Lecture 49 More on Phase Transition, binary system Critical point Tricritical point Binary system Osmosis pressure Mixture

Gibbs' phase rule The rule applies to non-reactive multi-component heterogeneous systems in thermodynamic equilibrium and is given by the equality 𝐹=𝐶−𝑃+2 𝐹 is the number of degrees of freedom 𝐶 is the number of components 𝑃 is the number of phases The number of degrees of freedom is the number of independent intensive variables

The composition of each phase is determined by 𝐶 – 1 intensive variables (such as mole fractions) in each phase The total number of variables is (𝐶–1)𝑃 + 2, where the extra two are temperature 𝑇 and pressure 𝑝.

The number of degrees of freedom 𝐹 = (𝐶–1)𝑃 + 2 – 𝐶(𝑃–1) = 𝐶 – 𝑃 + 2 Since the phases are in thermodynamic equilibrium with each other, the chemical potentials of the phases must be equal. The number of equality relationships determines the number of degrees of freedom. For example, the equation 𝜇 𝑙𝑖𝑞 (𝑇, 𝑝) = 𝜇 𝑣𝑎𝑝 (𝑇, 𝑝), defines temperature as a function of pressure or vice versa The number of constraints are 𝐶(𝑃–1), since the chemical potential of each component must be equal in all phases. The number of degrees of freedom 𝐹 = (𝐶–1)𝑃 + 2 – 𝐶(𝑃–1) = 𝐶 – 𝑃 + 2

Pure substances (one component) For pure substances 𝐶 = 1 so that 𝐹 = 3 – 𝑃 In a single phase (𝑃 = 1) condition, two variables (𝐹 = 2), such as temperature and pressure, can be chosen independently to be any pair of values consistent with the phase. However, if the temperature and pressure combination ranges to a point where the pure component undergoes a separation into two phases (𝑃 = 2), 𝐹 decreases from 2 to 1. When the system enters the two-phase region, it becomes no longer possible to independently control temperature and pressure.

Critical point Critical point is the end point of a phase equilibrium curve. The most prominent example is the liquid-vapor critical point, the end point of the pressure- temperature curve that designates conditions under which a liquid and its vapor can coexist. At the critical point, defined by a critical temperature 𝑇 𝑐 and a critical pressure 𝑝 𝑐 , phase boundaries vanish.

At the critical point, only one phase exists At the critical point, only one phase exists. The heat of vaporization is zero. There is an inflection point in the constant-temperature line (critical isotherm) on a PV diagram. This means that at the critical point: Above the critical point one has a state of matter that is continuously connected with (can be transformed without phase transition into) both the liquid and the gaseous state. It is called supercritical fluid.

Tricritical point It is possible that three phases, such as solid, liquid and vapour, can exist together in equilibrium (𝑃 = 3). If there is only one component, there are no degrees of freedom (𝐹 = 0) when there are three phases. Therefore, in a single-component system, this three-phase mixture can only exist at a single temperature and pressure, which is known as a triple point. Here there are two equations 𝜇 𝑠𝑜𝑙 (𝑇, 𝑝) = 𝜇 𝑙𝑖𝑞 (𝑇, 𝑝) = 𝜇 𝑣𝑎𝑝 (𝑇, 𝑝) which are sufficient to determine the two variables T and p. It is also possible for other sets of phases to form a triple point, for example in the water system there is a triple point where ice I, ice III and liquid can coexist.

If four phases of a pure substance were in equilibrium (𝑃 = 4), the phase rule would give 𝐹 = −1, which is meaningless, since there cannot be −1 independent variables. This explains the fact that four phases of a pure substance (such as ice I, ice III, liquid water and water vapour) are not found in equilibrium at any temperature and pressure.

Two-component systems For binary mixtures of two chemically independent components, 𝐶 = 2 so that 𝐹 = 4 – 𝑃. In addition to temperature and pressure, the other degree of freedom is the composition of each phase, often expressed as mole fraction or mass fraction of one component. Four thermodynamic variables which may describe the system include temperature (𝑇), pressure (𝑝), mole fraction of component 1 in the liquid phase ( 𝑥 1𝐿 ) mole fraction of component 1 in the vapour phase ( 𝑥 1𝑉 ). However since two phases are in equilibrium, only two of these variables can be independent (𝐹 = 2).

As an example, consider the system of two completely miscible liquids such as toluene and benzene, in equilibrium with their vapours. This system may be described by a boiling-point diagram which shows the composition (mole fraction) of the two phases in equilibrium as functions of temperature (at a fixed pressure).

Osmosis Osmosis is the spontaneous net movement of solvent molecules through a semi- permeable membrane into a region of higher solute concentration, in the direction that tends to equalize the solute concentrations on the two sides.

Osmosis and Osmotic Pressure

Osmosis and Osmotic Pressure

Uses of Colligative Properties Desalination:

Osmosis and Osmotic Pressure Isotonic: Solutions have equal concentration of solute, and so equal osmotic pressure. Hypertonic: Solution with higher concentration of solute. Hypotonic: Solution with lower concentration of solute.

Hypotonic – The solution on one side of a membrane where the solute concentration is less than on the other side. Hypotonic Solutions contain a low concentration of solute relative to another solution. Hypertonic – The solution on one side of a membrane where the solute concentration is greater than on the other side. Hypertonic Solutions contain a high concentration of solute relative to another solution.

Below are examples of red blood cells in different types of solutions and shows what happened to the red blood cells. Hypertonicity is the presence of a solution that causes cells to shrink. Hypotonicity is the presence of a solution that causes cells to swell. Isotonicity is the presence of a solution that produces no change in cell volume.

Physics of Osmosis Water tends to flow from where its chemical potential is higher to where it is lower OR Movement of water from region where net hydrostatic pressure is higher to a region where it is low across a semi permeable membrane The pressure to prevent transport is the osmotic pressure

Calculating osmotic pressure: Van’t Hoffs Equation The ideal gas law states 𝑃𝑉=𝑛𝑅𝑇 so the osmotic pressure Π is Π=𝑀𝑅𝑇 Where M is the molar concentration of 𝑀=𝑛/𝑉 Osmotic pressure can be gotten from the condition that the chemical potential of the solvent on both sides of the membrane is equal.

Counting particles The influence of the solute depends only on the number of particles Molecular and ionic compounds will produce different numbers of particles per mole of substance 1 mole of a molecular solid → 1 mole of particles 1 mole of NaCl → 2 moles of particles 1 mole of CaCl2 → 3 moles of particles

Gas Mixtures Dalton’s law of partial pressure states that the total pressure exerted by a gas mixture is the sum of the partial pressures exerted by each component of the mixture: 𝑝 𝑡𝑜𝑡𝑎𝑙 = 𝑝 𝑖

Vapor pressure Vapor pressure is the pressure of a vapor in equilibrium with its non-vapor phases (i.e., liquid or solid). It is a measure of the tendency of molecules and atoms to escape from a liquid or a solid. A liquid's atmospheric pressure boiling point corresponds to the temperature at which its vapor pressure is equal to the surrounding atmospheric pressure and it is often called the normal boiling point.