Heating Curves

Heat and temperature changes

Heat of Vaporization

Energy is required to convert a liquid into a gas (i.e. vaporization; a phase change). The energy required is called the heat (or enthalpy) of vaporization and is denoted as ΔH_vap and is generally given as energy per amount of substance (often kJ mol^–1). If enthalpy is given as a “per mole” quantity, we use the following equation to determine the amount of heat (q) required based on the amount of substance (n in moles) present.

\[q = n\Delta H_{\mathrm{vap}}\]

Water

\[\mathrm{H_2O}(l) \longrightarrow \mathrm{H_2O}(g) \qquad \Delta H_{\mathrm{vap}} = 44.01~\mathrm{kJ~mol^{-1}}\]

Substance	ΔH° (kJ mol–1)
H2O(l)	-285.83
H2O(g)	-241.82

Thermodynamic values from openStax

The enthalpy of the process is determined as follows:

\[\begin{align*} \Delta H_{\mathrm{vap}} &= \Delta H_{\mathrm{final}} - \Delta H_{\mathrm{initial}} \\ &= -241.82~\mathrm{kJ~mol^{-1}} - -285.83~\mathrm{kJ~mol^{-1}} \\ &= 44.01~\mathrm{kJ~mol^{-1}} \end{align*}\]

Exercise: Determine the amount of energy (q in kJ) required to transform the follow amounts of water from a liquid to a gas.

Amount	q (kJ)
1 mol
2 mol
0.5 mol
100 g

Example: 1 mol

\[\begin{align*} q &= n\Delta H \\ &= 1~\mathrm{mol} \left ( \frac{44.01~\mathrm{kJ}}{\mathrm{mol}} \right )\\ &= 44.01~\mathrm{kJ~mol^{-1}} \end{align*}\]

Methanol

\[\mathrm{CH_3OH}(l) \longrightarrow \mathrm{CH_3OH}(g)\]

Substance	ΔH° (kJ mol–1)
CH3OH(l)	-239.2
CH3OH(g)	-201

Thermodynamic values from openStax(Paul Flowers 2022)

Exercise: Determine the heat of vaporization (in kJ mol^–1) for methanol.

\[\Delta H_{\mathrm{vap}} = \phantom{38.2~\mathrm{kJ~mol^{-1}}}\]

Exercise: Determine the amount of energy (q in kJ) required to transform the follow amounts of water from a liquid to a gas.

Amount	q (kJ)
1 mol
2 mol
0.5 mol
100 g

Heat of Fusion

The heat of fusion is the amount of energy required to turn an amount of substance from a solid to a liquid (i.e. melting; a phase change) and is given in units similar to that of heat of vaporization.

\[q = n\Delta H_{\mathrm{fus}}\]

Water

\[\mathrm{H_2O}(s) \longrightarrow \mathrm{H_2O}(l) \qquad \Delta H_{\mathrm{fus}} = 6.01~\mathrm{kJ~mol^{-1}}\]

Exercise: Determine the amount of energy (q in kJ) required to transform the follow amounts of water from a solid to a liquid

Amount	q (kJ)
1 mol
2 mol
0.5 mol
100 g

Heating a Substance

Heat is required to heat a substance from one temperature to another without undergoing a phase change. This can be determined across a temperature range (ΔT) for an amount of substance (m) using the specific heat for that substance in the appropriate state (c in J g^–1 °C^–1).

\[q = mc\Delta T\] Below are the specific heats (in J g^–1 °C^–1) for water and methanol.

Substance	cs	cl	cg
Water	2.09	4.184	1.84
Methanol		2.531	1.376

See more enthalpies and specific heats here.

Water

Exercise: Determine the amount of heat required (in kJ) to heat the following amounts of water from 25 °C to 50 °C.

Amount	q (kJ)
1 mol
2 mol
0.5 mol
100 g

Note: Which specific heat will you use for this problem? What state of matter is water in for the specified temperature range? Is it critical to know the melting and boiling points of water to make the determination?

Example:

\[\begin{align*} q &= mc_{\mathrm{l}}\Delta T \\ &= 1~\mathrm{mol} \left ( \frac{18.02~\mathrm{g}}{\mathrm{mol}} \right ) \left ( \frac{4.184~\mathrm{J}}{\mathrm{g}~^{\circ}\mathrm{C}} \right ) \left ( 50~^{\circ}\mathrm{C} - 25~^{\circ}\mathrm{C} \right ) \left ( \frac{\mathrm{kJ}}{10^3~\mathrm{J}} \right )\\ &= 1.88~\mathrm{kJ~mol^{-1}} \end{align*}\]

Note: The amount of substance (given in moles) had to be converted to mass (in g) since the specific heat was given in g.

Methanol

Repeat the process for ethanol (CH₃OH) across a temperature range of 25 °C to 50 °C.

Amount	q (kJ)
1 mol
2 mol
0.5 mol
100 g

Note: Which specific heat will you use for this problem? What state of matter is water in for the specified temperature range? You have to know the melting and boiling points of methanol.

Additional Practice:

Repeat the exercise for water and methanol across the following temperature range:

• –110 °C to –80 °C

Combining it All

We can combine each individual concept into an overall heating curve problem that involves one or more phase change as well as the heating of a substance in a particular phase.

Water

Determine the amount of heat (in kJ mol^–1) required to heat 500.0 g water from –50 °C to 150 °C.

\[\mathrm{H_2O}(\phantom{s}) \longrightarrow \mathrm{H_2O}(\phantom{g})\]

Fill in the phase labels for this problem.

Heating the solid

\[\begin{equation*} \mathrm{H_2O}(\phantom{s})~\longrightarrow \mathrm{H_2O}(\phantom{s}) \\[1.5ex] T_{\mathrm{initial}} = -50.0 ^{\circ}\mathrm{C} ~\longrightarrow~ T_{\mathrm{melting}} = 0.0~^{\circ}\mathrm{C} \end{equation*}\]

Heat the solid up to its melting point. The normal melting point of water is 0.0 °C.

\[\begin{align*} q_1 &= mc\Delta T \end{align*}\]

Melting the Solid

\[\begin{equation*} \mathrm{H_2O}(\phantom{s})~\longrightarrow \mathrm{H_2O}(\phantom{l}) \\[1.5ex] T_{\mathrm{melting}} = 0.0~^{\circ}\mathrm{C} \end{equation*}\]

A phase transition occurs at the melting point of water.

\[\begin{align*} q_2 &= n\Delta H_{\mathrm{fus}} \end{align*}\]

Heating the liquid

\[\begin{equation*} \mathrm{H_2O}(\phantom{l})~\longrightarrow \mathrm{H_2O}(\phantom{l}) \\[1.5ex] T_{\mathrm{melting}} = 0.0~^{\circ}\mathrm{C} ~\longrightarrow~ T_{\mathrm{boiling}} = 100.0~^{\circ}\mathrm{C} \end{equation*}\]

Heat the solid up to its melting point. The normal melting point of water is 0.0 °C.

\[\begin{align*} q_3 &= mc\Delta T \end{align*}\]

Vaporizing the Liquid

\[\begin{equation*} \mathrm{H_2O}(\phantom{l})~\longrightarrow \mathrm{H_2O}(\phantom{g}) \\[1.5ex] T_{\mathrm{boiling}} = 100.0~^{\circ}\mathrm{C} \end{equation*}\]

A phase transition occurs at the boiling point of water.

\[\begin{align*} q_4 &= n\Delta H_{\mathrm{vap}} \end{align*}\]

Heating the gas

\[\begin{equation*} \mathrm{H_2O}(\phantom{l})~\longrightarrow \mathrm{H_2O}(\phantom{l}) \\[1.5ex] T_{\mathrm{boiling}} = 100.0~^{\circ}\mathrm{C} ~\longrightarrow~ T_{\mathrm{final}} = 150.0~^{\circ}\mathrm{C} \end{equation*}\]

Heat the gas up from the normal point boiling point of water to the final temperature of 150 °C.

\[\begin{align*} q_5 &= mc\Delta T \end{align*}\]

Combine all the heats

\[q_{\mathrm{tot}} = q_1 + q_2 + q_3 + q_4 + q_5\]

Methanol

Repeat the exercise for methanol. Note that you will need to look up the melting and boiling points first!

Additional Practice

1. Handout: Acetone
2. More practice problems