# Heating Curves

Lecture Notes

Heat and temperature changes

Eric Van Dornshuld https://dornshuld.chemistry.msstate.edu (Mississippi State University)https://chemistry.msstate.edu
2022-02-15

## 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 ΔHvap 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

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

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

### 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

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*}

### 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

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.

$\mathrm{H_2O}(\phantom{s}) \longrightarrow \mathrm{H_2O}(\phantom{g})$

#### Heating the solid

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

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

#### Melting the Solid

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

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

#### Heating the liquid

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

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

#### Vaporizing the Liquid

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

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

#### Heating the gas

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

\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!

Paul Flowers, Richard Langley, Klaus Theopold. 2022. “Chemistry 2e.” OpenStax. https://openstax.org/books/chemistry-2e/pages/g-standard-thermodynamic-properties-for-selected-substances.

### Citation

Dornshuld (2022, Feb. 15). Heating Curves. Retrieved from https://dornshuld.chemistry.msstate.edu/notes/ch10/heating-curve.html
@misc{dornshuld2022heating,
}