Question

Draw a heating curve (such as the one in Figure 12.36 ) for 1 mole of methanol beginning at \(170 \mathrm{~K}\) and ending at \(350 \mathrm{~K}\). Assume that the values given here are constant over the relevant temperature ranges. $$ \begin{array}{ll} \hline \text { Melting point } & 176 \mathrm{~K} \\ \text { Boiling point } & 338 \mathrm{~K} \\ \Delta \mathrm{H}_{\text {fus }} & 2.2 \mathrm{~kJ} / \mathrm{mol} \\ \Delta \mathrm{H}_{\text {vap }} & 35.2 \mathrm{~kJ} / \mathrm{mol} \\ \mathrm{C}_{\mathrm{s}, \text { solid }} & 105 \mathrm{~J} / \mathrm{K} \\ \mathrm{C}_{\mathrm{s}} \text { , liquid } & 81.3 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K} \\ \mathrm{C}_{\mathrm{s}, \text { gas }} & 48 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K} \\ \hline \end{array} $$

Short answer

The heating curve for 1 mole of methanol would show increases in temperature with constant slopes of 105 J/K, 81.3 J/K, and 48 J/K for the solid, liquid, and gas phases, respectively, and horizontal lines at the melting and boiling points indicating phase changes, with absorbed enthalpies of 2.2 kJ and 35.2 kJ.

Step-by-step solution

Plot Initial Phase (Solid Methanol)

Start by plotting a line segment for the solid phase of methanol from the initial temperature of 170 K to the melting point at 176 K. The slope of this line will be determined by the specific heat capacity of solid methanol, which is 105 J/K. Since heat is added at a constant rate, this part of the curve will be a straight line with a positive slope.

Indicate Phase Change from Solid to Liquid

At the melting point of 176 K, methanol undergoes a phase transition from solid to liquid. During this phase change, the temperature remains constant at 176 K until the enthalpy of fusion is absorbed, which is 2.2 kJ/mol. Plot this as a horizontal line segment since temperature does not change while methanol melts.

Plot Liquid Phase

Once the phase change is complete, continue plotting for the liquid phase of methanol, starting at 176 K and ending at the boiling point of 338 K. The slope here will be defined by the specific heat capacity of liquid methanol, which is 81.3 J/mol*K. The line for the liquid phase will again be a straight line but with a different slope than the solid phase due to the different heat capacity.

Indicate Phase Change from Liquid to Gas

At 338 K, the boiling point, methanol transitions from a liquid to a gas. Similarly to the melting point, the temperature remains constant during this phase change at 338 K while the enthalpy of vaporization, 35.2 kJ/mol, is absorbed. Represent this as another horizontal line on the heating curve.

Plot Gas Phase

Following the phase change to gas, plot the final segment for the gas phase from 338 K up to the final temperature of 350 K. The slope of the gas phase is determined by the specific heat capacity of methanol gas, which is 48 J/mol*K. This section of the curve will be the third straight line with a positive slope but will be relatively steeper than the liquid phase due to the lower heat capacity of the gas.

Complete the Heating Curve

Combine all the plotted segments to form the complete heating curve. The curve should consist of three straight lines with different slopes corresponding to the solid, liquid, and gas phases, as well as two horizontal lines representing the phase changes at the melting and boiling points.

Key concepts

Phase Change

Understanding phase change is essential when interpreting a heating curve. A phase change is the transformation of a substance from one state of matter—solid, liquid, or gas—to another. The most common phase changes are melting (solid to liquid), freezing (liquid to solid), vaporization (liquid to gas), condensation (gas to liquid), sublimation (solid to gas), and deposition (gas to solid).

On a heating curve, a phase change is represented by a horizontal line, showing that the temperature does not increase despite the ongoing addition of heat. This is because the energy added to the system is used to break intermolecular bonds rather than raise the temperature. Only once the phase change is complete will the temperature start to rise again, reflecting the continued input of heat into the system now acting to increase the kinetic energy of particles in the new phase.

Enthalpy of Fusion

The enthalpy of fusion is the amount of energy required to change a substance from solid to liquid state at its melting point. It's also known as the heat of fusion. During this phase change, the system's temperature remains constant because the energy supplied—quantified by the enthalpy of fusion—is used to overcome the attractions holding the particles in the solid-state, rather than to increase their kinetic energy. In the heating curve for methanol, the enthalpy of fusion is represented by the horizontal line at the melting point (176 K), with a value of 2.2 kJ/mol.

This energy barrier must be overcome for the substance to transition into a liquid, and it's a unique characteristic for each material. For example, the enthalpy of fusion for water is approximately 6 kJ/mol, significantly higher than that for methanol, reflecting the stronger hydrogen bonds in water that must be broken for it to melt.

Specific Heat Capacity

The concept of specific heat capacity is another critical element in understanding a heating curve. It refers to the quantity of heat needed to raise the temperature of one gram (or, as in this case, one mole) of a substance by one degree Celsius (or one Kelvin). Specific heat capacity is a property intrinsic to a substance's phase; for instance, liquid methanol has a different specific heat capacity than solid or gaseous methanol.

In the heating curve for methanol, the specific heat capacities given for the solid, liquid, and gas phases determine the slopes of the curve in their respective temperature ranges. As the specific heat capacity decreases from solid (105 J/K) to liquid (81.3 J/mol*K) and to gas (48 J/mol*K), it takes less heat to raise the substance's temperature, resulting in steeper slopes on the heating curve. High specific heat capacity indicates a substance that resists changes in temperature when heat is added or removed, a characteristic that has wide-ranging implications from cooking to climate science.