Thermochemistry: State Functions, Enthalpy, and Hess's Law

State Functions and Path Dependent Functions

Definition and Examples

In thermochemistry, a state function is a property of a system that depends only on its current state, not on the path taken to reach that state. State functions are typically denoted by capital letters in chemistry.

• Examples of State Functions: Pressure (P), Volume (V), Internal Energy (U), Enthalpy (H), Entropy (S)

• Path Dependent Functions: Properties that depend on the process or path taken, such as heat (q) and work (w), are denoted by lowercase letters.

Analogy: The altitude of a mountain peak is a state function—it depends only on the difference in elevation between the base and the peak, not on the trail taken. The trail itself is a path dependent function.

First Law of Thermodynamics

Conservation of Energy

The First Law of Thermodynamics states that energy can be converted from one form to another, but the total energy of the universe remains constant. This is known as the principle of conservation of energy.

Internal Energy (U)

Definition and Transfer

Internal Energy (U) is a state function representing the total kinetic and potential energy within a system. According to the first law, the internal energy of a system changes only if energy is transferred in or out as heat (q) or work (w).

• Heat (q): Energy transferred due to temperature difference.

• Work (w): Energy transferred by macroscopic motion (e.g., compression, expansion).

Equation:

$\Delta U = q + w$

• Heat transfer in (endothermic): $+q$

• Heat transfer out (exothermic): $-q$

• Work transfer in: $+w$

• Work transfer out: $-w$

Example: If a gas absorbs 9 J of heat and does 25 J of work by expanding, $\Delta U = 9\,\text{J} + (-25\,\text{J}) = -16\,\text{J}$.

Enthalpy (H)

Definition and Significance

Enthalpy (H) is a state function defined as:

$H = U + PV$

While the absolute value of enthalpy has limited physical significance, changes in enthalpy ($\Delta H$) are crucial for understanding the heat exchanged during chemical reactions.

Changes in Enthalpy ($\Delta H$)

• $\Delta H = \Delta U + \Delta(PV)$

• Under constant pressure: $\Delta H = q_p$ (heat at constant pressure)

Changes in enthalpy allow us to calculate the heat of reaction by knowing the initial and final states of the system.

Calorimetry

Measuring Heat Changes

• Coffee cup calorimeter (constant pressure): Measures $q_p = \Delta H$

• Bomb calorimeter (constant volume): Measures $q_v = \Delta U$

Thermochemical Equations

Energy Changes in Chemical Reactions

Thermochemical equations show the enthalpy change associated with a chemical reaction. For example:

$\mathrm{CH}_4(g) + 2\mathrm{O}_2(g) \rightarrow \mathrm{CO}_2(g) + 2\mathrm{H}_2O(g) \qquad \Delta H^\circ = -802\,\text{kJ}$

• Negative $\Delta H$ indicates an exothermic reaction (energy released).

• Positive $\Delta H$ indicates an endothermic reaction (energy absorbed).

• $\Delta H$ scales with the reaction coefficients.

Example Calculation: How much energy is released when 128.5 g of methane is combusted?

$128.5\,\text{g CH}_4 \times \frac{1\,\text{mol CH}_4}{16.0\,\text{g CH}_4} \times \frac{-802\,\text{kJ}}{1\,\text{mol CH}_4} = -6440\,\text{kJ}$

Standard Molar Enthalpies of Formation ($\Delta H_f^\circ$)

Definition and Table

The standard molar enthalpy of formation ($\Delta H_f^\circ$) is the enthalpy change when 1 mol of a substance is formed from its elements in their most stable states under standard conditions (P = 1 bar, T = 298 K, concentration = 1 mol/L).

Substance	$\Delta H_f^\circ$ (kJ/mol)
H2O (g)	-242
CO2 (g)	-393
CH3OH (g)	-201.5
O2 (g)	0.0

Additional info: See the full table for more substances and their standard enthalpies of formation.

Formation Reactions

Writing Formation Equations

A formation reaction creates 1 mol of a compound from its elements in their standard states. For example:

• $\mathrm{H}_2(g) + \frac{1}{2}\mathrm{O}_2(g) \rightarrow \mathrm{H}_2O(l)$, $\Delta H_f^\circ = -242\,\text{kJ}$

• $\mathrm{C(graphite)} + \mathrm{O}_2(g) \rightarrow \mathrm{CO}_2(g)$, $\Delta H_f^\circ = -393\,\text{kJ}$

By definition, $\Delta H_f^\circ = 0$ for elements in their standard states (e.g., O2(g), Na(s), C(graphite)).

Hess's Law

Calculating Enthalpy Changes

Hess's Law states that if a reaction is the sum of two or more reactions, the overall enthalpy change ($\Delta H$) is the sum of the enthalpy changes of the individual reactions. This is possible because enthalpy is a state function.

Example:

• Given: (1) $2\mathrm{H}_2(g) + \mathrm{N}_2(g) \rightarrow \mathrm{N}_2\mathrm{H}_4(g)$, $\Delta H_1 = +95.4\,\text{kJ}$ (2) $2\mathrm{NH}_3(g) \rightarrow \mathrm{N}_2\mathrm{H}_4(g) + \mathrm{H}_2(g)$, $\Delta H_2 = 187.6\,\text{kJ}$

• To find $\Delta H$ for $3\mathrm{H}_2(g) + \mathrm{N}_2(g) \rightarrow 2\mathrm{NH}_3(g)$, sum the appropriate reactions.

Using Standard Enthalpy Values

General Formula

For any reaction, the enthalpy change can be calculated using:

$\Delta H^\circ = \sum \Delta H_f^\circ(\text{products}) - \sum \Delta H_f^\circ(\text{reactants})$

Example: Calculate $\Delta H^\circ$ for the combustion of naphthalene, C10H8(l), given $\Delta H^\circ$[CO2(g)] = -393.5 kJ/mol and $\Delta H^\circ$[H2O(l)] = -285.7 kJ/mol.