BCHEM259 Physical Chemistry I Solutions PDF 2024

BCHEM259: Physical Chemistry I Solutions I & II Dr. Elliot Sarpong Menkah Faculty of Physical and Computational Sciences Chemistry Department...

BCHEM259: Physical Chemistry I Solutions I & II Dr. Elliot Sarpong Menkah Faculty of Physical and Computational Sciences Chemistry Department February 24, 2024 Elliot S. Menkah, Ph.D. (KNUST) Solutions I January, 2024 1 / 10  Phase Equilibria Phase Equilibria - Phase diagrams - Melting and freezing points - Boiling point - Phase diagrams of One-Component systems - Phase diagrams of Multi-Component systems Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 2 / 10  Phase Diagrams A phase diagram is a diagram that shows the state of a system under varying conditions. The state of a system under varying T and P is termed the P-T diagram. In a one-component system’s p − T phase diagram, volume and the molar amount n are kept constant. line shows two(2) phases coexist thermodynamically (phase boundaries). Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 3 / 10  Phase Diagrams A phase diagram is a diagram that shows the state of a system under varying conditions. The state of a system under varying T and P is termed the P-T diagram. In a one-component system’s p − T phase diagram, volume and the molar amount n are kept constant. line shows two(2) phases coexist thermodynamically (phase boundaries). Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 3 / 10  Phase Diagrams A phase diagram is a diagram that shows the state of a system under varying conditions. The state of a system under varying T and P is termed the P-T diagram. In a one-component system’s p − T phase diagram, volume and the molar amount n are kept constant. line shows two(2) phases coexist thermodynamically (phase boundaries). Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 3 / 10  Phase Diagrams A phase diagram is a diagram that shows the state of a system under varying conditions. The state of a system under varying T and P is termed the P-T diagram. In a one-component system’s p − T phase diagram, volume and the molar amount n are kept constant. line shows two(2) phases coexist thermodynamically (phase boundaries). Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 3 / 10  Phase Diagrams The boiling equilibrium condition is most easily represented as a line on a plot of the pressure, p against the temperature, T A line of positive gradient, so any point on this line corresponds to the situation where liquid and vapor are at equilibrium. Away from the line, the equilibrium condition no longer applies; above the line (at increased pressure and/or decreased temperature), only liquid exists, whereas below the line (at increased temperature and/or decreased pressure) there is only vapor. The line terminates at c, the critical point where liquid and gas are indistinguishable. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 3 / 10  Phase Diagrams The boiling equilibrium condition is most easily represented as a line on a plot of the pressure, p against the temperature, T A line of positive gradient, so any point on this line corresponds to the situation where liquid and vapor are at equilibrium. Away from the line, the equilibrium condition no longer applies; above the line (at increased pressure and/or decreased temperature), only liquid exists, whereas below the line (at increased temperature and/or decreased pressure) there is only vapor. The line terminates at c, the critical point where liquid and gas are indistinguishable. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 3 / 10  Phase Diagrams The boiling equilibrium condition is most easily represented as a line on a plot of the pressure, p against the temperature, T A line of positive gradient, so any point on this line corresponds to the situation where liquid and vapor are at equilibrium. Away from the line, the equilibrium condition no longer applies; above the line (at increased pressure and/or decreased temperature), only liquid exists, whereas below the line (at increased temperature and/or decreased pressure) there is only vapor. The line terminates at c, the critical point where liquid and gas are indistinguishable. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 3 / 10  Phase Diagrams The boiling equilibrium condition is most easily represented as a line on a plot of the pressure, p against the temperature, T A line of positive gradient, so any point on this line corresponds to the situation where liquid and vapor are at equilibrium. Away from the line, the equilibrium condition no longer applies; above the line (at increased pressure and/or decreased temperature), only liquid exists, whereas below the line (at increased temperature and/or decreased pressure) there is only vapor. The line terminates at c, the critical point where liquid and gas are indistinguishable. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 3 / 10  Phase Diagram Line shows two(2) phases coexist thermodynamically (phase boundaries). Boiling point is the temperature at which the vapor pressure of the liquid is equal to the external pressure. When pressure is kept at 1 atm. Melting point is the temperature at which the liquid and solid phases coexist and equals the freezing temperature when pressure is kept at 1 atm. Critical point is when liquid and vapor merge to form a supercritical fluid Triple point is where 3 phases coexist. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 4 / 10  Phase Diagram Why is the phase diagram important? These diagrams allow prediction of the nature of the phase(s) present for any condition of p and T (any state point on the diagram). , p, and T, which are both intensive variables, are all that is required to specify the state of the system The state is its molecular disposition or the number and amount of the phases present and their composition This is because although the pressure, p, the molar volume, Vm , and the temperature, T, are the three intensive variables used to define the state of a single substance system, these are linked by an equation of state Knowledge of only two of these variables is necessary, as the equation of state can be used to calculate the third and specify the state. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 4 / 10  Phase Diagram The solid phase is favored at low temperatures and high pressures (low volumes and entropies). The gas (or vapor) is favored at high temperatures and low pressures (increased volumes and entropies). The liquid is favored at intermediate conditions, simplifying phase diagram labeling. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 4 / 10  Phase Diagram The solid phase is favored at low temperatures and high pressures (low volumes and entropies). The gas (or vapor) is favored at high temperatures and low pressures (increased volumes and entropies). The liquid is favored at intermediate conditions, simplifying phase diagram labeling. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 4 / 10  Phase Diagram The solid phase is favored at low temperatures and high pressures (low volumes and entropies). The gas (or vapor) is favored at high temperatures and low pressures (increased volumes and entropies). The liquid is favored at intermediate conditions, simplifying phase diagram labeling. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 4 / 10  Melting and Freezing Points Melting and Freezing Points A(s) ⇌ A(l) (⋆) (1) When a pure solid species is heated through its melting point (melting temperature), the solid changes to a liquid. At the melting point, an equilibrium is established between the solid and liquid phases. The melting point and the freezing point are identical for a pure substance, but not necessarily for a mixture. Thus, a pure substance’s melting and freezing temperatures are identical; in this case, the terms can be used interchangeably. dG = Vdp − SdT Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 5 / 10  Melting and Freezing Points Melting and Freezing Points A(s) ⇌ A(l) (⋆) (1) At the melting point, equilibrium for pure species, A : A(s) ⇌ A(l) (⋆) ∆G = 0... then, (28) dp ∆S = (29) dT ∆V Entropy and volume changes are positive This means entropy increases at equilibrium as the solid melts, and volume increases due to melting. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 5 / 10  Melting and Freezing Points Melting and Freezing Points dp In this case, is positive(+), and increasing the pressure increases the dT melting temperature. A notable exception to this is water, as solid water (ice) has an open-hydrogen-bonded structure, which occupies more volume than liquid water. dp ∆V is negative(-) and dT is positive(+). Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 5 / 10  Boiling and Condensation Points Boiling and Condensation Point A(l) ⇌ A(g ) (⋆) dp ∆S = (30) dT ∆V Entropy and volume changes are positive Volume change, ∆V in a liquid-to-gas transition is larger than from solid-to-liquid transition. Volume occupied by 1 mole of gas is much larger than volume occupied by 1 mole of liquid. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 6 / 10  Boiling and Condensation Points A(l) ⇌ A(g ) (⋆) dp ∆S = (30) dT ∆V Volume change, ∆V in a liquid-to-gas transition is larger than from dp solid-to-liquid transition, this implies that dT has a larger impact on a Boiling-Condensation process than on a Melting-Freezing process. dp A changed/increase in dT whiles attaining equilibrium will the increase pressure which will compress the vapor volume. A compression of the vapour volume will increase its density, whilst the increase in temperature tends to weaken the liquid intermolecular forces, decreasing its density. Eventually, at the critical point, characterized by a critical pressure and a critical temperature, the densities of vapor and liquid become equal, the two phases are indistinguishable and there is no longer any measurable phase transition. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 7 / 10  Phase Diagram - CO2 Phase Diagram of Carbon dioxide, CO2 CO2 is solid as dry ice and can also sublime at 5 atm. At this point, the solid and liquid phases coexists. To form a liquid, pressure is required An Increase in Pressure results in increased melting point. It can be a supercritical fluid where it shows the properties of both a liquid and gas. A solid CO2 is stable at higher Temp because it has a higher density. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 8 / 10  Phase Diagram - CO2 Phase Diagram of Carbon dioxide, CO2 CO2 is solid as dry ice and can also sublime at 5 atm. At this point, the solid and liquid phases coexists. To form a liquid, pressure is required An Increase in Pressure results in increased melting point. It can be a supercritical fluid where it shows the properties of both a liquid and gas. A solid CO2 is stable at higher Temp because it has a higher density. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 8 / 10  Phase Diagram - CO2 Phase Diagram of Carbon dioxide, CO2 CO2 is solid as dry ice and can also sublime at 5 atm. At this point, the solid and liquid phases coexists. To form a liquid, pressure is required An Increase in Pressure results in increased melting point. It can be a supercritical fluid where it shows the properties of both a liquid and gas. A solid CO2 is stable at higher Temp because it has a higher density. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 8 / 10  Phase Diagram - CO2 Phase Diagram of Carbon dioxide, CO2 CO2 is solid as dry ice and can also sublime at 5 atm. At this point, the solid and liquid phases coexists. To form a liquid, pressure is required An Increase in Pressure results in increased melting point. It can be a supercritical fluid where it shows the properties of both a liquid and gas. A solid CO2 is stable at higher Temp because it has a higher density. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 8 / 10  Phase Diagram - CO2 Phase Diagram of Carbon dioxide, CO2 CO2 is solid as dry ice and can also sublime at 5 atm. At this point, the solid and liquid phases coexists. To form a liquid, pressure is required An Increase in Pressure results in increased melting point. It can be a supercritical fluid where it shows the properties of both a liquid and gas. A solid CO2 is stable at higher Temp because it has a higher density. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 8 / 10  Phase Diagram - H2 O Triple point is formed at 273.15 K (zero point on the Celsius scale) and 611 Pa. Water is more stable with a negative curve because it is denser than ice at higher temperatures. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 9 / 10  Phase Diagram - H2 O Triple point is formed at 273.15 K (zero point on the Celsius scale) and 611 Pa. Water is more stable with a negative curve because it is denser than ice at higher temperatures. Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 9 / 10  Gibbs Phase Rule Gibbs Phase Rule F=C−P+2 (31) A useful concept, especially when applied to multi-component (multi-substance) systems. The number of degrees of freedom, F , of the system. This is the minimum number of intensive variables that can be varied without changing the number of phases in the system. C = components (compounds or chemical entities) P = physical states F = number of independent variables to vary (T , P, V , n) The Gibbs phase rule allows calculation of the number of intensive parameters that can be varied independently (F ) without disturbing equilibrium for a given number of phases (P) and number of component (C ) system Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 10 / 10  Gibbs Phase Rule Gibbs Phase Rule F=C−P+2 (31) A useful concept, especially when applied to multi-component (multi-substance) systems. The number of degrees of freedom, F , of the system. This is the minimum number of intensive variables that can be varied without changing the number of phases in the system. C = components (compounds or chemical entities) P = physical states F = number of independent variables to vary (T , P, V , n) The Gibbs phase rule allows calculation of the number of intensive parameters that can be varied independently (F ) without disturbing equilibrium for a given number of phases (P) and number of component (C ) system Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 10 / 10  Gibbs Phase Rule Gibbs Phase Rule F=C−P+2 (31) A useful concept, especially when applied to multi-component (multi-substance) systems. The number of degrees of freedom, F , of the system. This is the minimum number of intensive variables that can be varied without changing the number of phases in the system. C = components (compounds or chemical entities) P = physical states F = number of independent variables to vary (T , P, V , n) The Gibbs phase rule allows calculation of the number of intensive parameters that can be varied independently (F ) without disturbing equilibrium for a given number of phases (P) and number of component (C ) system Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 10 / 10  Gibbs Phase Rule Gibbs Phase Rule F=C−P+2 (31) A useful concept, especially when applied to multi-component (multi-substance) systems. The number of degrees of freedom, F , of the system. This is the minimum number of intensive variables that can be varied without changing the number of phases in the system. C = components (compounds or chemical entities) P = physical states F = number of independent variables to vary (T , P, V , n) The Gibbs phase rule allows calculation of the number of intensive parameters that can be varied independently (F ) without disturbing equilibrium for a given number of phases (P) and number of component (C ) system Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 10 / 10  Gibbs Phase Rule Gibbs Phase Rule F=C−P+2 (31) A useful concept, especially when applied to multi-component (multi-substance) systems. The number of degrees of freedom, F , of the system. This is the minimum number of intensive variables that can be varied without changing the number of phases in the system. C = components (compounds or chemical entities) P = physical states F = number of independent variables to vary (T , P, V , n) The Gibbs phase rule allows calculation of the number of intensive parameters that can be varied independently (F ) without disturbing equilibrium for a given number of phases (P) and number of component (C ) system Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 10 / 10  Gibbs Phase Rule Gibbs Phase Rule F=C−P+2 (31) A useful concept, especially when applied to multi-component (multi-substance) systems. The number of degrees of freedom, F , of the system. This is the minimum number of intensive variables that can be varied without changing the number of phases in the system. C = components (compounds or chemical entities) P = physical states F = number of independent variables to vary (T , P, V , n) The Gibbs phase rule allows calculation of the number of intensive parameters that can be varied independently (F ) without disturbing equilibrium for a given number of phases (P) and number of component (C ) system Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 10 / 10  Gibbs Phase Rule Gibbs Phase Rule F=C−P+2 (31) A useful concept, especially when applied to multi-component (multi-substance) systems. The number of degrees of freedom, F , of the system. This is the minimum number of intensive variables that can be varied without changing the number of phases in the system. C = components (compounds or chemical entities) P = physical states F = number of independent variables to vary (T , P, V , n) The Gibbs phase rule allows calculation of the number of intensive parameters that can be varied independently (F ) without disturbing equilibrium for a given number of phases (P) and number of component (C ) system Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 10 / 10  Gibbs Phase Rule F=C−P+2 (31) For a double phase; F = 1 - 2 + 2 = 1; only one variable can be changed. e.g. T or P (univariant) For a single phase; F = 1 - 1 + 2 = 2; two degrees of freedom For a triple phase; F =1−3+2=0 (32) Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 10 / 10  Gibbs Phase Rule F=C−P+2 (31) For a double phase; F = 1 - 2 + 2 = 1; only one variable can be changed. e.g. T or P (univariant) For a single phase; F = 1 - 1 + 2 = 2; two degrees of freedom For a triple phase; F =1−3+2=0 (32) Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 10 / 10  Gibbs Phase Rule F=C−P+2 (31) For a double phase; F = 1 - 2 + 2 = 1; only one variable can be changed. e.g. T or P (univariant) For a single phase; F = 1 - 1 + 2 = 2; two degrees of freedom For a triple phase; F =1−3+2=0 (32) Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 10 / 10  Gibbs Phase Rule F=C−P+2 (31) For a double phase; F = 1 - 2 + 2 = 1; only one variable can be changed. e.g. T or P (univariant) For a single phase; F = 1 - 1 + 2 = 2; two degrees of freedom For a triple phase; F =1−3+2=0 (32) Elliot S. Menkah, Ph.D. (KNUST) Phase Equilibria January, 2024 10 / 10