Chapter 12.2, Problem 31SSC

To determine

To Calculate:The amount of thermal energy needed to raise the temperature of ice.

Answer to Problem 31SSC

  $20.8\text{kJ}$

Explanation of Solution

Given:

Mass of ice, $m=50\text{g}$

Initial temperature of ice, $T_1=-{20.0}^{\text{o}}\text{C}$

Final temperature of water, $T_2={10.0}^{\text{o}}\text{C}$

Formula Used:

Heat required to raise the temperature of a substance can be obtained by:

  $Q=mc\Delta T$

Where, m is the mass, c is the specific heat capacity and $\Delta T$ is the difference in temperature.

Heat required to melt the ice:

  $Q\text{'}=m{L}_f$

Where, m is the mass of ice and $L_f$ is the latent heat of fusion.

Calculations:

1. Amount of heat required to the temperature of the ice from $T_1=-{20.0}^{\text{o}}\text{C}$ to $T\text{'}={0.0}^{\text{o}}\text{C}$ is:

Specific heat of ice, $c_i=2.03\text{J/g°C}$

  $\begin{array}{c}{Q}_1=m{c}_i\Delta T\\ =\left(50\text{g}\right)\left(2.03\text{J/g°C}\right)\left(0°\text{C}-\left(-20°\text{C}\right)\right)\\ =2030\text{J}\end{array}$

Latent heat of fusion of ice, $L_f=334\text{J/g}$

Heat required to melt the ice:

  $\begin{array}{c}{Q}_2=m{L}_f\\ =50\text{g}\times \text{334J/g}\\ \text{=16700}\text{J}\end{array}$

Specific heat capacity of water is $c=4.18\text{J/g}\cdot \text{°C}$

Heat required to raise the temperature from $0°\text{C}$ to $10°\text{C}$ .

Specific heat capacity of water is, $c_w=4.184\text{J/g}\cdot \text{°C}$

  $\begin{array}{c}{Q}_3=m{c}_w\Delta T\\ =\left(50\text{g}\right)\left(4.184\text{J/g}\cdot \text{°C}\right)\left(10°\text{C-0}°\text{C}\right)\\ =2092\text{J}\end{array}$

The total amount of thermal energy needed is:

  $\begin{array}{c}Q=Q_1+Q_2+Q_3\\ =2030\text{J}+16700\text{J}+2092\text{J}\\ \text{=20822}\text{J}\\ =20.8\text{kJ}\end{array}$

Conclusion:The amount of thermal energy needed to raise the temperature of ice is: $20.8\text{kJ}$