17.12: MultiStep Problems with Changes of State
Which takes more heat – melting or boiling?
You have a cube of ice. Which process will take more energy – the melting of that ice cube or the conversion of the water to steam? The short answer is that more energy is needed to convert the water to steam. The long answer is really a question: how do you get from one point to the other? What is the temperature of the ice? What is the mass of that ice cube? A lot goes into taking the material from the starting point to the endpoint.
MultiStep Problems with Changes of State
Heating curves show the phase changes that a substance undergoes as heat is continuously absorbed.
Heating curve of water.
The specific heat of a substance allows us to calculate the heat absorbed or released as the temperature of the substance changes. It is possible to combine that type of problem with a change of state to solve a problem involving multiple steps. The Figure above shows ice at 30°C being converted in a fivestep process to gaseous water (steam) at 140°C. It is now possible to calculate the heat absorbed during that entire process. The process and the required calculation is summarized below.
 Ice is heated from 30°C to 0°C. The heat absorbed is calculated by using the specific heat of ice and the equation \begin{align*}\Delta H = c_p \times m \times \Delta T\end{align*}
ΔH=cp×m×ΔT .  Ice is melted at 0°C. The heat absorbed is calculated by multiplying the moles of ice by the molar heat of fusion.
 Water at 0°C is heated to 100°C. The heat absorbed is calculated by using the specific heat of water and the equation \begin{align*}\Delta H = c_p \times m \times \Delta T\end{align*}
ΔH=cp×m×ΔT .  Water is vaporized to steam at 100°C. The heat absorbed is calculated by multiplying the moles of water by the molar heat of vaporization.
 Steam is heated from 100°C to 140°C. The heat absorbed is calculated by using the specific heat of steam and the equation \begin{align*}\Delta H = c_p \times m \times \Delta T\end{align*}
ΔH=cp×m×ΔT .
Sample Problem: MultiStep Problems using a Heating Curve
Calculate the total amount of heat absorbed (in kJ) when 2.00 mol of ice at 30.0°C is converted to steam at 140.0°C. The required specific heats can be found in the table in "Heat Capacity and Specific Heat".
Step 1: List the known quantities and plan the problem.
Known
 2.00 mol ice = 36.04 g ice

\begin{align*}c_p (\text{ice}) = 2.06 \text{ J/g}^\circ \text{C}\end{align*}
cp(ice)=2.06 J/g∘C 
\begin{align*}c_p(\text{water}) = 4.18 \text{ J/g}^\circ \text{C}\end{align*}
cp(water)=4.18 J/g∘C 
\begin{align*}c_ p(\text{steam}) = 1.87 \text{ J/g}^\circ \text{C}\end{align*}
cp(steam)=1.87 J/g∘C 
\begin{align*}\Delta H_{\text{fus}} = 6.01 \text{ kJ/mol}\end{align*}
ΔHfus=6.01 kJ/mol 
\begin{align*}\Delta H_{\text{vap}}= 40.7 \text{ kJ/mol}\end{align*}
ΔHvap=40.7 kJ/mol
Unknown

\begin{align*}\Delta H_{\text{total}}= ? \text{ kJ}\end{align*}
ΔHtotal=? kJ
Follow the steps previously described. Note that the mass of the water is needed for the calculations that involve the specific heat, while the moles of water is needed for the calculations that involve changes of state. All heat quantities must be in kilojoules so that they can be added together to get a total for the fivestep process.
Step 2: Solve.

\begin{align*}\Delta H_1=2.06 \text{ J/g}^\circ \text{C} \times 36.04 \text{ g} \times 30^\circ \text{C} \times \frac{1 \text{ kJ}}{1000 \text{ J}}=2.23 \text{ kJ}\end{align*}
ΔH1=2.06 J/g∘C×36.04 g×30∘C×1 kJ1000 J=2.23 kJ 
\begin{align*}\Delta H_2=2.00 \text{ mol} \times \frac{6.01 \text{ kJ}}{1 \text{ mol}}=12.0 \text{ kJ}\end{align*}
ΔH2=2.00 mol×6.01 kJ1 mol=12.0 kJ 
\begin{align*}\Delta H_3=4.18 \text{ J/g}^\circ \text{C} \times 36.04 \text{ g} \times 100^\circ \text{C} \times \frac{1 \text{ kJ}}{1000 \text{ J}}=15.1 \text{ kJ}\end{align*}
ΔH3=4.18 J/g∘C×36.04 g×100∘C×1 kJ1000 J=15.1 kJ  \begin{align*}\Delta H_4=2.00 \text{ mol} \times \frac{40.7 \text{ kJ}}{1 \text{ mol}}=81.4 \text{ kJ}\end{align*}
 \begin{align*}\Delta H_5=1.87 \text{ J/g}^\circ \text{C} \times 36.04 \text{ g} \times 40^\circ \text{C} \times \frac{1 \text{ kJ}}{1000 \text{ J}}=2.70 \text{ kJ} \end{align*}
\begin{align*}\Delta H_{\text{total}}= \Delta H_1+ \Delta H_2+\Delta H_3+ \Delta H_4+ \Delta H_5=113.4 \text{ kJ}\end{align*}
Step 3: Think about your result.
The total heat absorbed as the ice at 30°C is heated to steam at 140°C is 113.4 kJ. The largest absorption of heat comes during the vaporization of the liquid water.
Summary
 Multistep calculations for changes of state are described.
Practice
Questions
Watch the video below and answer the following questions:
http://www.youtube.com/watch?v=zz4KbvF_X0
 What is the specific heat of ice?
 Why do you need the mass of ice?
 What happens if you leave out the energy for a phase change?
Review
Questions
 Why are two different sets of units used?
 What other units problem do you need to be aware of?
 What would you need to know to do calculations like this for acetone?
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Image Attributions
 Perform energy calculations involving multiple changes of state.