A45.0 gblock of tungsten at $\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{°}\mathbf{C}$and a 25.0 gblock of silver at$\mathbf{-}\mathbf{120}\mathbf{.}\mathbf{0}\mathbf{°}\mathbf{C}$are placed together in an insulated container. (See Table 18-3 for specific heats.) (a) What is the equilibrium temperature? What entropy changes do (b) the tungsten, (c) the silver, and (d) the tungsten–silver system undergo in reaching the equilibrium temperature?

a) Mass of the tungsten block,$m_1=0.045\mathrm{kg}$

b) Mass of silver,$m_2=0.025\mathrm{kg}$

c) Temperature of tungsten,$T_1=30°\mathrm{C}=303\mathrm{K}$

d) Temperature of silver,$T_2=-120\mathrm{m}°\mathrm{C}=153\mathrm{k}$

e) Specific heat of tungsten,$c_1=134\mathrm{J}/\mathrm{kg}.\mathrm{K}$

f) Specific heat of silver,$c_2=236\mathrm{J}/\mathrm{kg}.\mathrm{K}$

Using the concept of thermal equilibrium, the net heat absorbed or released by the system can be calculated. When two objects are made in contact within an insulated container, they achieve a state of equilibrium at a given temperature. In this case, the heat released by the hot body is equal to the heat absorbed by the cold body.

From the second law of thermodynamics, the entropy change of a body for a reversible process can be given as the net heat released by the body to the system per the temperature of the system.

Formulae:

The heat absorbed or released by a body,$Q=mc∆T$ ……(i)

The entropy of a body, $∆S={\int }_{{\mathrm{T}}_1}^{{\mathrm{T}}_2}\frac{\mathrm{dQ}}{\mathrm{T}}$ ……(ii)

Let the equilibrium temperature of the two materials be $T_f$.

For the equilibrium state of the tungsten-silver system, the net heat absorbed or released by the system can be given and equilibrium temperature can be calculated as follows:

$$\begin{gathered} \sum _{}Q=0 \\ {Q}_{tunsten}+Q_{silver}=0 \\ {m}_1{c}_1\left(T_f-T_1\right)m_2{c}_2\left(T_f-T_2\right)=0 \\ \left(m_1{c}_1+m_2{c}_2\right)T_f=m_1{c}_1{T}_1+m_2{c}_2{T}_2 \\ {T}_f=\frac{m_1{c}_1{T}_1+m_2{c}_2{T}_2}{\left(m_1{c}_1+m_2{c}_2\right)} \\ =\frac{\left(0.045\mathrm{kg}\right)\left(134\mathrm{J}/\mathrm{kg}.\mathrm{K}\right)\left(303\mathrm{K}\right)+\left(0.025\mathrm{kg}\right)\left(\left(236\mathrm{J}/\mathrm{kg}.\mathrm{K}\right)\right)\left(153\mathrm{K}\right)}{\left(0.045\mathrm{kg}\right)\left(134\mathrm{J}/\mathrm{kg}.\mathrm{K}\right)+\left(0.025\mathrm{kg}\right)\left(\left(236\mathrm{J}/\mathrm{kg}.\mathrm{K}\right)\right)} \\ =228.8\mathrm{K} \\ =-44.2°\mathrm{C} \end{gathered}$$

Hence, the equilibrium temperature is $-44.2°\mathrm{C}$.

Substituting equation (i) in equation (ii), the entropy change of tungsten element within the insulated container can be given as follows:

$$\begin{gathered} ∆S={\int }_{T_1}^{T_1}\frac{m_1{c}_{1}dt}{T} \\ =m_1{c}_1{\int }_{T_1}^{T_1}\frac{dT}{T} \\ =m_1{c}_{1}In{\left[T\right]}_{T_1}^{T_1} \\ =m_1{c}_{1}In\left[\frac{T_f}{T_1}\right] \end{gathered}$$

For the given values, we have-

$$\begin{gathered} ∆\mathrm{S}=\left(0.045\mathrm{kg}\right)\left(134\mathrm{J}/\mathrm{kg}.\mathrm{K}\right)\mathrm{In}\left[\frac{229\mathrm{K}}{303\mathrm{K}}\right] \\ =1.69\mathrm{J}/\mathrm{K} \end{gathered}$$

Hence, the value of entropy change is -1.69 J/K.

The entropy change of silver element within the insulted container can be given as follows:

$$\begin{gathered} ∆S={\int }_{T_1}^{T_1}\frac{m_2{c}_{2}dt}{T} \\ =m_2{c}_2{\int }_{T_1}^{T_1}\frac{dT}{T} \\ =m_2{c}_{2}In{\left[T\right]}_{T_1}^{T_1} \\ =m_2{c}_{2}In\left[\frac{T_f}{T_1}\right] \end{gathered}$$

For the given values-

$$\begin{gathered} ∆\mathrm{S}=\left(0.025\mathrm{kg}\right)\left(236\mathrm{J}/\mathrm{kg}.\mathrm{K}\right)\mathrm{In}\left[\frac{229\mathrm{K}}{153\mathrm{K}}\right] \\ =2.38\mathrm{J}/\mathrm{K} \end{gathered}$$

Hence, the value of entropy change is 2.38 J/K.

The entropy change of the tungsten-silver system is the net energy released by both the materials and that is given as follows:

$$\begin{gathered} ∆{\mathrm{S}}_{\mathrm{system}}=\left(2.38-1.69\right)\mathrm{J}/\mathrm{K} \\ =0.69\mathrm{J}/\mathrm{K} \end{gathered}$$

Hence, the net entropy change for the system is 0.69 J/K.