Chapter 21, Problem 84PQ

(a)

To determine

The amount of heat that need to flow into Zan to completely melt his body.

Answer to Problem 84PQ

The amount of heat that need to flow into Zan to completely melt his body is $\underset{\_}{2.33\times {10}^7\text{J}}$.

Explanation of Solution

Zan is initially existing as ice at temperature $-10.00\text{°C}$ ($263.15\text{K}$). The final state is water at $0\text{°C}$ ($273.15\text{}\text{K}$). The mass of the body is $65.88\text{kg}$, the specific heat of ice is $2100\text{J/kg}\cdot \text{K}$, and the latent heat of fusion of ice is $3.33\times {10}^5\text{J/kg}$.

In order Zan to completely melt his body, first the $-10.00\text{°C}$ ice has to become $0\text{°C}$ ice, then the $0\text{°C}$ ice has to become $0\text{°C}$ water.

Write the expression for the heat corresponding to temperature change.

  $Q_1=mc\left(T_f-T_i\right)$                                                                                                     (I)

Here, $Q_1$ is the heat, $m$ is the mass, $c$ is the specific heat, $T_f$ is the final temperature, and $T_i$ is the initial temperature.

Write the expression for the heat required for phase transition from solid to liquid.

  $Q_2=m{L}_F$                                                                                                               (II)

Here, $Q_2$ is the heat required for phase transition, and $L_F$ is the latent heat of fusion.

Write the expression for the total heat absorbed by the body in the process.

  $Q_{\text{tot}}=Q_1+Q_2$                                                                                                        (III)

Conclusion:

Substitute $65.88\text{kg}$ for $m$, $2100\text{J/kg}\cdot \text{K}$ for $c$, $273.15\text{}\text{K}$ for $T_f$ and $263.15\text{}\text{K}$ for $T_i$ in equation (I) to find $Q_1$.

  $\begin{array}{c}{Q}_1=\left(65.88\text{kg}\right)\left(2100\text{J/kg}\cdot \text{K}\right)\left(273.15\text{}\text{K}-263.15\text{}\text{K}\right)\\ =1.38\times {10}^6\text{J}\end{array}$

Substitute $65.88\text{kg}$ for $m$, $3.33\times {10}^5\text{J/kg}$ for $L_F$ in equation (II) to find $Q{2}_{}$.

  $\begin{array}{c}{Q}_2=\left(65.88\text{kg}\right)\left(3.33\times {10}^5\text{J/kg}\right)\\ =2.19\times {10}^7\text{J}\end{array}$

Substitute $1.38\times {10}^6\text{J}$ for $Q_1$, and $2.19\times {10}^7\text{J}$ for $Q_2$ in equation (III) to find $Q_{\text{tot}}$.

  $\begin{array}{c}{Q}_{\text{tot}}=1.38\times {10}^6\text{J}+2.19\times {10}^7\text{J}\\ =2.33\times {10}^7\text{J}\end{array}$

Therefore, the amount of heat that need to flow into Zan to completely melt his body is $\underset{\_}{2.33\times {10}^7\text{J}}$.

(b)

To determine

The amount of additional heat required to make the water form of Zan’s body to become vapor at $100\text{°C}$.

Answer to Problem 84PQ

The amount of additional heat required to make the water form of Zan’s body to start vaporizing at $100\text{°C}$ is $\underset{\_}{2.76\times {10}^7\text{J}}$.

Explanation of Solution

Initial temperature of water is $0\text{°C}$ ($273.15\text{K}$), its mass is $65.88\text{kg}$, and the final temperature is $100.0\text{°C}$ ($373.15\text{K}$).

Write the expression for the heat corresponding to the temperature change.

  $Q=mc\left(T_f-T_i\right)$

Conclusion:

Substitute $65.88\text{kg}$ for $m$, $2100\text{J/kg}\cdot \text{K}$ for $c$, $373.15\text{}\text{K}$ for $T_f$ and $273.15\text{}\text{K}$ for $T_i$ in the above equation to find $Q$.

  $\begin{array}{c}Q=\left(65.88\text{kg}\right)\left(2100\text{J/kg}\cdot \text{K}\right)\left(373.15\text{}\text{K}-273.15\text{}\text{K}\right)\\ =2.76\times {10}^7\text{J}\end{array}$

Therefore, the amount of additional heat required to make the water form of Zan’s body to start vaporizing at $100\text{°C}$ is $\underset{\_}{2.76\times {10}^7\text{J}}$.

(c)

To determine

The amount of additional heat required to make the $100\text{°C}$ water form of Zan’s body to vapor at $100\text{°C}$.

Answer to Problem 84PQ

The amount of additional heat required to make the $100\text{°C}$ water form of Zan’s body to vapor at $100\text{°C}$ is $\underset{\_}{1.486\times {10}^8\text{J}}$.

Explanation of Solution

The mass of water is $65.88\text{kg}$, it undergoes phase transition from liquid to vapor at $100.0\text{°C}$.

Write the expression for the heat corresponding to the phase transition from liquid to vapor.

  $Q=m{L}_V$

Here, $L_V$ is the latent heat of vaporization (for water $L_V=2.256\times {10}^6\text{J/kg}$).

Conclusion:

Substitute $65.88\text{kg}$ for $m$, and $2.256\times {10}^6\text{J/kg}$ for $L_V$ in the above equation to find $Q$.

  $\begin{array}{c}Q=\left(65.88\text{kg}\right)2.256\times {10}^6\text{J/kg}\\ =1.486\times {10}^8\text{J}\end{array}$

Therefore, the amount of additional heat required to make the $100\text{°C}$ water form of Zan’s body to vapor at $100\text{°C}$ is $\underset{\_}{1.486\times {10}^8\text{J}}$.

(d)

To determine

The average kinetic energy of one of water vapor molecule that make up Zan.

Answer to Problem 84PQ

The average kinetic energy of one of water vapor molecule that make up Zan is $\underset{\_}{6.88\times {10}^{-21}}$.

Explanation of Solution

The temperature of the water vapor is $100.0\text{°C}$ ($373.15\text{K}$).

Write the expression for the average kinetic energy of one molecule of water vapor.

  $K_{\text{av}}=\frac{3}{2}{k}_{\text{B}}T$

Here, $K_{\text{av}}$ is the average kinetic energy, $k_{\text{B}}$ is the Boltzmann constant, and $T$ is the temperature.

Conclusion:

Substitute $1.23\times {10}^{-23}\text{J/K}$ for $k_{\text{B}}$, and $373.15\text{K}$ for $T$ in the above equation to find $K_{\text{av}}$.

  $\begin{array}{c}{K}_{\text{av}}=\frac{3}{2}\left(1.23\times {10}^{-23}\text{J/K}\right)\left(373.15\text{K}\right)\\ =6.88\times {10}^{-21}\text{J}\end{array}$

Therefore, the average kinetic energy of one of water vapor molecule that make up Zan is $\underset{\_}{6.88\times {10}^{-21}}$.