Enthalpy

H \equiv E + P V

Differential (from $d E = T d S - P d V + μ d N$ ):

d H = d E + P d V + V d P = T d S + V d P + μ d N

$H = H (S, P, N)$ . Maxwell relations:

{(\frac{\partial H}{\partial S})}_{P, N} = T, {(\frac{\partial H}{\partial P})}_{S, N} = V, {(\frac{\partial H}{\partial N})}_{P, S} = μ

Enthalpy is the constant-pressure analogue of energy: at constant $P$ , $Δ H = Q$ .

Heat capacities

C_{V} = {(\frac{\partial Q}{\partial T})}_{V} = {(\frac{\partial E}{\partial T})}_{V} e x t (n o w o r k a t c o n s t a n t V e x t)

At constant $P$ : gas must expand to maintain pressure, so work is done and energy goes down. Total energy change: $Δ E = Q - P Δ V$ , so $Δ H = Q$ . Thus:

C_{P} = {(\frac{\partial H}{\partial T})}_{P}

For monatomic ideal gas ( $P V = N k_{B} T$ , $E = \frac{3}{2} N k_{B} T$ ):

H = E + P V = \frac{5}{2} N k_{B} T ⟹ C_{P} = \frac{5}{2} N k_{B}

Enthalpy in chemistry (constant $P$ )

Chemistry and biology happen at constant $P$ (open containers, solutions, cells). Heat released in a reaction = $Δ H$ .

Three approaches to compute reaction enthalpy (best → worst):

Standard enthalpy of reaction $Δ_{r} H^{\circ}$ — look it up directly (defined at reference conditions, e.g. 298 K, 1 atm)
Enthalpies of formation + Hess's law — $Δ_{r} H^{\circ} = \sum Δ_{f} H^{\circ} (e x t p r o d u c t s) - \sum Δ_{f} H^{\circ} (e x t r e a c t a n t s)$ ; enthalpy of formation of an element in standard state = 0 by definition
Bond enthalpies — last resort; sum up enthalpies per bond type (tabulated for gases only, not reliable for liquids/solids)

Example — hydrogenation of ethene:

e x t H_{2} + e x t C_{2} e x t H_{4} o e x t C_{2} e x t H_{6}, Δ_{r} H^{\circ} = - 136.3 e x t k J / m o l (e x o t h e r m i c)

Via formation enthalpies: $Δ_{r} H^{\circ} = - 84 - 52.4 = - 136.4$ kJ/mol ✓
Via bond enthalpies: $- 124$ kJ/mol — close but not great (bonds aren't identical across molecules)

Hess's law: enthalpies are state functions, so they can be added and subtracted freely.

Enthalpy vs energy: when does the difference matter?

Δ H - Δ E = Δ (P V) \approx (Δ n_{e} x t g a s) R T

where $Δ n_{e} x t g a s$ = change in moles of gas in the reaction.

Solids/liquids: $Δ (P V) ≪ Δ H$ (< 1%) — $H \approx E$
- Example: aragonite → calcite, $Δ (P V) \sim 10^{- 4}$ kJ/mol vs $Δ H = 0.21$ kJ/mol
Gases: $Δ (P V) \sim$ few % — can matter for reaction kinetics
- Example: $3 e x t H_{2} + e x t N_{2} o 2 e x t {N H}_{3}$ , $Δ n = - 2$ , $Δ (P V) = - 4.9$ kJ/mol ≈ 5% of $Δ H$
Rule of thumb: $(Δ n) R T \approx 2.48 e x t k J / m o l$ per mole of gas change at room $T$

Heat capacities

Enthalpy in chemistry (constant P)

Enthalpy vs energy: when does the difference matter?

Enthalpy in chemistry (constant $P$ )