The entropy S of a certain object (not an Einstein solid) is the following function of the internal energy $\mathbf{E}\mathbf{:}\mathbf{S}\mathbf{=}{\mathbf{bE}}^{\mathbf{1}\mathbf{/}\mathbf{2}}$, where b is a constant. (a) Determine the internal energy of this object as a function of the temperature.

(b) What is the specific heat of this object as a function of the temperature?

Entropy, the measure of a system's thermal energy per unit temperature that is unavailable for doing useful work. Because work is obtained from ordered molecular motion, the amount of entropy is also a measure of the molecular disorder, or randomness, of a system.

The relation that connects entropy, energy and temperature is given as follows:

$$\begin{gathered} \partial S=\frac{\partial E}{T} \\ T=\frac{\partial E}{\partial S}.........\left(1\right) \end{gathered}$$

Here, $\partial S$is the change in entropy in $\mathrm{J}/\mathrm{kg},\partial \mathrm{E}$is change in internal energy in

Joules and T is temperature in Kelvin.

Theexpression for entropy is

$S=b\sqrt{E}$

Here, b is constant and E is internal energy.

Rearrange the expression for E.

$E=\frac{S^2}{b^2}.........\left(2\right)$

Substitute equation (2) in equation (1).

$$\begin{gathered} T=\frac{\partial }{\partial S}\left(\frac{S^2}{b^2}\right) \\ =\frac{1}{b^2}\frac{\partial }{\partial S}\left(S^2\right) \\ =\frac{1}{b^2}\left(2S\right) \end{gathered}$$

Rearrange the expression for S.

$S=\frac{1}{2}{b}^{2}T$

Substitute$b-\sqrt{E}\mathrm{for}\mathrm{S}$

$$\begin{gathered} b\sqrt{E}=\frac{1}{2}{b}^{2}T \\ \sqrt{E}=\frac{1}{2}bT \\ E=\frac{1}{2}{b}^2{T}^2 \end{gathered}$$

Thus, the expression for energy in terms of temperature is

$E=\frac{1}{2}{b}^2{T}^2$

The internal energy of a system having N number of atoms is proportional to its temperature T.

$E=C{N}_{atoms}T$

Here, C is the specific heat, T is the temperature in Kelvin

Apply partial derivatives to both sides.

$$\begin{gathered} \partial E=C{N}_{atoms}\partial T \\ \frac{\partial E}{\partial T}=C{N}_{atoms} \end{gathered}$$

Here,$\partial E$is change in internal energy in joules.

Substitute $\frac{1}{4}{b}^2{T}^2\mathrm{for}E$

$$\begin{gathered} C{N}_{atoms}=\frac{\partial }{\partial T}\left(\frac{1}{4}{b}^2{T}^2\right) \\ C=\frac{b^2}{2}\left(\frac{T}{N_{atoms}}\right) \end{gathered}$$

Thus, the expression foe specific heat in terms of temperature is

$C=\frac{b^2}{2}\left(\frac{T}{N_{atoms}}\right)$