Thermochemical Equations

Standard enthalpy of reaction from standard enthalpies of formation. The standard enthalpy of a reaction is the enthalpy change when all reactants and products are in their standard states (denoted with the superscript $^0$, as in $\Delta H^0$). Once every substance's own $\Delta H_f^0$ is known, a reaction's $\Delta H_r^0$ can be calculated WITHOUT ever running that specific reaction in a calorimeter: it is simply the sum of the products' formation enthalpies minus the sum of the reactants', each weighted by its stoichiometric coefficient. For a general reaction $aA+bB\rightarrow cC+dD$:

Worked example (Problem 7.2). The standard enthalpy change for the combustion of ethanol, $C_2H_5OH(l)+3O_2(g)\rightarrow2CO_2(g)+3H_2O(l)$, given the formation enthalpies of $C_2H_5OH(l)$, $CO_2(g)$ and $H_2O(l)$ are $-277$, $-393.5$ and $-285.5$ kJ mol$^{-1}$ respectively (and $O_2(g)$, an element in its standard state, contributes zero):

$\Delta H_r^0 = [2(-393.5)+3(-285.5)] - [(-277)+0] = [-787-856.5]-[-277] = -1643.5+277 = -1366.5\ \text{kJ}$ …

Heat (enthalpy) of combustion, $\Delta H_C$, is defined as "the change in enthalpy of a system when one mole of the substance is completely burnt in excess air or oxygen." Because combustion always releases energy, $\Delta H_C$ is ALWAYS negative -- there is no such thing as an endothermic combustion.

Worked examples: methane's heat of combustion,

and the combustion of carbon,

$C(s)+O_2(g)\rightarrow CO_2(g),\qquad \Delta H_C=-394.55\ \text{kJ mol}^{-1}$ …

Molar heat capacity. When heat $q$ is supplied to a system, the constituent molecules absorb it as extra kinetic energy, raising the system's temperature from $T_1$ to $T_2$. This temperature rise is directly proportional to the heat absorbed and inversely proportional to the mass of substance present:

The proportionality constant $c$ here is the heat capacity. When $m=1$ kg and $(T_2-T_1)=1$ K, this is called the specific heat capacity: the heat absorbed by one kilogram of a substance to raise its temperature by one kelvin at a specified temperature. When instead 1 MOLE of substance is used, the same quantity is the molar heat capacity $c_m$: the heat absorbed by one mole of substance to raise its temperature by 1 kelvin. Its SI unit is $\text{J K}^{-1}\text{mol}^{-1}$.

Molar heat capacity can be measured under two different conditions, giving two distinct values: at constant volume ($C_V$) or at constant pressure ($C_P$). Starting from the first law, $\Delta U=q+w=q-P\,dV$, so $q=\Delta U+P\,dV$ (7.19); differentiating with respect to temperature at CONSTANT volume ($dV=0$):

so $C_V$ is the rate of change of internal energy with temperature at constant volume. Similarly, the molar heat capacity at constant pressure is defined as the rate of change of enthalpy with temperature at constant pressure:

Relation between $C_P$ and $C_V$ for an ideal gas. Starting from $H=U+PV$ (7.8), and for 1 mole of ideal gas $PV=nRT$ (7.22), so $H=U+nRT$ (7.23). Differentiating with respect to $T$:

(for one mole, simply $C_P-C_V=R$). Physically: at constant pressure the system additionally has to do expansion work against its surroundings as it warms, so it needs MORE heat than at constant volume for the identical temperature rise -- which is exactly why $C_P$ is always greater than $C_V$.

Calculating $\Delta U$ and $\Delta H$ from heat capacities. For one mole of ideal gas, $C_V=dU/dT \Rightarrow dU=C_V\,dT$; for a finite change, $\Delta U=C_V\Delta T=C_V(T_2-T_1)$, and for $n$ moles, $\Delta U=nC_V(T_2-T_1)$ (7.25). Identically, $\Delta H=nC_P(T_2-T_1)$ (7.26). …