Solubility and Complex-Ion Equilibria

Solubility equilibria are heterogeneous equilibria that describe the dissolution and precipitation of slightly soluble ionic compounds. Examples are commonly encountered in many chemical and biological processes. This chapter deals with the application of chemical equilibrium to heterogeneous systems involving saturated solutions of slightly soluble ionic compounds. When a slightly soluble salt is dissolved in water, it eventually reaches a point of saturation where equilibrium is established between the undissolved salt and its solution.

Measuring Chemical Quantities: The Mole

A mole is defined as the amount of a given substance that contains the same number of atoms, molecules, or formula units as there are atoms in 12 g of carbon-12. For example, one mole of glucose contains the same number of glucose molecules as there are carbon atoms in 12 g of carbon-12. The number of atoms in exactly 12 g of carbon-12 has been determined to be 6.02×1023. This number, 6.02×1023, is called Avogadro’s number (NA). Therefore, a mole is the amount of a substance that contains Avogadro’s number of atoms, ions, molecules, or particles. For example: 1 mol He atoms = 6.02×1023 atoms 1 mol CH3 OH molecules = 6.02×1023 molecules 1 mol SO2−4 ions = 6.02×1023 ions The term molar mass is now commonly used as a general term for both formula mass and molecular mass. The molar mass of any substance is the mass in grams of one mole of the substance, and it is numerically equal to its formula mass (expressed in amu). For example, the formula mass of glucose, C6H12O6, is 180.0 amu. So the molar mass or the mass in grams of 1 mol of glucose is 180.0 g. In terms of chemical arithmetic, the mole is the most important number in chemistry. It provides useful stoichiometric information about reactants and products in any given chemical reaction. The quantities commonly encountered in chemical problems include the number of moles of a substance; the number of atoms, molecules, or formula units of a substance; and the mass in grams. These quantities are related and can be readily interconverted with the aid of the molar mass and Avogadro’s number. Calculations based on the mole can be carried out by using conversion factors, or with simple equations based on the conversion factor.

Radioactivity and Nuclear Reactions

Nuclide: an atom containing a specified number of protons and neutrons in its nucleus—in other words, any particular atom under discussion. Unstable nuclide: one that will spontaneously disintegrate or emit radiation, thus giving off energy and altering to some new form (often another element). The new form may also be unstable; often it will be stable, that is, with no tendency to disintegrate. Unstable nuclides are also referred to as radioactive. Radioactivity: the spontaneous emission of radiation by elements with unstable nuclei. Radionuclide: a radioactive (that is, unstable) nuclide. Radioisotope: another more commonly seen term for radionuclide. Radioactive decay: the process whereby a radionuclide is converted to another form (usually another element) by emitting radiation. Parent nuclide: a nuclide undergoing radioactive decay. Daughter nuclide: the nuclide produced when a parent nuclide decays. Activity: the rate at which a sample of the material decays, usually expressed as the number of disintegrations per unit time. Naturally radioactive elements decay spontaneously by emitting alpha particles, beta particles, and gamma radiation. Other elements can be induced to decay by bombarding them with high-energy particles; this is known as artificial radioactivity. Like chemical reactions, equations representing nuclear reactions must be balanced. However, the method for balancing nuclear equations differs from that used for chemical equations. To balance a nuclear equation, the sum of the atomic numbers or particle charges (subscripts) and the sum of the mass numbers (superscripts) on both sides of the equation must be equal. When a nucleus undergoes alpha decay, it emits a particle that is identical to a helium nucleus, with an atomic number of 2 and a mass number of 4. Since the emission of an alpha particle from the nucleus results in a loss of 2 protons and 2 neutrons, when writing a nuclear reaction involving an alpha decay, subtract 4 from the mass number and 2 from the atomic number.

Oxidation and Reduction Reactions

Oxidation-reduction reactions, or redox reactions, occur in many chemical and biochemical systems. The process involves the complete or partial transfer of electrons from one atom to another. Oxidation and reduction processes are complementary. For every oxidation, there is always a corresponding reduction process. This is because for a substance to gain electrons in a chemical reaction, another substance must be losing these electrons. Oxidation is defined as a process by which an atom or ion loses electrons. This can occur in several ways: • Addition of oxygen or other electronegative elements to a substance:. . . 2 Mg(s)+O2(g) → 2 MgO(s) . . .2 Mg(s)+O2(g) → MgCl2 (s). . . • Removal of hydrogen or other electropositive elements from a substance: . . . H2S(g)+Cl2(g) → 2 HCl(g)+S(s) . . .Here, H2S is oxidized. • The direct removal of electrons from a substance: . . . 2 FeCl2 (s)+Cl2(g) → 2 FeCl3 (s) . . . Fe2+ → Fe3+ +e− . . . Reduction is defined as the process by which an atom or ion gains electrons. This can occur in the following ways: • Removal of oxygen or other electronegative elements from a substance: . . . MgO(s)+H2(g) → Mg(s)+H2O(g). . . • Addition of hydrogen or other electropositive elements to a substance: . . . H2(g)+Br2(g) → 2 HBr(g). . . 2 Na(s)+Cl2(g) → 2 NaCl(s). . . Here, chlorine (Cl2) is reduced. • The addition of electrons to a substance: . . . Fe3+ +e− → Fe2+ . . . Oxidation number or oxidation state is a number assigned to the atoms in a substance to describe their relative state of oxidation or reduction. These numbers are used to keep track of electron transfer in chemical reactions. Some general rules are used to determine the oxidation number of an atom in free or combined state. 1. Any atom in an uncombined (or free) element (e.g., N2, Cl2, S8, O2, O3, and P4) has an oxidation number of zero. 2. Hydrogen has an oxidation number of +1 except in metal hydrides (e.g., NaH, MgH2) where it is −1. 3. Oxygen has an oxidation number of −2 in all compounds except in peroxides (e.g., H2O2, Na2O2) where it is –1.

Ideal Solutions and Colligative Properties

Colligative properties of solutions are those that depend only on the number of solute particles (molecules or ions) in the solution rather than on their chemical or physical properties. The colligative properties that can be measured experimentally include: • Vapor pressure depression • Boiling point elevation • Freezing point depression • Osmotic pressure Noncolligative properties, on the other hand, depend on the identity of the dissolved species and the solvent. Examples include solubility, surface tension, and viscosity. The addition of a solute to a solvent typically causes the vapor pressure of the solvent (above the resulting solution) to be lower than the vapor pressure above the pure solvent. As the concentration of the solute in the solution changes, so does the vapor pressure of the solvent above a solution. The vapor pressure of a solution of a nonvolatile solute is always lower than that of the pure solvent. For example, an aqueous solution of NaCl has a lower vapor pressure than pure water at the same temperature. The addition of solute to a pure solvent depresses the vapor pressure of the solvent. This observation, first made by Raoult, is now commonly known as Raoult’s law. The law states that the lowering of vapor pressure of a solution containing non-volatile solute is proportional to the mole fraction of the solute.

Solution Chemistry

A solution is a homogeneous mixture of two or more substances. It is usually made up of a solute and a solvent. Generally, Solute+Solvent = Solution A solute is any substance that is dissolved in a solvent. For example, when granulated sugar dissolves in water to give a clear sugar solution, the sugar is the solute, while water is the solvent. Relative to the solvent, a solute is usually present in small amounts. A solvent is any substance in which a solute dissolves. It is usually the part of the solution that is present in the largest amount. Two liquids are said to be miscible if they form a single phase (homogeneous solution) or dissolve in each other in all proportions. For example, ethanol and water are miscible. If two liquids do not form a single phase (or do not dissolve in each other) in any appreciable amount, they are said to be immiscible. For example, water and oil are immiscible. When mixed, they separate into two distinct layers. Substances that are only slightly soluble in a given solvent are said to be insoluble. An aqueous solution is one in which water is the solvent. A dilute solution is one that contains a small amount of solute compared to the maximum amount the solvent can dissolve at that temperature. A concentrated solution is one that contains a large amount of solute compared to the maximum amount the solvent can dissolve at that temperature. A saturated solution is one that is in equilibrium with undissolved solute at a given temperature and pressure: Solute(solid) ⇌ Solute(dissolved) In other words, it contains the maximum amount of solute that can be dissolved at that particular temperature. An unsaturated solution contains less solute than the maximum amount (saturated solution) possible at the same temperature. A supersaturated solution is a solution that contains more solute than the saturated solution at the same temperature. This type of solution is very unstable. When it is agitated, or a speck of the solute is added to it, the excess solute will begin to crystallize out rapidly from the solution until the concentration becomes equal to that of the saturated solution.

Gas Laws

Volumes and densities of gases vary significantly with changes in pressure and temperature. This means that measurements of the volumes of gases will likely vary from one laboratory to another. To correct for this, scientists have adopted a set of standard conditions of temperature and pressure (STP) as a reference point in reporting all measurements involving gases. They are 0°C (or 273 K) and 760mmHg or 1 atm (or 1.013×105 N m−2 in S.I. units). Therefore standard temperature and pressure, as used in calculations involving gases, are defined as 0°C (or 273 K) and 1 atmosphere (or 760 torr). (Note: For calculations involving the gas laws, temperature must be in K.) Boyle’s law states that the volume of a given mass of gas at constant temperature is inversely proportional to the pressure. The law can be expressed in mathematical terms: V α 1/P or PV = k at constant n and T Since P×V = constant, problems dealing with P–V relationships can be solved by using the simplified equation: P1V1 = P2V2 Here P1, V1 represent one set of conditions and P2, V2 represent another set of conditions for a given mass of gas. Charles’s law states that the volume of a given mass of gas is directly proportional to its absolute temperature. So if the absolute temperature is doubled, say from 300 K to 600 K, the volume of the gas will also double. A plot of the volume of a gas versus its temperature (K) gives a straight line. A notable feature of such a plot is that the volume of all gases extrapolates to zero at the same temperature, −273.2◦C. This point is defined as 0 K, and is called absolute zero. Thus, the relationship between the Kelvin and Celsius temperature scales is given as: K = 0°C + 273. Scientists believe that the absolute zero temperature, 0 K, cannot be attained, although some laboratories have reported producing 0.0001 K.

Chemical Thermodynamics

Chemical thermodynamics is the study of the energy changes and transfers associated with chemical and physical transformations. Energy is the ability to do work or to transfer heat. A spontaneous process is one that can occur on its own without any external influence. A spontaneous process always moves a system in the direction of equilibrium. When a process or reaction cannot occur under the prescribed conditions, it is nonspontaneous. The reverse of a spontaneous process or reaction is always nonspontaneous. Heat (q) is the energy transferred between a system and its surroundings due to a temperature difference. Work (w) is the energy change when a force (F) moves an object through a distance (d). Thus. . . W = F ×d. . . . A system is a specified part of the universe (e.g., a sample or a reaction mixture we are studying). Everything outside the system is referred to as the surroundings. The universe is the system plus the surroundings. A state function is a thermodynamic quantity that defines the present state or condition of the system. Changes in state function quantities are independent of the path (or process) used to arrive at the final state from the initial state. Examples of state functions include enthalpy change (ΔH), entropy change, (ΔS) and free energy change, (ΔG). The internal energy of a system is the sum of the kinetic and potential energies of the particles making up the system. While it is not possible to determine the absolute internal energy of a system, we can easily measure changes in internal energy (which correspond to energy given off or absorbed by the system). The change in internal energy, . . . ΔE, is: ΔE = Efinal –Einitial. . . . The first law of thermodynamics, also called the law of conservation of energy, states that the total amount of energy in the universe is constant, that is, energy can neither be created nor destroyed. It can only be converted from one form into another. In mathematical terms, the law states that the change in internal energy of a system, ΔE, equals q+w. That is,. . . ΔE = q+w. . . In other words, the change in E is equal to the heat absorbed (or emitted) by the system, plus work done on (or by) the system.

Ionic Equilibria and pH

Water is a weak acid. At 25°C, pure water ionizes to form a hydrogen ion and a hydroxide ion: H2O ⇋ H+ + OH− Hydration of the proton (hydrogen ion) to form hydroxonium ion is ignored here for simplicity. This equilibrium lies mainly to the left; that is, the ionization happens only to a slight extent. We know that 1 L of pure water contains 55.6 mol. Of this, only 10−7 mol actually ionizes into equal amounts of [H+] and [OH−], i.e., [H+] = [OH−] = 10−7M Because these concentrations are equal, pure water is neither acidic nor basic. A solution is acidic if it contains more hydrogen ions than hydroxide ions. Similarly, a solution is basic if it contains more hydroxide ions than hydrogen ions. Acidity is defined as the concentration of hydrated protons (hydrogen ions); basicity is the concentration of hydroxide ions. Pure water ionizes at 25°C to produce 10−7 M of [H+] and 10−7 M of [OH−]. The product Kw = [H+]×[OH−] = 10−7 M×10−7 M= 10−14 M is known as the ionic product of water. Note that this is simply the equilibrium expression for the dissociation of water. This equation holds for any dilute aqueous solution of acid, base, and salt. The pH of a solution is defined as the negative logarithm of the molar concentration of hydrogen ions. The lower the pH, the greater the acidity of the solution. Mathematically: pH=−log10[ H+] or −log10[H3O+] This can also be written as: pH = log10 1/[H+] or log10 1/[H3O+] Taking the antilogarithm of both sides and rearranging gives: [H+] = 10−pH This equation can be used to calculate the hydrogen ion concentration when the pH of the solution is known.

Volumetric Analysis

Volumetric analysis is a chemical analytical procedure based on measurement of volumes of reaction in solutions. It uses titration to determine the concentration of a solution by carefully measuring the volume of one solution needed to react with another. In this process, a measured volume of a standard solution, the titrant, is added from a burette to the solution of unknown concentration. When the two substances are present in exact stoichiometric ratio, the reaction is said to have reached the equivalence or stoichiometric point. In order to determine when this occurs, another substance, the indicator, is also added to the reaction mixture. This is an organic dye which changes color when the reaction is complete. This color change is known as the end point; ideally, it will coincide with the equivalence point. For various reasons, there is usually some difference between the two, though if the indicator is carefully chosen, the difference will be negligible. A typical titration is based on a reaction of the general type aA+bB → products where A is the titrant, B the substance titrated, and a:b is the stoichiometric ratio between the two. Some indicators include Litmus, Methyl Orange, Methyl Red, Phenolphthalein, and Thymol Blue. Titration can be applied to any of the following chemical reactions: • Acid–base • Complexation • Oxidation–reduction • Precipitation Only acid–base and oxidation–reduction titration will be treated here, though the fundamental principles are the same in all cases. Acid–base titration involves measuring the volume of a solution of the acid (or base) that is required to completely react with a known volume of a solution of a base (or acid). The relative amounts of acid and base required to reach the equivalence point depend on their stoichiometric coefficients. It is therefore critical to have a balanced equation before attempting calculations based on acid–base reactions. Below we define some of the common terms associated with acid–base reactions. A molar solution is one that contains one mole of the substance per liter of solution. For example, a molar solution of sodium hydroxide contains 40 g (NaOH=40 g/mol) of the solute per liter of solution. As described in chapter 13, the concentration of a solution expressed in moles per liter of solution is known as the molarity of the solution.