will give,

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

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

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Suppose that the entropy of a certain substance (not anEinstein solid) is given by $S = a \sqrt{E}$ , where ais a constant. Whatis the energy Eas 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 energy E as a function of time T.

One App. One Place for Learning.

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Suppose that the entropy of a certain substance (not anEinstein solid) is given byS=aE, where ais a constant. Whatis the energy Eas 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 energy E as a function of time T.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Force

Medical Physics

Translational Dynamics

Famous Physicists

Thermodynamics

Atoms and Radioactivity

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 energy Eas a function of the temperature T?