Chapter 10.21, Problem 1QP

(a)

To determine

The amount of sugar dissolve in $\text{1000}\text{g}$ of water at $\text{80°C}\left(\text{176°F}\right)$.

Answer to Problem 1QP

The amount of sugar dissolve in $\text{1000}\text{g}$ of water at $\text{80°C}\left(\text{176°F}\right)$ is $\text{2846}\text{g}$.

Explanation of Solution

Given:

The amount of water, $m_{\text{water}}=1000\text{g}$.

The temperature for dissolving of sugar into water, $T=80\text{°C}$.

Explanation:

Express the amount of sugar.

$C_{\text{sugar}}\left(\text{wt\%}\right)=\frac{m_{\text{sugar}}}{m_{\text{sugar}}+m_{\text{water}}}\times 100$ (I)

Here, mass of sugar is $m_{\text{sugar}}$ and maximum concentration of sugar is $C_{\text{sugar}}\left(\text{wt\%}\right)$.

Conclusion:

Refer Figure 9.1, "the solubility of sugar in sugar water syrup" and show the concentration of sugar at temperature of $\text{80°C}$ as in Figure (1).

From Figure (1), write the concentration of sugar at temperature of $\text{80°C}$.

$C_{\text{sugar}}\left(\text{wt\%}\right)=74\text{wt}\%$

Substitute $74\text{wt}\%$ for $C_{\text{sugar}}\left(\text{wt\%}\right)$ and $\text{1000}\text{g}$ for $m_{\text{water}}$ in Equation (I).

$\begin{array}{l}74=\frac{m_{\text{sugar}}}{m_{\text{sugar}}+1000}\times 100\\ 74m_{\text{sugar}}+74000=100m_{\text{sugar}}\\ 26m_{\text{sugar}}=74000\\ m_{\text{sugar}}=2846\text{g}\end{array}$

Hence, the amount of sugar dissolve in $\text{1000}\text{g}$ of water at $\text{80°C}\left(\text{176°F}\right)$ is $\text{2846}\text{g}$.

(b)

To determine

The composition of the saturated liquid solution (in wt% sugar) at $\text{20°C}$.

Answer to Problem 1QP

The composition of the saturated liquid solution (in wt% sugar) at $\text{20°C}$ is $\text{64wt\%}$.

Explanation of Solution

Given:

The temperature for dissolving of sugar into water, $T=20\text{°C}\left(68\text{°F}\right)$.

Explanation:

From Figure (1), write the concentration of sugar at temperature of $\text{20°C}$.

$C_{\text{sugar}}\left(\text{wt\%}\right)=64\text{wt}\%$

Hence, the composition of the saturated liquid solution (in wt% sugar) at $\text{20°C}$ is $\text{64wt\%}$.

(c)

To determine

The amount of solid sugar will come out of solution upon cooling at $\text{20°C}$.

Answer to Problem 1QP

The amount of solid sugar will come out of solution upon cooling at $\text{20°C}$ is $\text{1068}\text{g}$.

Explanation of Solution

Given:

The temperature for dissolving of sugar into water, $T=20\text{°C}\left(68\text{°F}\right)$.

Explanation:

Express the concentration of sugar at $20\text{°C}$

$C_{\text{sugar}}\left(\text{wt\%}\right)=\frac{m{\text{'}}_{\text{sugar}}}{m{\text{'}}_{\text{sugar}}+m_{\text{water}}}\times 100$ (II)

Here, mass of sugar is $m_{\text{sugar}}$ and maximum concentration of sugar is $C_{\text{sugar}}\left(\text{wt\%}\right)$.

Express the amount of solid sugar will come out of solution upon cooling at $\text{20°C}$.

$m\text{'}{\text{'}}_{\text{sugar}}=m_{\text{sugar}}-m{\text{'}}_{\text{sugar}}$ (III)

Conclusion:

Substitute $64\text{wt}\%$ for $C_{\text{sugar}}\left(\text{wt\%}\right)$ and $\text{1000}\text{g}$ for $m_{\text{water}}$ in Equation (II).

$\begin{array}{l}64=\frac{m{\text{'}}_{\text{sugar}}}{m{\text{'}}_{\text{sugar}}+1000}\times 100\\ 64m{\text{'}}_{\text{sugar}}+64000=100m{\text{'}}_{\text{sugar}}\\ 36m{\text{'}}_{\text{sugar}}=64000\\ m{\text{'}}_{\text{sugar}}=1778\text{g}\end{array}$

Substitute $2846\text{g}$ for $m_{\text{sugar}}$ and $1778\text{g}$ for $m{\text{'}}_{\text{sugar}}$ in Equation (III).

$\begin{array}{c}m\text{'}{\text{'}}_{\text{sugar}}=2846\text{g}-1778\text{g}\\ =\text{1068g}\end{array}$

Hence, the amount of solid sugar will come out of solution upon cooling at $\text{20°C}$ is $\text{1068}\text{g}$.