Chapter 12, Problem 12.142P

a.

Interpretation Introduction

Interpretation:

The time to heat the sample to its melting point is to be calculated.

Concept introduction:

Specific heat capacity $\left(c\right)$ of a substance is the amount of heat needed to raise the temperature of $1\text{g}$ of a substance by $1\text{K}$. The formula to calculate heat required is as follows:

  $q=\left(\text{mass}\right)\left(c\right)\left(T_2-T_1\right)$        (1)

Here,

$T_2$ is the final temperature.

$T_1$ is the initial temperature.

$q$ is the heat released or absorbed.

$c$ is the specific heat capacity of the substance.

The formula to calculate heat at phase change is as follows:

  $q=\left(\text{mass}\right)\left(\Delta H\right)$        (2)

Here,

$\Delta H$ is the enthalpy change.

Answer to Problem 12.142P

The time to heat the sample to its melting point is $1.1\min$.

Explanation of Solution

Substitute $25\text{g}$ for mass, $1.0\text{J/g}\cdot °\text{C}$ for $c$, $-20.0°\text{C}$ for $T_2$ and $-40.0°\text{C}$ for $T_1$ in the equation (1) to calculate the heat required to increase the temperature of sample from $-40.0°\text{C}$ to $-20.0°\text{C}$.

  $\begin{array}{c}q=\left(25\text{g}\right)\left(1.0\text{J/g}\cdot °\text{C}\right)\left(\left(-20.0°\text{C}\right)-\left(-40.0°\text{C}\right)\right)\\ =500\text{J}\end{array}$

Constant rate of heating is $\text{450}\text{J/min}$. The time to heat the sample to its melting point is calculated as follows:

  $\begin{array}{c}\text{Time}=\left(\frac{\text{500}\text{J}}{\text{450}\text{J/min}}\right)\\ =1.1111\min \\ \approx 1.1\min .\end{array}$

Conclusion

The given sample reaches its melting point by heating for $1.1\min$.

(b)

Interpretation Introduction

Interpretation:

The time taken to melt the sample is to be calculated.

Concept introduction:

Specific heat capacity $\left(c\right)$ of a substance is the amount of heat needed to raise the temperature of $1\text{g}$ of a substance by $1\text{K}$. The formula to calculate heat required is as follows:

  $q=\left(\text{mass}\right)\left(c\right)\left(T_2-T_1\right)$        (1)

Here,

$T_2$ is the final temperature.

$T_1$ is the initial temperature.

$q$ is the heat released or absorbed.

$c$ is the specific heat capacity of the substance.

The formula to calculate heat at phase change is as follows:

  $q=\left(\text{mass}\right)\left(\Delta H\right)$        (2)

Here,

$\Delta H$ is the enthalpy change.

Answer to Problem 12.142P

The time taken to melt the sample is $10\min$.

Explanation of Solution

Substitute $25\text{g}$ for mass and $180\text{J/g}$ for $\Delta H_{\text{fus}}^{°}$ in the equation (2) to change the phase of solid sample at $-20.0°\text{C}$ to liquid sample at $-20.0°\text{C}$.

  $\begin{array}{c}q=\left(25\text{g}\right)\left(180\text{J/g}\right)\\ =4500\text{ J}\end{array}$

Constant rate of heating is $\text{450}\text{J/min}$. The time taken to melt the sample is calculated as follows:

  $\begin{array}{c}\text{Time}=\left(\frac{4500\text{ J}}{\text{450}\text{J/min}}\right)\\ =10\min \end{array}$

Conclusion

The time taken to melt $25\text{g}$ sample is $10\min$.

(c)

Interpretation Introduction

Interpretation:

A curve of temperature vs. time for the entire heating process is to be calculated.

Concept introduction:

Specific heat capacity $\left(c\right)$ of a substance is the amount of heat needed to raise the temperature of $1\text{g}$ of a substance by $1\text{K}$. The formula to calculate heat required is as follows:

  $q=\left(\text{mass}\right)\left(c\right)\left(T_2-T_1\right)$        (1)

Here,

$T_2$ is the final temperature.

$T_1$ is the initial temperature.

$q$ is the heat released or absorbed.

$c$ is the specific heat capacity of the substance.

The formula to calculate heat at phase change is as follows:

  $q=\left(\text{mass}\right)\left(\Delta H\right)$        (2)

Here,

$\Delta H$ is the enthalpy change.

Answer to Problem 12.142P

The curve of temperature vs. time for the entire heating process is as follows:

Explanation of Solution

Substitute $25\text{g}$ for mass, $2.5\text{J/g}\cdot °\text{C}$ for $c$, $85°\text{C}$ for $T_2$ and $-20.0°\text{C}$ for $T_1$ in the equation (1) to calculate the heat required to increase the temperature of the sample from $-20.0°\text{C}$ to $85°\text{C}$.

  $\begin{array}{c}{q}_1=\left(25\text{g}\right)\left(2.5\text{J/g}\cdot °\text{C}\right)\left(85°\text{C}-\left(-20.0°\text{C}\right)\right)\\ =6562.5\text{J}\end{array}$

Constant rate of heating is $\text{450}\text{J/min}$. The time to increase the temperature of the sample from $-20.0°\text{C}$ to $85°\text{C}$ is calculated as follows:

  $\begin{array}{c}\text{Time}=\left(\frac{6562.5\text{J}}{\text{450}\text{J/min}}\right)\\ \approx 15\min \end{array}$

Substitute $25\text{g}$ for mass and $500\text{J/g}$ for $\Delta H_{\text{vap}}$ in the equation (2) to change the phase of the liquid sample at $85°\text{C}$ to gas sample at $85°\text{C}$.

  $\begin{array}{c}{q}_2=\left(25\text{g}\right)\left(500\text{J/g}\right)\\ =12500\text{J}\end{array}$

Constant rate of heating is $\text{450}\text{J/min}$. The time to change the phase of the liquid sample at $85°\text{C}$ to gas sample at $85°\text{C}$ is calculated as follows:

  $\begin{array}{c}\text{Time}=\left(\frac{12500\text{J}}{\text{450}\text{J/min}}\right)\\ \approx 28\min \end{array}$

Substitute $25\text{g}$ for mass, $0.5\text{J/g}\cdot °\text{C}$ for $c$, $100°\text{C}$ for $T_2$ and $85°\text{C}$ for $T_1$ in the equation (1) to calculate the heat required to increase the temperature of the sample from $85°\text{C}$ to $100°\text{C}$.

  $\begin{array}{c}{q}_3=\left(25\text{g}\right)\left(0.5\text{J/g}\cdot °\text{C}\right)\left(100°\text{C}-85°\text{C}\right)\\ =187.5\text{J}\end{array}$

Constant rate of heating is $\text{450}\text{J/min}$. The time to increase the temperature of the sample from $85°\text{C}$ to $100°\text{C}$ is calculated as follows:

  $\begin{array}{c}\text{Time}=\left(\frac{187.5\text{J}}{\text{450}\text{J/min}}\right)\\ \approx 0.4\min .\end{array}$

The curve of temperature vs. time for the entire heating process is as follows:

Conclusion

The temperature versus time plot for the given sample represents the phase change.