Chemical Thermodynamics

• \( U \)

The definition depends greatly on the system.

In chemistry, the system is often a bulk of matter. Internal energy represents bonding energy, intermolecular energy, dispersion as well as the kinetic energy of molecules.

Keep in mind the first law of thermodynamics: \[ dU = \delta Q + \delta W, \] and the relation between state functions: \[ dU = TdS - pdV + \sum_i \mu_i dN_i. \]

• \( H := U + pV \)

Chemical reaction is mostly done in isobaric condition in which: \[ dH = \delta Q + Vdp = \delta Q. \]

• \( A := U - TS \)

A for Arbeit.

• \( G := U - TS + pV = H - TS = A + pV \)

Relative activity \( a_i \) of a species \( i \) is defined as: \[ a_i := \exp \left( \frac{\mu_i - \mu_i^{\circleddash}}{RT} \right) \] where \( \mu_i \) is the molar chemical potential of the species \( i \).

Equivalently, \[ \mu_i = \mu_i^{\circleddash} + RT\ln a_i. \]

Activity is indicates the deviation from the standard chemical potential \( \mu_i^{\circleddash} \). In particular, it contains the entropy of mixing: \[ \Delta S_{\rm mix} = -R \sum_i n_i \ln x_i. \] The Gibbs free energy per mole is given under isothermal condition:

\begin{align*} \left(\frac{\partial G}{\partial n_i}\right)_{T, p, N_{j\neq i}} &= \left( \frac{\partial G^{\circleddash}}{\partial n_i} \right)_{T, p, N_{j\neq i}} + \left( \frac{\partial \Delta H_{\rm mix}}{\partial n_i}\right)_{T, p, N_{j\neq i}}- T \left( \frac{\partial \Delta S_{\rm mix}}{\partial n_i} \right)_{T, p, N_{j\neq i}} \\ &= \mu_i^{\circleddash} + RT\ln \gamma_i + RT\ln x_i. \end{align*}

The Helmholtz energy change can happen due to the intermolecular interaction which is not present in ideal case. Yes, \( n_i \) dependence of \( x_i \) is properly considered. Try differentiate it.

There are different activity coefficients for each intensive quantity of amount.

\[ a_{i} = \gamma_{x,i} x_i = \gamma_{c,i} \frac{c_i}{c^{\circleddash}} \] where \( x_i \) is the mole fraction of the species \( i \), \( c_{i} \) is the molarity, and \( c^{\circleddash} \) is the standard concentration (usually 1 mol/L).

\[ a_i = \frac{f_i}{p^{\circleddash}} \]

\[ \mathrm{pH} := - \log a_{\rm H^+}. \]

Yes, this is more exact definition.

For a reaction \[ 0 \to \sum_i\nu_i \mathrm{X}_i, \] the equilibrium constant \( K \) is defined as: \[ K := \prod_i a_i^{\nu_i}. \]

\[ \mu := \frac{1}{2}\sum_{\text{species } i} c_iz_i^2 \] where \( c_i \) is the concentration, and \( z_i \) is the charge number.

For each pair of indistinguishable electrons, the system is stabilized by the amount of exchange energy \( \Pi_e \).

The absolute hardness \( \eta \) is defined by: \[ \eta := \frac{I - A}{2} \] where \( I \) is the ionization energy, and \( A \) is the electron affinity.

Notice that it is half the gap between HOMO and LUMO, with the center being the absolute electronegativity \( \chi \): \[ \chi := \frac{I + A}{2}. \]

If \( \eta \) is high, it is hard for the electron in the HOMO to "squish", that is, to be in a superposition of excited states and ground state.

• SALC

• Water activity - Wikipedia