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ChemistryEdu Logo Thermodynamics | Enthalpy#

Enthalpy (H)#

  • The heat content of a system at constant pressure is called enthalpy.
  • Enthalpy of a system can be represented as a function of pressure, P and temperature, T.
\[H = f(P,\ T)\]
\[Partially\ differentiating\ both\ sides:\]
\[dH = {({dH \over dP})}_T dP + {({dH \over dT})}_P dT\]
\[Here, {({dH \over dP})}_T = Change\ in\ H\ w.r.t\ P\ at\ constant\ temperature\]
\[Here, {({dH \over dT})}_P = Change\ in\ H\ w.r.t\ T\ at\ constant\ pressure\]
\[Since,\ pressure\ is\ constant.\ So,\ dP = 0\]
\[dH = {({dH \over dT})}_P dT\]
\[For\ 1\ mole\ of\ ideal\ gas:\]
\[dH = C_pdT\]
\[For\ n\ moles\ of\ ideal\ gas:\]
\[dH = nC_pdT\]
\[Integrating\ both\ sides:\]
\[ \int\limits_{H_1}^{H_2} dH = \int\limits_{T_1}^{T_2} nC_pdT\]
\[H_2 - H_1 = \int\limits_{T_1}^{T_2} nC_pdT\]
\[ΔH = \int\limits_{T_1}^{T_2} nC_pdT\]
\[Here,\ C_p = Molar\ heat\ capacity\ at\ constant\ pressure\]
  • If ΔH is negative, then heat is released and reaction is exothermic.
  • If ΔH is positive, then heat is absorbed and reaction is endothermic.

Relation between Internal Energy (U) and Enthalpy (H)#

  • The relation between U and H is given by the following equation:
\[H = U + PV\]
\[Differentiating\ both\ sides:\]
\[dH = dU + d(PV)\]
  • If pressure P is constant (isobaric process), we can write:
\[dH = dU + d(PV)\]
\[dH = dU + PdV\]
\[Integrating\ both\ sides:\]
\[\int\limits_{H_1}^{H_2} dH = \int\limits_{U_1}^{U_2} dU + P \int\limits_{V_1}^{V_2} dV\]
\[H_2 - H_1 = (U_2 - U_1) + P(V_2 - V_1)\]
\[ΔH = ΔU + P ΔV\]
  • If Volume V is constant (isochoric process), we can write:
\[dH = dU + d(PV)\]
\[dH = dU + VdP\]
\[Integrating\ both\ sides:\]
\[\int\limits_{H_1}^{H_2} dH = \int\limits_{U_1}^{U_2} dU + V \int\limits_{P_1}^{P_2} dP\]
\[H_2 - H_1 = (U_2 - U_1) + V(P_2 - P_1)\]
\[ΔH = ΔU + V ΔP\]
  • For a gaseous phase chemical reaction containing ideal gases:
\[dH = dU + d(PV)\]
\[dH = dU + d(nRT)\]

Since, most chemical reactions occur at constant temperature:

\[dH = dU + RT(dn)\]
\[Integrating\ both\ sides:\]
\[\int\limits_{H_1}^{H_2} dH = \int\limits_{U_1}^{U_2} dU + RT \int\limits_{n_1}^{n_2} dn\]
\[H_2 - H_1 = (U_2 - U_1) + RT (n_2 - n_1)\]
\[ΔH = ΔU + Δn_g RT\]

Here, Δng = Gaseous moles of products - Gaseous moles of reactants