Environmental Chemistry

Solubility product (Ksp)

The solubility product, K_{sp}, is the equilibrium constant for the dissolution of a sparingly soluble ionic solid. It equals the product of the molar concentrations of the ions, each raised to its stoichiometric coefficient. For a salt that releases one cation and one anion, such as \text{CaSO}_4 \rightleftharpoons \text{Ca}^{2+} + \text{SO}_4^{2-}, the expression is K_{sp} = [\text{Ca}^{2+}][\text{SO}_4^{2-}]. In pure water the two ion concentrations are equal, so the solubility is s = \sqrt{K_{sp}}. Sulphate is the conjugate base of a strong acid, so its concentration is essentially unaffected by pH near neutrality, and a neutral pH does not change the calculated solubility. The reaction quotient Q is computed from the actual ion concentrations and compared with K_{sp}. Q < K_{sp} means the solution is unsaturated, Q = K_{sp} means equilibrium, and Q > K_{sp} means precipitation. For metal hydroxides, pH fixes [\text{OH}^-] = 10^{-(14-\text{pH})} at 25 °C. Because hydroxide appears squared in Q for \text{Zn(OH)}_2, a small change in pH has a large effect on Q. The ratio Q/K_{sp} decides whether the solid will form. The common-ion effect follows from Le Chatelier's principle. Adding an ion already present in the solid, such as carbonate to calcium carbonate, shifts the equilibrium back to the solid and lowers the concentration of the other ion. A salt that shares no ion with the solid raises the ionic strength, which lowers the activity coefficients and increases the apparent solubility, so the free calcium ion concentration can increase. Solubility in water in general depends on polarity. Water is a bent molecule in which oxygen is more electronegative than hydrogen, so the shared electrons lie closer to oxygen and the molecule carries a permanent dipole. Polar and ionic solutes dissolve readily, while non-polar compounds dissolve poorly. Aromatic rings reduce solubility because they are hydrophobic, whereas charged or polar functional groups increase it.

Alkalinity titration

Alkalinity is the capacity of water to neutralize acid, and it is mainly due to bicarbonate, carbonate and hydroxide ions. It is determined by titrating with a strong acid such as \text{H}_2\text{SO}_4 to two end points. The phenolphthalein end point, near pH 8.3, measures the hydroxide and carbonate fraction. The methyl orange end point, near pH 4.5, measures the total alkalinity. Colourless phenolphthalein indicates that no hydroxide and no carbonate-based alkalinity is present, while a yellow colour with methyl orange shows that bicarbonate-type alkalinity exists. Such a sample therefore contains bicarbonate but no hydroxide. The acid consumed to reach the total end point is found from normality times volume, so \text{meq} = N \times V_{\text{mL}}. Alkalinity in meq/L equals the meq of acid divided by the sample volume in litres. Reporting this value as mg/L as NaHCO_3 multiplies by the equivalent mass of 84 mg/meq, while reporting as CaCO_3 uses 50 mg/meq. For example, 10 mL of 0.02 N acid neutralized in a 100 mL sample gives 0.2\ \text{meq}/0.1\ \text{L} = 2 meq/L, which equals 168 mg/L as NaHCO_3.

Ionic strength of water

The ionic strength, I, measures the total ionic environment of a solution and controls the activity coefficients of ions. It is defined as I = \frac{1}{2}\sum c_i z_i^2, where c_i is the molar concentration of each ion and z_i its charge number. The charge enters squared, so a divalent ion contributes four times as much per mole as a monovalent ion. Neutral, un-ionized species contribute nothing. Each ion must be counted separately, including the anion of every salt, because the anion's charge also contributes to I. For example, a 0.03 M solution of \text{MgCl}_2 has \text{Mg}^{2+} = 0.03 M with z^2 = 4 and \text{Cl}^- = 0.06 M with z^2 = 1. Therefore I = \frac{1}{2}(0.12 + 0.06) = 0.09 M. Mixed solutions of salts are handled the same way, by summing the c_i z_i^2 terms for all ions before taking one half.

Water quality parameters

Water quality parameters are reported for specific chemical forms of a constituent, and the form determines what the result means. Total Kjeldahl nitrogen (TKN) measures reduced nitrogen only. It is the sum of organic nitrogen and ammoniacal nitrogen, because digestion in hot concentrated sulphuric acid converts organic nitrogen to ammonium, which is counted together with the ammonia already present. Nitrite and nitrate are oxidized forms that are not reduced under Kjeldahl conditions, so they are excluded from TKN and must be measured separately. Urban environmental quality also includes the urban heat island effect. Trees and vegetation reduce urban temperatures through shading and evapotranspiration, and aerosols such as PM2.5 alter the surface radiation balance, so they can change the intensity of the heat island. The effect is usually strongest at night because urban materials release stored heat after sunset. Reflective roofs reduce the effect by increasing albedo, so a statement that they intensify the heat island is false.

Water hardness causes and ions

Water hardness is caused by polyvalent metal cations, chiefly calcium and magnesium, with strontium, iron and manganese contributing in smaller amounts. The anions that accompany these cations, such as bicarbonate, sulphate, chloride and nitrate, do not themselves cause hardness. Carbonate is an anion and is likewise not a hardness-causing species. Hardness is reported as mg/L of CaCO_3 equivalent. Each cation concentration is multiplied by the ratio of the equivalent mass of CaCO_3 (50 mg/meq) to the equivalent mass of the ion. The factors are 100/40 = 2.5 for calcium and 100/24 for magnesium, and total hardness is the sum of the two contributions. For example, 12 mg/L of Mg^{2+} and 20 mg/L of Ca^{2+} give 12 \times 100/24 + 20 \times 100/40 = 50 + 50 = 100 mg/L as CaCO_3. Sodium and chloride ions are not included in the hardness sum.

First-order degradation kinetics

First-order degradation follows the rate law -\frac{dC}{dt} = kC, which integrates to C = C_0 e^{-kt}. The concentration therefore decays exponentially, and the rate constant has units of inverse time. In a batch reactor the rate constant follows from two concentrations measured at known times, using k = \frac{1}{t}\ln\frac{C_0}{C}. The time needed to reach a target concentration is t = \frac{1}{k}\ln\frac{C_0}{C}. For a given fractional reduction, the time depends only on the ratio C_0/C and not on the starting concentration. A tenfold reduction therefore takes the same time whatever the initial value. The same constant can be used to size a plug flow reactor, where the space time for a given conversion is \tau = \frac{1}{k}\ln\frac{C_0}{C} and the volume is V = Q\tau. When the lab and field temperatures are equal, the rate constant from a batch test transfers directly to the full-scale design.

Oxidation states of nitrogen

Oxidation states are assigned by requiring that the algebraic sum of the states in a species equals its charge, with oxygen normally -2 and hydrogen +1. For the ammonium ion, x + 4(+1) = +1 gives x = -3, and the same nitrogen state appears in ammonia. For the nitrite ion, x + 2(-2) = -1 gives x = +3. For nitric oxide, x + (-2) = 0 gives x = +2. Nitrogen in nitrate is +5 and in molecular nitrogen is 0. Nitrogen states therefore span from -3 to +5 along the nitrogen cycle, and nitrification and denitrification are oxidation and reduction steps between these values.

pH and hydrogen ion concentration

The pH of water is defined as \text{pH} = -\log_{10}[\text{H}^+], where the hydrogen ion concentration is in mol/L. A concentration given in mass units must first be converted to molar units by dividing the mass per litre by the atomic or molecular weight. For example, 10 mg/L of hydrogen with an atomic weight of 1 g/mol equals 10^{-2} mol/L, which gives a pH of 2. Each unit change in pH corresponds to a tenfold change in hydrogen ion concentration. A sample at pH 4 therefore has a hydrogen ion concentration 100 times that of a sample at pH 6. This relationship should be kept in mind when comparing the acidity of samples.

Sodium adsorption ratio

The sodium adsorption ratio (SAR) is an index of the risk that sodium will disperse soil clay when irrigation water is applied. It is defined as \text{SAR} = \dfrac{[\text{Na}^+]}{\sqrt{([\text{Ca}^{2+}] + [\text{Mg}^{2+}])/2}}, with all concentrations in meq/L. Each mg/L value must first be converted to meq/L by dividing by the equivalent mass, which is 23 for sodium, 20 for calcium and 12 for magnesium. Potassium is excluded from the formula even when it is present in the water. For example, a water containing 46 mg/L of Na^+, 40 mg/L of Ca^{2+} and 24 mg/L of Mg^{2+} has 2.0 meq/L of each ion. Its SAR is therefore 2.0/\sqrt{2.0} \approx 1.41. A high SAR means that sodium dominates the exchangeable cations, which raises the risk of damage to soil structure.

Specific conductance and TDS

Specific conductance is the electrical conductance of a water sample normalized to unit length and unit cross-sectional area, and it is usually reported in µS/cm at 25 °C. The current is carried by every dissolved charged species, so the reading reflects the combined contribution of cations and anions. It is therefore an indirect indicator of total dissolved solids (TDS). TDS is estimated by multiplying the conductance by an empirical conversion factor that must be calibrated for the type of water, because different ions have different mobilities. Un-ionized molecules do not conduct current and do not register in the measurement. Conductance therefore underestimates the dissolved solids when a large part of them is neutral.

Gibbs free energy and spontaneity

The Gibbs free energy change, \Delta G, determines the spontaneous direction of a process at constant temperature and pressure. A negative \Delta G means the forward process is spontaneous. A positive \Delta G means the forward process is non-spontaneous, so the reverse process is favoured. When \Delta G = 0 the system is at equilibrium, with forward and reverse rates equal and no net change in composition. A zero value therefore does not mean that the reaction proceeds in one direction only. The value depends on the standard change and the reaction quotient through \Delta G = \Delta G^\circ + RT\ln Q. At equilibrium Q = K and \Delta G^\circ = -RT\ln K, so the standard free energy change is linked to the equilibrium constant.

Phosphorus cycle

The phosphorus cycle differs from the carbon and nitrogen cycles because phosphorus has no significant gaseous phase. It cycles through rocks, soils, water and organisms, and it moves between ecosystems mainly by water rather than by air. Phosphorus is often the limiting nutrient in freshwater and many terrestrial ecosystems, so its shortage restricts growth. Excess phosphorus from fertilizer, sewage and runoff stimulates algal blooms and eutrophication. Phosphate in rocks and soil minerals such as apatite is inorganic, whereas organic phosphates occur in living matter and decomposing organic material. Phosphorus fertilizer applied on land is carried by rivers into the ocean, which is a major human alteration of the cycle.

CO2 Henry's law and carbonate equilibria

Henry's law states that the dissolved concentration of a gas is proportional to its partial pressure in the gas phase, [\text{CO}_2(aq)] = K_H\,p_{\text{CO}_2}, where K_H is the Henry's law constant. For carbon dioxide at 25 °C the constant is about 3.4\times10^{-2} M/atm, so the dissolved concentration follows directly from the partial pressure. The dissolved CO_2 forms carbonic acid, which dissociates in two steps with K_1 \approx 4.3\times10^{-7} M and K_2 \approx 4.7\times10^{-11} M. In neutral water, [\text{H}^+] = 10^{-7} M, the bicarbonate concentration is K_1[\text{H}_2\text{CO}_3^*]/[\text{H}^+]. The carbonate concentration is smaller than bicarbonate by the factor K_2/[\text{H}^+], so bicarbonate usually dominates the dissolved inorganic carbon. The total dissolved CO_2 is the sum of the three species, and it is the quantity obtained from the atmospheric partial pressure.