Chemical Energetics Summary - Definitions and Formulas

The Fundamental Principle of Hess' Law and Heat Change Determination

Hess' Law is the central governing principle of chemical energetics, stating that the enthalpy change accompanying a chemical reaction is identical regardless of the specific route or intermediate steps by which the chemical change occurs, provided that the initial and final states of the chemical system remain the same.

To determine the enthalpy change ($\Delta H$) experimentally, one must first calculate the heat change ($q$). Assuming 100% efficiency, the formula is expressed as:

$q = m_{soln} C_{soln} \Delta T + C_{calorimeter} \Delta T$

An alternative expression for this calculation, using volume, is:

$q = V_{soln} C_{soln} \Delta T + C_{calorimeter} \Delta T$

In these equations, $m_{soln}$ refers to the mass of the solution in grams ($g$), while $V_{soln}$ represents the total volume of the solution in cubic centimeters ($cm^3$). The term $C_{soln}$ denotes the specific heat capacity of the solution, typically measured in $J\,g^{-1}\,K^{-1}$ or $J\,cm^{-3}\,K^{-1}$. The term $C_{calorimeter}$ represents the heat capacity of the calorimeter itself, measured in $J\,K^{-1}$. If the system is not perfectly efficient, the total heat change ($q$) must be adjusted using the following equation:

$\text{total heat change, } q = \frac{100}{\% \text{ efficiency}} \times [(m_{soln} C_{soln} \Delta T) + (C_{calorimeter} \Delta T)]$

For many standard calculations, the heat capacity of the calorimeter ($C_{calorimeter}$) is assumed to be zero. Once the heat change is determined, the enthalpy change is calculated as:

$\Delta H = \pm \frac{q}{n}$

Here, $n$ represents the amount of substance in moles ($mol$) as defined by the specific enthalpy change being studied.

Standard Enthalpy Change of Reaction and Calculation Methods

The standard enthalpy change of reaction ($\Delta H_r^{\circ}$) is defined as the heat change when molar quantities of reactants, exactly as specified in the balanced chemical equation, react completely to form products under standard conditions of $298\,K$ and $1\,bar$. This can be determined experimentally using the limiting reagent as follows:

$\Delta H_r = \pm \frac{\text{heat change, } q}{n_{limiting\,reagent}} \times \text{stoichiometric coefficient of limiting reagent}$

Hess' Law allows for the calculation of $\Delta H_r^{\circ}$ using alternative thermodynamic data. If the standard enthalpy changes of combustion ($\Delta H_c^{\circ}$) are provided for all reactants and products, the formula is:

$\Delta H_r = \sum m \Delta H_c^{\circ}(\text{reactants}) - \sum n \Delta H_c^{\circ}(\text{products})$

If the standard enthalpy changes of formation ($\Delta H_f^{\circ}$) are provided, the formula shifts to:

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

In both cases, $m$ and $n$ represent the stoichiometric coefficients found in the thermochemical equation. Additionally, bond energy (BE) data can be utilized to estimate the reaction enthalpy:

$\Delta H_r = \sum E(\text{all bonds in reactants}) - \sum E(\text{all bonds in products})$

Specific Enthalpy Changes of Neutralisation, Combustion, and Formation

The standard enthalpy change of neutralisation ($\Delta H_n^{\circ}$) is an exothermic process defined as the heat evolved when $1\,mol$ of water is formed from the reaction of an acid and an alkali under standard conditions ($298\,K$ and $1\,bar$). This value is always negative. For example, the reaction between aqueous sodium hydroxide and sulfuric acid is represented as:

$NaOH(aq) + \frac{1}{2}H_2SO_4(aq) \rightarrow \frac{1}{2}Na_2SO_4(aq) + H_2O(l)$

$\Delta H_n = -57.3\,kJ\,mol^{-1}$

Experimentally, it is determined by:

$\Delta H_n = -\frac{\text{heat change, } q}{n_{water\,formed}}$

The standard enthalpy change of combustion ($\Delta H_c^{\circ}$) is also exothermic and is defined as the heat evolved when $1\,mol$ of a substance is completely burnt in excess oxygen under standard conditions ($298\,K$ and $1\,bar$). For ethanol, the thermochemical equation is:

$CH_3CH_2OH(l) + 3O_2(g) \rightarrow 2CO_2(g) + 3H_2O(l)$

The experimental determination is:

$\Delta H_c = -\frac{\text{heat change, } q}{n_{substance\,burnt}}$

The standard enthalpy change of formation ($\Delta H_f^{\circ}$) is the heat change when $1\,mol$ of a substance is formed from its constituent elements in their standard states under standard conditions of $298\,K$ and $1\,bar$. An example is the formation of nitrobenzene:

$6C(s) + \frac{5}{2}H_2(g) + \frac{1}{2}N_2(g) + O_2(g) \rightarrow C_6H_5NO_2(l)$

Crucially, the $\Delta H_f^{\circ}$ of an element in its standard state is always $0$ because the element is formed from itself (e.g., $N_2(g) \rightarrow N_2(g)$).

Atomic-Scale Energy Changes: Bond Energy, Atomisation, Ionisation, and Electron Affinity

Bond energy ($E(X-Y)$) is defined as the heat absorbed when $1\,mol$ of a specific covalent bond between two atoms in the gaseous state is broken. This is always an endothermic process ($E(X-Y) > 0$). An example is the average bond energy of the carbon-hydrogen bond:

$CH_4(g) \rightarrow C(g) + 4H(g)$

The standard enthalpy change of atomisation ($\Delta H_a^{\circ}$) is the heat absorbed (always positive) when $1\,mol$ of gaseous atoms is formed from its element in its standard state under standard conditions ($298\,K$, $1\,bar$). For example, the atomisation of solid iodine:

$\frac{1}{2}I_2(s) \rightarrow I(g)$

Ionisation Energy (IE) refers to the energy required to remove electrons. The 1st IE is the heat absorbed to remove $1\,mol$ of electrons from $1\,mol$ of gaseous atoms to form $1\,mol$ of singly charged gaseous cations. This is always positive:

$Mg(g) \rightarrow Mg^+(g) + e^{-}$

The 2nd IE is the heat absorbed to remove $1\,mol$ of electrons from $1\,mol$ of singly charged gaseous cations to form $1\,mol$ of doubly charged gaseous cations:

$Mg^+(g) \rightarrow Mg^{2+}(g) + e^{-}$

Electron Affinity (EA) describes the energy change when electrons are added. The 1st EA is the heat evolved (always negative) when $1\,mol$ of electrons is added to $1\,mol$ of gaseous atoms:

$O(g) + e^{-} \rightarrow O^{-}(g)$

Subsequent electron affinities, such as the 2nd EA, are always positive (endothermic) because of the repulsion between the incoming electron and the existing negative charge on the anion:

$O^{-}(g) + e^{-} \rightarrow O^{2-}(g)$

Energetics of Ionic Solids and Aqueous Solutions

Lattice energy (LE) is defined as the heat evolved when $1\,mol$ of a solid ionic compound is formed from its constituent gaseous ions. This is always an exothermic process ($LE < 0$). For calcium bromide, the process is:

$Ca^{2+}(g) + 2Br^{-}(g) \rightarrow CaBr_2(s)$

The magnitude of lattice energy is modeled by the following relationship:

$|\text{lattice energy}| \propto \frac{|q_+ \times q_-|}{r_+ + r_-}$

In this relationship, $q_+$ and $q_-$ represent the charges of the cation and anion respectively, while $r_+$ and $r_-$ are their ionic radii. Note that the effect of ionic charge ($q_+ \times q_-$) generally has a greater impact on the lattice energy than the ionic radii ($r_+ + r_-$).

Standard enthalpy change of hydration ($\Delta H_{hyd}^{\circ}$) is the heat evolved when $1\,mol$ of gaseous ions is completely dissolved in sufficient water at $298\,K$ and $1\,bar$ such that no further heat change occurs. This is always negative:

$Ca^{2+}(g) \rightarrow Ca^{2+}(aq)$

The magnitude of the hydration enthalpy of an ion is proportional to its charge density:

$|\Delta H_{hyd}^{\circ} \text{ of an ion}| \propto |\text{charge density of ion}|$

$\text{Charge density of ion} \propto \frac{\text{charge of ion}}{\text{size of ion}}$

Finally, the standard enthalpy change of solution ($\Delta H_{soln}^{\circ}$) is the heat change when $1\,mol$ of a substance is completely dissolved in sufficient water at $298\,K$ and $1\,bar$ such that no further heat change occurs upon dilution. For calcium chloride:

$CaCl_2(s) \rightarrow Ca^{2+}(aq) + 2Cl^{-}(aq)$

Experimentally, it is calculated as:

$\Delta H_{soln} = \pm \frac{\text{heat change, } q}{\text{substance dissolved}}$

Through Hess' Law, the enthalpy change of solution can be related to lattice energy and hydration enthalpy:

$\Delta H_{soln} = -LE + \sum \Delta H_{hyd}^{\circ}(\text{ions})$