States of Matter Equations of State Ideal GasDeviations Van der WaalsVirial Series Kinetic Molecular Model Corresponding States Fluids Reduced Variables Condensed Phases Berth., R-K Mixtures

Hi Chem.412 students, Due to a last minute appointment, there is a good chance that I will not be able to make the 9:00 a.m. class on time tomorrow (Wednesday). Therefore, I am substituting the Wednesday 9 am lecture on the next topic “Nature of Matter and Mystery of the Universe” with the following You-Tube videos: (Click on the hyperlinks to see them in sequence) Wednesday afternoon and evening labs go on as scheduled. Video #1 (explanation of the Big Bang, ~5.5 minutes) S. Hawking: Big BangS. Hawking: Big Bang Video #2 (How to find particles, ~17 min) Particle HuntersParticle Hunters Video #3 (A Rap on the LHC, ~4.5 min) HadronsHadrons [Please be somewhat skeptical and don’t take any offense regarding comments after these (free) videos … these are “uncontrolled” public comments that can be at times insensitive and offensive!] Please watch them before Friday’s class since I will be skipping the beginning parts of the next powerpoint (States of Matter). Wednesday afternoon and evening labs go on as scheduled. Dr. Ng. 9/11/13 – Lec sub

Matter Three States of Matter LiquidGas Microscopic Molecular Size Molecular Shape Velocity/Momentum Intermolecular Forces Macroscopic Temperature Pressure Viscosity Density Solid CyberChem: Big Bang S. Hawking: Big Bang

Mystery of our Universe: A Matter of Family ? Quarks Fermions - Particles Leptons HadronsHadrons neutron proton e -  -  - [ ] nuclidesatoms elements mixturescompounds moleculescomplexes homogeneousheterogeneous Bosons – Force carriers Strong (gluon) Weak (+W, -W, Z) Electromag. (photon) Gravity (graviton) Three families 1)u d e - e 2) c s  -  3) t b  - 

Mystery of our Universe: Quarks Particle Hunters Big Bang Theory physics episodes

We can combine these into a general gas law: The Ideal Gas Equation Boyle’s Law: Charles’s Law: Avogadro’s Law:

R = gas constant, then The ideal gas equation is: R = L·atm/mol·K = J/mol·K J = kPa·L = kPa·dm 3 = Pa·m 3 Real Gases behave ideally at low P and high T. The Ideal Gas Equation

Calculate the number of air molecules in 1.00 cm 3 of air at 757 torr and 21.2 o C. Mathcad

Calculate the number of air molecules in 1.00 cm 3 of air at 757 torr and 21.2 o C. MathcadF12

Low P  Ideal

High T  Ideal

Gas Densities and Molar Mass The density of a gas behaving ideally can be determined as follows: The density of a gas was measured at 1.50 atm and 27°C and found to be 1.95 g/L. Calculate the molecular weight of the gas? If the gas is a homonuclear diatomic, what is this gas?Calculatewhat is Plotting data of density versus pressure (at constant T) can give molar mass. Density of an Ideal-Gas Mathcad

Density of an Ideal-Gas Derivation of :

Plotting data of density versus pressure (at constant T) can give molar mass.

Molar Mass of an Non-Ideal Gas Generally, density changes with P at constant T, use power series: First-order approximation: Plotting data of ρ/P vs. P (at constant T) can give molar mass. Deviation of Density from Ideal

Plotting data of ρ/P vs. P (at constant T) can give molar mass.

Dalton’s Law: in a gas mixture the total pressure is given by the sum of partial pressures of each component: Each gas obeys the ideal gas equation: Ideal Gas Mixtures and Partial Pressures Density?

Partial Pressures and Mole Fractions Let n i be the number of moles of gas i exerting a partial pressure P i, then where χ i is the mole fraction. Ideal Gas Mixtures and Partial Pressures CyberChem (diving) videoCyberChem (diving) video:

The van der Waals Equation General form of the van der Waals equation: Real Gases: Deviations from Ideal Behavior Corrects for molecular volume Corrects for molecular attraction

Real Gases: Deviations from Ideal Behavior Berthelot Dieterici Redlick-Kwong

The van der Waals Equation  Calculate the pressure exerted by 15.0 g of H 2 in a volume of 5.00 dm 3 at 300. K. Calculate

The van der Waals Equation  Calculate the molar volume of H 2 gas at 40.0 atm and 300. K. Calculate

The van der Waals Equation Can solve for P and T, but what about V? Let: V m = V/n { molar volume, i.e. n set to one mole} Cubic Equation in V, not solvable analytically! Use Newton’s Iteration Method: Mathcad: Text Solution Mathcad: Matrix Solution

Picture

Postulates: –Gases consist of a large number of molecules in constant random motion. –Volume of individual molecules negligible compared to volume of container. –Intermolecular forces (forces between gas molecules) negligible. Kinetic Energy => Root-mean-square Velocity => Kinetic Molecular Theory

Kinetic Molecular Model – Formal Derivation Preliminary note: Pressure of gas caused by collisions of molecules with rigid wall. No intermolecular forces, resulting in elastic collisions. Consideration of Pressure: Identify F=(∆p/∆t) ≡ change in momentum wrt time.

x z y Wall of Unit Area A Consider only x-direction: ( m=molecule ) ( w=wall ) BeforeAfter p m =mup m ’ =-mu p w =0p w ’ =?

Assumption: On average, half of the molecules are hitting wall and other not. In unit time => half of molecules in volume (Au) hits A If there are N molecules in volume V, then number of collisions with area A in unit time is: And since each collision transfers 2mu of momentum, then Total momentum transferred per unit time = p w ’ x (# collisions)

Mean Square Velocity: In 3-D, can assume isotropic distribution: Substituting [eqn 3] into [eqn 2b] gives:

Mathcad

Molecular Effusion and Diffusion The lower the molar mass, M, the higher the rms. Kinetic Molecular Theory

Concept of Virial Series Define: Z = compressibility factor Virial Series: Expand Z upon molar concentration [ n/V ] or [ 1/V m ] B=f(T) => 2 nd Virial Coeff., two-molecule interactions C=f(T) => 3 rd Virial Coeff., three-molecule interactions Virial Series tend to diverge at high densities and/or low T.

Concept of Virial Series – vdw example

Phase Changes

Critical Temperature and Pressure Gases liquefied by increasing pressure at some temperature. Critical temperature: the minimum temperature for liquefaction of a gas using pressure. Critical pressure: pressure required for liquefaction. Phase Changes

Critical Temperature and Pressure Phase Changes

Phase Diagrams

The Phase Diagrams of H 2 O and CO 2 Phase Diagrams

Reduced Variables

PVT Variations among Condensed Phases Brief Calculus Review

PVT Variations among Condensed Phases

Brief Calculus Review – F13 -1 Mathcad

Brief Calculus Review – F13 - 2

Brief Calculus Review – F13 - 3

Brief Calculus Review – F13 - 4

Brief Calculus Review – F14 -1

Brief Calculus Review – F14 -2

Brief Calculus Review – F14 -3

Brief Calculus Review – F14 -4

Exact and Partial Differentials: Tutorial A right-circular cylinder has a base radius ( r ) of 2.00 cm and a height ( h ) of 5.00 cm. (a) Find the “approximate change” in the volume (  V ) of the cylinder if r is increased by 0.30 cm and h is decreased by 0.40 cm. Express the answer in terms of  cm 3. This is the “differential” volume change. Then compare to the “real” volume change from algebraic calculations of initial and final volumes. (b)Repeat for r increase of 0.10 cm and h decrease of 0.10 cm. (c)Repeat for r increase of cm and h decrease of cm. What is your conclusion regarding the comparisons?

A right-circular cylinder has a base radius ( r ) of 2.00 cm and a height ( h ) of 5.00 cm.

Mathcad-file

A right-circular cylinder has a base radius ( r ) of 2.00 cm and a height ( h ) of 5.00 cm.

Differential Algebra r / cmh / cm  r / cm  h / cm  V /  *cm 3 V1V2V'=V2-V1DiffDiff% E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-07

States of Matter Equations of State Ideal GasDeviations Van der WaalsVirial Series Kinetic Molecular Model Corresponding States Fluids Reduced Variables Condensed Phases Berth., R-K Mixtures