Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Solution Chemistry

Field: Chemical Thermodynamics

The study of chemical reactions in solution.

Sequence of Expressions

For a species ii in a solution, the activity aia_i is defined by the product of its activity coefficient γi\gamma_i and its effective concentration CiC_i: ai=γiCi a_i = \gamma_i C_i where CiC_i is the molar concentration. The activity coefficient γi\gamma_i quantifies the deviation from ideal behavior, such that for an ideal solution, γi=1\gamma_i = 1. The chemical potential μi\mu_i is then given by: μi=μi∘+RTln⁡ai=μi∘+RTln⁡(γiCi) \mu_i = \mu_i^\circ + R T \ln a_i = \mu_i^\circ + R T \ln (\gamma_i C_i)
Consider the dissolution equilibrium of a sparingly soluble salt AmBnA_mB_n: AmBn(s)⇌mAn+(aq)+nBm−(aq) A_mB_n(s) \rightleftharpoons m A^{n+}(aq) + n B^{m-}(aq) The Solubility Product Constant (KspK_{sp}) is defined by the activities of the dissolved ions at equilibrium: Ksp=(aAn+)m(aBm−)n K_{sp} = \left( a_{A^{n+}} \right)^m \left( a_{B^{m-}} \right)^n Assuming ideal behavior and using molar concentrations CC: Ksp=(γAn+CAn+)m(γBm−CBm−)n K_{sp} = (\gamma_{A^{n+}} C_{A^{n+}})^m (\gamma_{B^{m-}} C_{B^{m-}})^n
Define the chemical potential μi\mu_i of component ii in a system as the partial molar Gibbs free energy: \nμi=(∂G−∂ni)T,P,nj≠i\mu_i = \left(\frac{\partial G}{-\partial n_i}\right)_{T, P, n_{j\neq i}} \nAlternatively, it can be expressed in terms of activity aia_i and standard state μi0\mu_i^0: \nμi=μi0+RTln⁡(ai)\mu_i = \mu_i^0 + R T \ln(a_i)
Define the change in Gibbs Free Energy for a process occurring at constant pressure and temperature as: ΔG=ΔH−TΔS \Delta G = \Delta H - T \Delta S where ΔH\Delta H is the change in enthalpy, ΔS\Delta S is the change in entropy, and TT is the absolute temperature. Alternatively, for a reaction aA+bB→cC+dDaA + bB \rightarrow cC + dD, the change is: ΔG=∑iνiΔGi∘=−RT∑iνiΔSi∘ni \Delta G = \sum_{i} \nu_i \Delta G_i^\circ = -R T \sum_{i} \nu_i \frac{\Delta S_i^\circ}{n_i}
The relationship between the equilibrium constant KK and temperature TT is described by the integrated form of the Van't Hoff equation: d(ln⁡K)d(1/T)=−ΔH∘R \frac{d(\ln K)}{d(1/T)} = -\frac{\Delta H^\circ}{R} where ΔH∘\Delta H^\circ is the standard enthalpy change of the reaction, and RR is the universal gas constant. Integrating this yields: ln⁡(K2K1)=−ΔH∘R(1T2−1T1) \ln \left(\frac{K_2}{K_1}\right) = -\frac{\Delta H^\circ}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)
Define the activity coefficient γi\gamma_i for component ii in a solution such that, under the ideal approximation, γi=1\gamma_i = 1. Consequently, the chemical potential μi\mu_i is given by: \nμi=μi0+RTln⁡(xi)\mu_i = \mu_i^0 + R T \ln(x_i) \nwhere μi0\mu_i^0 is the standard chemical potential, RR is the universal gas constant, TT is the absolute temperature, and xix_i is the mole fraction of component ii.
Let JJ be the molar flux (mol⋅m−2⋅s−1\text{mol} \cdot \text{m}^{-2} \cdot \text{s}^{-1}) of a species across a membrane, and let DD be the diffusion coefficient (m2⋅s−1\text{m}^2 \cdot \text{s}^{-1}). The flux JJ is proportional to the negative gradient of the concentration CC with respect to position xx: \nJ=−DdCdxJ = -D \frac{dC}{dx}
Define the Hildebrand solubility parameter δ\delta (in MPa1/2\text{MPa}^{1/2}) as the square root of the cohesive energy density (EcohE_{coh}) of a pure substance: \nδ=Ecoh\delta = \sqrt{E_{coh}} \nFor two components, the limit of miscibility is approached when the difference in their solubility parameters is minimized, i.e., Δδ=∣δ1−δ2∣→0\Delta \delta = |\delta_1 - \delta_2| \rightarrow 0.
For a solution containing multiple volatile components ii, the partial vapor pressure PiP_i of component ii is proportional to its mole fraction xix_i and its pure component vapor pressure Pi∘P_i^\circ: Pi=xiPi∘ P_i = x_i P_i^\circ The total vapor pressure PtotalP_{total} of the solution is the sum of the partial pressures: Ptotal=∑iPi=∑ixiPi∘ P_{total} = \sum_{i} P_i = \sum_{i} x_i P_i^\circ
Consider a solution where the concentration of solute ii is low, such that the activity coefficient γi\gamma_i approaches unity. The activity aia_i of component ii is then approximated by its mole fraction xix_i: \nai≈xia_i \approx x_i \nThis leads to the simplified expression for the chemical potential: \nμi≈μi0+RTln⁡(xi)\mu_i \approx \mu_i^0 + R T \ln(x_i)