09/14 Thermochemistry

Energy Transfer and the First Law

Thermochemistry examines how energy changes accompany chemical reactions and physical processes. A system is the portion of the universe being studied, such as a reacting mixture; the surroundings include everything outside the system. Energy can cross the system boundary as heat or work, but total energy is conserved.

• Heat is energy transferred because of a temperature difference.

• Work is energy transferred when a force causes matter to move. For a reacting gas, the most common type is pressure–volume work.

• Internal energy is the total microscopic energy of the system.

Using the chemistry sign convention:

• $q>0$: the system absorbs heat.

• $q<0$: the system releases heat.

• $w>0$: work is done on the system.

• $w<0$: work is done by the system.

The first law of thermodynamics is:

For pressure–volume work:

During expansion, $\Delta V>0$, so $w<0$: the system does work on the surroundings. During compression, $\Delta V<0$, so $w>0$.

For example, if a system absorbs $125\ \mathrm{J}$ of heat and does $40\ \mathrm{J}$ of work on the surroundings, then:

The internal energy increases by $85\ \mathrm{J}$. Energy, heat, and work are measured in joules, while chemical reaction energies are commonly reported in kilojoules.

Takeaway: Energy is conserved, and the first law tracks how energy enters or leaves a system as heat and work.

Temperature, Heat Capacity, and Specific Heat

Temperature describes the average kinetic energy of particles. It is not the same as heat: heat is energy in transit, whereas temperature describes the thermal state of matter.

The heat required to change the temperature of a sample is:

Here, $m$ is mass, $c$ is specific heat capacity, and $\Delta T=T_{\text{final}}-T_{\text{initial}}$. Specific heat capacity is the heat required to raise the temperature of $1\ \mathrm{g}$ of a substance by $1\ ^\circ\mathrm{C}$ or $1\ \mathrm{K}$. For liquid water, a commonly used value is:

A positive temperature change gives positive $q$, meaning that the sample absorbs heat. A negative temperature change gives negative $q$, meaning that the sample releases heat.

For $75.0\ \mathrm{g}$ of water warming from $20.0\ ^\circ\mathrm{C}$ to $35.0\ ^\circ\mathrm{C}$:

The water absorbs $4.71\ \mathrm{kJ}$.

Takeaway: Use $q=mc\Delta T$ when the mass, specific heat, and temperature change are known.

Enthalpy and Reaction Heat

Enthalpy is defined as:

For a process at constant pressure in which the only work is pressure–volume work:

This relationship makes enthalpy particularly useful for reactions carried out in open containers at approximately constant atmospheric pressure.

• An exothermic process releases heat, so $q_p=\Delta H<0$.

• An endothermic process absorbs heat, so $q_p=\Delta H>0$.

A thermochemical equation must contain a balanced chemical equation and the physical states of all substances because the enthalpy change depends on the amounts and states involved. For example:

The value applies to the reaction exactly as written. Reversing the equation changes the sign of $\Delta H$, and multiplying every coefficient by a factor multiplies $\Delta H$ by the same factor. Enthalpy is an extensive property, so the energy change scales with the amount of material.

Takeaway: At constant pressure, reaction heat is represented by $\Delta H$; its sign identifies whether the process is exothermic or endothermic.

Calorimetry and Measuring Reaction Heat

Calorimetry determines heat transfer from measured temperature changes. In an insulated experiment, energy released by the reaction is absorbed by the solution and/or the calorimeter:

Therefore:

Coffee-cup calorimetry

A coffee-cup calorimeter operates approximately at constant pressure and is commonly used for aqueous reactions. If the calorimeter heat capacity is negligible:

If the calorimeter absorbs significant heat, include its heat capacity:

For a reaction that warms $100.0\ \mathrm{g}$ of solution from $21.5\ ^\circ\mathrm{C}$ to $28.0\ ^\circ\mathrm{C}$, using $c=4.184\ \mathrm{J\ g^{-1}\ ^\circ C^{-1}}$:

Thus:

The reaction is exothermic. To obtain a molar enthalpy, divide the reaction heat by the moles of the limiting reactant or by the moles of reaction represented by the balanced equation.

Bomb calorimetry

A bomb calorimeter operates at constant volume and is useful for combustion reactions. Because $\Delta V\approx0$, pressure–volume work is negligible, so the measured heat is related most directly to internal energy:

The reaction heat is calculated from the heat absorbed by the water and the bomb:

Takeaway: Calorimetry uses the temperature change of the surroundings to determine the heat released or absorbed by a reaction.

Hess’s Law and Reaction Pathways

Hess’s law follows from the fact that enthalpy is a state function: its change depends only on the initial and final states, not on the pathway between them.

To combine thermochemical equations:

1. Reverse an equation when necessary; change the sign of $\Delta H$.

2. Multiply an equation by a factor when necessary; multiply $\Delta H$ by the same factor.

3. Add the equations and their enthalpy changes.

4. Cancel substances that appear on both sides.

For example, consider:

Adding the equations cancels $\mathrm{CO(g)}$:

Thus:

Takeaway: Manipulate chemical equations first, then apply the same operations to their enthalpy changes.

Standard Enthalpies of Formation

The standard enthalpy of formation $\Delta H_f^\circ$ is the enthalpy change when exactly $1\ \mathrm{mol}$ of a compound forms from its constituent elements in their standard states.

The formation reaction must produce exactly one mole of the compound. For carbon dioxide:

The standard enthalpy of formation of an element in its most stable standard state is defined as zero. Examples include:

• $\Delta H_f^\circ[\mathrm{O_2(g)}]=0$

• $\Delta H_f^\circ[\mathrm{C(graphite)}]=0$

• $\Delta H_f^\circ[\mathrm{H_2(g)}]=0$

Standard-state data generally refer to a pressure of $1\ \mathrm{bar}$, and tabulated values are commonly given at $298.15\ \mathrm{K}$ unless another temperature is specified.

Takeaway: Formation enthalpy values describe formation of exactly one mole from elements in their standard states, with elemental reference values set to zero.

Calculating Reaction Enthalpy from Formation Data

For a balanced reaction, calculate the standard reaction enthalpy by subtracting the total formation enthalpy of the reactants from that of the products:

The coefficient $n$ is the stoichiometric coefficient in the balanced equation. Include every substance, even those with a formation enthalpy of zero.

For methane combustion:

Using $\Delta H_f^\circ[\mathrm{CH_4(g)}]=-74.8\ \mathrm{kJ\ mol^{-1}}$, $\Delta H_f^\circ[\mathrm{CO_2(g)}]=-393.5\ \mathrm{kJ\ mol^{-1}}$, $\Delta H_f^\circ[\mathrm{H_2O(l)}]=-285.8\ \mathrm{kJ\ mol^{-1}}$, and $\Delta H_f^\circ[\mathrm{O_2(g)}]=0$:

Therefore, combustion of $1\ \mathrm{mol}$ of methane as written releases $890.3\ \mathrm{kJ}$. For $0.250\ \mathrm{mol}$ of methane:

The negative sign indicates that the system releases energy.

Takeaway: Balance the equation, multiply each formation enthalpy by its coefficient, and apply products minus reactants.

A Quantitative Problem-Solving Strategy

A reliable thermochemistry calculation follows a consistent sequence:

1. Write and balance the chemical equation, including physical states.

2. Identify the requested quantity: heat $q$, internal-energy change $\Delta U$, enthalpy change $\Delta H$, or energy per mole.

3. Select the appropriate relationship, such as $\Delta U=q+w$, $q=mc\Delta T$, $q_{\text{rxn}}=-q_{\text{surroundings}}$, Hess’s law, or the formation-enthalpy equation.

4. Convert units consistently, such as grams to moles or joules to kilojoules.

5. Track signs carefully: heat absorbed by the system is positive, and heat released by the system is negative.

6. Apply stoichiometry to scale the energy to the amount reacting.

7. Check whether the result is chemically reasonable. Combustion and many acid–base neutralizations are usually exothermic, but the sign must ultimately be determined from data.

8. Report appropriate significant figures and units, such as $\mathrm{kJ\ mol^{-1}}$ for a molar reaction enthalpy or $\mathrm{kJ}$ for the reaction as written.

Final checklist: A complete answer should show the balanced equation, the chosen relationship, substitutions with units, the sign interpretation, and the final result at an appropriate precision.