Aqueous Solutions, Acids & Bases

Solubility, colligative properties, acid-base equilibria, and electrochemistry

Solubility & Dissolution Thermodynamics

A solute dissolves when the Gibbs free energy of solution is negative:

$$\Delta G_{\text{soln}} = \Delta H_{\text{soln}} - T\Delta S_{\text{soln}} < 0$$

The enthalpy of solution $\Delta H_{\text{soln}}$ can be decomposed into the lattice energy (energy to break the crystal apart) and the hydration energy (energy released when ions are solvated). For ionic compounds: $\Delta H_{\text{soln}} = -U_{\text{lattice}} + \Delta H_{\text{hydration}}$.

Even endothermic dissolution can occur spontaneously if the entropy increase ($T\Delta S_{\text{soln}}$) is large enough, as when a highly ordered crystal disperses into a disordered solution.

Colligative Properties

Colligative properties depend only on the number of solute particles, not their identity. For a non-volatile solute:

Boiling Point Elevation

$$\Delta T_b = iK_b m$$

$K_b$ is the ebullioscopic constant, $m$ is molality, and $i$ is the van't Hoff factor.

Freezing Point Depression

$$\Delta T_f = iK_f m$$

$K_f$ is the cryoscopic constant. For water,$K_f = 1.86$ °C/m and $K_b = 0.512$ °C/m.

Brønsted-Lowry Acid-Base Theory

A Brønsted acid donates a proton (H⁺); a Brønsted base accepts a proton. Every acid has a conjugate base, and vice versa.

$$\text{pH} = -\log[\text{H}^+]$$

$$K_w = [\text{H}^+][\text{OH}^-] = 1.0 \times 10^{-14} \;\text{at 25°C}$$

Henderson-Hasselbalch Equation

For a buffer solution containing a weak acid HA and its conjugate base A⁻:

$$\text{pH} = \text{p}K_a + \log\frac{[\text{A}^-]}{[\text{HA}]}$$

Buffers resist pH changes upon addition of small amounts of acid or base. Maximum buffering capacity occurs when $[\text{A}^-] = [\text{HA}]$, i.e., when $\text{pH} = \text{p}K_a$.

Solubility Product

For a sparingly soluble salt $M_aX_b \rightleftharpoons aM^{b+} + bX^{a-}$:

$$K_{sp} = [M^{b+}]^a[X^{a-}]^b$$

Electrochemistry: The Nernst Equation

The Nernst equation relates the cell potential to the standard potential and the reaction quotient:

$$E = E^\circ - \frac{RT}{nF}\ln Q$$

At 25°C, this simplifies to:

$$E = E^\circ - \frac{0.05916}{n}\log Q$$

where $n$ is the number of electrons transferred, $F = 96485$ C/mol is Faraday's constant, and $Q$ is the reaction quotient. At equilibrium, $E = 0$ and $Q = K$, giving$\Delta G^\circ = -nFE^\circ = -RT\ln K$.

Python: pH Titration Curve Simulator

Simulates titration curves for strong acid/strong base and weak acid/strong base systems. The weak acid case uses the Henderson-Hasselbalch equation in the buffer region and hydrolysis calculations at the equivalence point.

pH Titration Curves

Python

Simulates strong acid and weak acid titration with NaOH, showing equivalence points and buffer regions

script.py102 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
# ─────────────────────────────────────────────────
# pH Titration Curve Simulator
# Strong acid/strong base and weak acid/strong base
# ─────────────────────────────────────────────────
Kw = 1.0e-14  # water autoionization constant at 25C

def strong_acid_strong_base(V_acid, C_acid, C_base, V_base_arr):
    """pH for titration of strong acid with strong base."""
    pH = np.zeros_like(V_base_arr, dtype=float)
    for i, Vb in enumerate(V_base_arr):
        n_acid = C_acid * V_acid
        n_base = C_base * Vb
        V_total = V_acid + Vb
        if n_base < n_acid:
            H_plus = (n_acid - n_base) / V_total
            pH[i] = -np.log10(H_plus)
        elif abs(n_base - n_acid) < 1e-12:
            pH[i] = 7.0
        else:
            OH_minus = (n_base - n_acid) / V_total
            pH[i] = 14.0 + np.log10(OH_minus)
    return pH

def weak_acid_strong_base(V_acid, C_acid, Ka, C_base, V_base_arr):
    """pH for titration of weak acid with strong base."""
    pH = np.zeros_like(V_base_arr, dtype=float)
    for i, Vb in enumerate(V_base_arr):
        n_acid = C_acid * V_acid
        n_base = C_base * Vb
        V_total = V_acid + Vb
        if n_base < n_acid - 1e-12:
            # Buffer region: Henderson-Hasselbalch
            nHA = n_acid - n_base
            nA = n_base
            if nA < 1e-15:
                # Pure weak acid
                CHA = nHA / V_total
                H = (-Ka + np.sqrt(Ka**2 + 4*Ka*CHA)) / 2
                pH[i] = -np.log10(H)
            else:
                pH[i] = -np.log10(Ka) + np.log10(nA / nHA)
        elif abs(n_base - n_acid) < 1e-12:
            # Equivalence point: hydrolysis of conjugate base
            Cb = n_acid / V_total
            Kb = Kw / Ka
            OH = np.sqrt(Kb * Cb)
            pH[i] = 14.0 + np.log10(OH)
        else:
            OH_minus = (n_base - n_acid) / V_total
            pH[i] = 14.0 + np.log10(OH_minus)
    return pH

# Parameters
V_acid = 0.050   # 50 mL
C_acid = 0.100   # 0.1 M
C_base = 0.100   # 0.1 M NaOH
Ka = 1.8e-5      # acetic acid

V_base = np.linspace(0.001, 0.100, 1000)  # 0 to 100 mL

pH_strong = strong_acid_strong_base(V_acid, C_acid, C_base, V_base)
pH_weak = weak_acid_strong_base(V_acid, C_acid, Ka, C_base, V_base)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(13, 5))

# Strong acid / strong base
ax1.plot(V_base * 1000, pH_strong, 'cyan', linewidth=2)
ax1.axhline(y=7, color='gray', linestyle='--', alpha=0.5, label='pH = 7')
ax1.axvline(x=50, color='red', linestyle=':', alpha=0.7, label='Equivalence')
ax1.set_xlabel('Volume NaOH added (mL)', fontsize=12)
ax1.set_ylabel('pH', fontsize=12)
ax1.set_title('Strong HCl + Strong NaOH', fontsize=14)
ax1.set_ylim(0, 14)
ax1.legend(fontsize=10)
ax1.grid(True, alpha=0.3)

# Weak acid / strong base
pKa = -np.log10(Ka)
ax2.plot(V_base * 1000, pH_weak, 'lime', linewidth=2)
ax2.axhline(y=pKa, color='orange', linestyle='--', alpha=0.5,
            label=f'pKa = {pKa:.2f}')
ax2.axvline(x=50, color='red', linestyle=':', alpha=0.7, label='Equivalence')
ax2.axvline(x=25, color='yellow', linestyle=':', alpha=0.5, label='Half-equiv.')
ax2.set_xlabel('Volume NaOH added (mL)', fontsize=12)
ax2.set_ylabel('pH', fontsize=12)
ax2.set_title('Weak CH$_3$COOH + Strong NaOH', fontsize=14)
ax2.set_ylim(0, 14)
ax2.legend(fontsize=10)
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight')

print(f"Strong acid-strong base equivalence pH: 7.00")
print(f"Weak acid pKa = {pKa:.2f}")
print(f"Weak acid equivalence point pH: {pH_weak[np.argmin(np.abs(V_base - 0.050))]:.2f}")
print("Plot saved.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Nernst Equation Calculator

Computes cell potential for several electrochemical cells at various concentrations using the Nernst equation.

Nernst Equation Calculator

Fortran

Computes cell potential for Daniell cell and Ag/AgCl electrode at varying concentrations

nernst_calculator.f9057 lines

program nernst_calculator
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: R = 8.314_dp      ! J/(mol*K)
  real(dp), parameter :: F = 96485.0_dp    ! C/mol
  real(dp), parameter :: T = 298.15_dp     ! K (25 C)
  real(dp) :: E0, E_cell, Q, n_elec
  real(dp) :: conc_ratio
  integer :: i

write(*,'(A)') '================================================'
  write(*,'(A)') '     Nernst Equation: E = E0 - (RT/nF) ln(Q)'
  write(*,'(A)') '================================================'
  write(*,'(A)') ''

! Cell 1: Zn/Cu Daniel cell
  ! Zn(s) + Cu2+(aq) -> Zn2+(aq) + Cu(s)
  E0 = 1.10_dp   ! V
  n_elec = 2.0_dp
  write(*,'(A)') '--- Daniell Cell: Zn | Zn2+ || Cu2+ | Cu ---'
  write(*,'(A,F6.3,A)') 'Standard potential E0 = ', E0, ' V'
  write(*,'(A)') ''
  write(*,'(A15, A15, A15)') '[Zn2+]/[Cu2+]', 'Q', 'E (V)'
  write(*,'(A)') '---------------------------------------------'

do i = -3, 3
    conc_ratio = 10.0_dp ** i
    Q = conc_ratio
    E_cell = E0 - (R * T) / (n_elec * F) * log(Q)
    write(*,'(ES15.2, ES15.2, F15.4)') conc_ratio, Q, E_cell
  end do

write(*,'(A)') ''

! Cell 2: Ag/AgCl half-cell
  ! AgCl(s) + e- -> Ag(s) + Cl-(aq)
  E0 = 0.222_dp  ! V
  n_elec = 1.0_dp
  write(*,'(A)') '--- Ag/AgCl Reference Electrode ---'
  write(*,'(A,F6.3,A)') 'Standard potential E0 = ', E0, ' V'
  write(*,'(A)') ''
  write(*,'(A15, A15)') '[Cl-] (M)', 'E (V)'
  write(*,'(A)') '------------------------------'

do i = -4, 1
    conc_ratio = 10.0_dp ** i
    ! Q = [Cl-] for this half-reaction
    Q = conc_ratio
    E_cell = E0 - (R * T) / (n_elec * F) * log(Q)
    write(*,'(ES15.2, F15.4)') conc_ratio, E_cell
  end do

write(*,'(A)') ''
  write(*,'(A,F8.4,A)') 'RT/F at 298.15 K = ', R*T/F*1000, ' mV'
  write(*,'(A,F8.5,A)') '(RT/F)*ln(10) = ', R*T/F*log(10.0_dp)*1000, &
    ' mV  (Nernst factor)'
end program nernst_calculator