States of Matter

Matter S. Hawking: Big Bang CyberChem: Big Bang

Mystery of our Universe: A Matter of Family ? Bosons – Force carriers Fermions - Particles Strong (gluon) Weak (+W , -W , Z) Electromag. (photon) Gravity (graviton) Quarks Leptons Hadrons neutron proton e- - - [  ] nuclides atoms Three families u d e- e c s -  t b  -  elements compounds mixtures molecules complexes homogeneous heterogeneous

Mystery of our Universe: Quarks Big B T physics: QM Gravity vs. strings Particle Hunters

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

The Ideal Gas Equation R = gas constant, then The ideal gas equation is: R = 0.08206 L·atm/mol·K = 8.3145 J/mol·K J = kPa·L = kPa·dm3 = Pa·m3 Real Gases behave ideally at low P and high T.

Calculate the number of air molecules in 1 Calculate the number of air molecules in 1.00 cm3 of air at 757 torr and 21.2 oC. Mathcad

Calculate the number of air molecules in 1 Calculate the number of air molecules in 1.00 cm3 of air at 757 torr and 21.2 oC. F12 Mathcad

Low P  Ideal

High T  Ideal

Density of an Ideal-Gas Mathcad 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? Plotting data of density versus pressure (at constant T) can give molar mass.

Density of an Ideal-Gas Derivation of :

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

The density of a gas was measured at 1 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?

Deviation of Density from Ideal Molar Mass of a 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.

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

Ideal Gas Mixtures and Partial Pressures 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: Density?

Density?

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

Real Gases: Deviations from Ideal Behavior The van der Waals Equation General form of the van der Waals equation: 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 H2 in a volume of 5.00 dm3 at 300. K .

The van der Waals Equation Calculate the molar volume of H2 gas at 40.0 atm and 300. K .

The van der Waals Equation Can solve for P and T , but what about V? Let: Vm = 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

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Kinetic Molecular Theory 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 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.

Before After pm=mu pm’=-mu pw=0 pw’=? Wall of Unit Area A z y x Consider only x-direction: ( m=molecule ) ( w=wall ) Before After pm=mu pm’=-mu pw=0 pw’=?

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 = pw’ x (# collisions)

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

Mathcad

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

Concept of Virial Series Define: Z = compressibility factor Virial Series: Expand Z upon molar concentration [ n/V ] or [ 1/Vm ] B=f(T) => 2nd Virial Coeff., two-molecule interactions C=f(T) => 3rd 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

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 Diagrams

Phase Diagrams The Phase Diagrams of H2O and CO2

Reduced Variables

PVT Variations among Condensed Phases Brief Calculus Review

PVT Variations among Condensed Phases

PVT Variations among Condensed Phases

Brief Calculus Review – F15 -1 Mathcad

Brief Calculus Review – F15 -2 Mathcad

Brief Calculus Review – F15 -3 Mathcad

Brief Calculus Review – F15 -4 Mathcad

Brief Calculus Review – F15 - 5 Mathcad

Brief Calculus Review – F15 - 6 Mathcad

Brief Calculus Review – F15 - 7 Mathcad

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  cm3 . This is the “differential” volume change. Then compare to the “real” volume change from algebraic calculations of initial and final volumes. Repeat for r increase of 0.10 cm and h decrease of 0.10 cm. Repeat for r increase of 0.001 cm and h decrease of 0.001 cm. What is your conclusion regarding the comparisons?

A right-circular cylinder has a base radius ( r ) of 2 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 A right-circular cylinder has a base radius ( r ) of 2.00 cm and a height ( h ) of 5.00 cm.

  Differential Algebra r / cm h / cm Dr / cm Dh / cm DV / p*cm3 V1 V2 V'=V2-V1 Diff Diff% 2.00 5.00 0.300000 -0.400000 4.400000 20.00000 24.33400 4.334000 6.6000E-02 1.52E+00 0.100000 -0.100000 1.600000 21.60900 1.609000 9.0000E-03 5.59E-01 0.030000 -0.040000 0.440000 20.43966 0.439664 3.3600E-04 7.64E-02 0.010000 -0.010000 0.160000 20.16010 0.160099 9.9000E-05 6.18E-02 0.003000 -0.004000 0.044000 20.04400 0.043997 3.0360E-06 6.90E-03 0.000300 -0.000400 4.40000E-03 20.00440 4.39997E-03 3.0036E-08 6.83E-04 0.000030 -0.000040 4.40000E-04 20.00044 3.0003E-10 6.82E-05 3.00E-06 -4.00E-06 4.40000E-05 20.00004 2.9994E-12 6.82E-06 3.00E-07 -4.00E-07 4.40000E-06 3.3846E-14 7.69E-07 3.00E-08 -4.00E-08 4.40000E-07 2.6741E-15 6.08E-07

States of Matter