Chapter 22, Problem 71PQ

A system consisting of 10.0 g of water at a temperature of 20.0°C is converted into ice at –10.0°C at constant atmospheric pressure. Calculate the total change in entropy of the system. Assume the specific heat of water is 4.19 × 10³ J/(kg · K), the specific heat of ice is 2.10 × 10³ J/(kg · K), and the latent heat of fusion is 3.33 × 10⁵ J/kg.

To determine

The total change in entropy of the system.

Answer to Problem 71PQ

The total change in entropy of the system is $-15.9\text{J}/\text{K}$.

Explanation of Solution

Write the expression to calculate the entropy change when $293\text{K}$ or $20\text{°C}$ water convert to $273\text{K}$ or $0\text{°C}$.

  $\Delta s=m{c}_p\ln \left(\frac{T_f}{T_i}\right)$                                                                                                       (I)

Here, $\Delta s$ is the change in entropy, m is the mass of the water, $c_p$ is the specific heat capacity of the water, $T_f$ is the final temperature and $T_i$ is the initial temperature.

Write the expression to calculate the entropy change for the transformation of phase from ice to water.

  $\Delta s_1=\frac{-m{L}_f}{T}$                                                                                                              (II)

Here, $\Delta s_1$ is the entropy change of the phase transformation, T is the temperature of the ice and $L_f$ is the latent heat of fusion of the ice.

Write the expression to calculate the change in entropy of the ice to decrease the temperature from $273\text{K}$ or $0\text{°C}$ to $263\text{K}$ or $-10.0\text{°C}$.

  $\Delta s_2=m{c}_p{}^{\prime }\ln \left(\frac{T_f{}^{\prime }}{T_i{}^{\prime }}\right)$                                                                                                 (III)

Here, $\Delta s_2$ is the change in entropy for the ice, $c_p{}^{\prime }$ is the specific heat of ice, $T_f{}^{\prime }$ is the final temperature and $T_i{}^{\prime }$ is the initial temperature.

Write the expression to calculate the entropy change for the whole process.

  $\Delta s_{\text{tot}}=\Delta s+\Delta s_1+\Delta s_2$                                                                                              (IV)

Here, $\Delta s_{\text{tot}}$ is the total entropy change of the whole process.

Conclusion:

Substitute $10.0\text{g}$ for m, $4.19\times {10}^3\text{J}/\text{kg}\cdot \text{K}$ for, $c_p$, $293\text{K}$ for $T_i$ and $273\text{K}$ for $T_f$ in the above equation to calculate $\Delta s$.

  $\begin{array}{c}\Delta s=\left(10.0\text{g}\right)\left(\frac{{10}^{-3}\text{kg}}{1\text{g}}\right)\left(4.19\times {10}^3\text{J}/\text{kg}\cdot \text{K}\right)\ln \left(\frac{273\text{K}}{293\text{K}}\right)\\ =-2.96\text{J}/\text{K}\end{array}$

Substitute $10.0\text{g}$ for m, $3.33\times {10}^5\text{J}/\text{kg}$ for $L_f$ and $273\text{K}$ for T in the equation (II) to calculate $\Delta s_1$.

  $\begin{array}{c}\Delta s_1=\frac{-\left(10.0\text{g}\left(\frac{{10}^{-3}\text{kg}}{1\text{g}}\right)\right)\left(3.33\times {10}^5\text{J}/\text{kg}\right)}{273\text{K}}\\ =-12.2\text{J}/\text{K}\end{array}$

Substitute $10.0\text{g}$ for m, $2.10\times {10}^3\text{J}/\text{kg}\cdot \text{K}$ for $c_p{}^{\prime }$, $263\text{K}$ for $T_f{}^{\prime }$ and $273\text{K}$ for $T_i{}^{\prime }$ in the above equation (III) to calculate . $\Delta s_2$

  $\begin{array}{c}\Delta s_2=\left(10.0\text{g}\left(\frac{{10}^{-3}\text{kg}}{1\text{g}}\right)\right)\left(2.10\times {10}^3\text{J}/\text{kg}\cdot \text{K}\right)\ln \left(\frac{263\text{K}}{273\text{K}}\right)\\ =-0.78\text{J}/\text{K}\end{array}$

Substitute $-2.96\text{J}/\text{K}$ for $\Delta s$, $-12.2\text{J}/\text{K}$ for $\Delta s_1$ and $-0.78\text{J}/\text{K}$ for $\Delta s_2$ in the above equation to calculate $\Delta s_{\text{tot}}$.

  $\begin{array}{c}\Delta s_{\text{tot}}=-2.96\text{J}/\text{K}+\left(-12.2\text{J}/\text{K}\right)+\left(-0.78\text{J}/\text{K}\right)\\ =-15.9\text{J}/\text{K}\end{array}$

Therefore, the total change in entropy of the system is $-15.9\text{J}/\text{K}$.