states of matter

Formula for Density (D or ρ) of an ideal gas

D = (PM)/(RT), where P is pressure, M is molar mass, R is gas constant, and T is temperature.

Common units vs. SI units for gas density

Common: g/L (P in atm, R = 0.0821); SI: kg/m³ (P in Pascal, R = 8.314).

Relationship between Density and Molar Mass under similar P and T

D ∝ M

Boyle’s Law statement and constant conditions

V ∝ 1/P or PV = K at constant temperature (T) and moles (n).

Charles’s Law statement and constant conditions

V ∝ T or V/T = K at constant pressure (P) and moles (n).

Gay-Lussac’s (Amonton's) Law statement and constant conditions

P ∝ T or P/T = K at constant volume (V) and moles (n).

Avogadro’s Law statement and constant conditions

V ∝ n at constant temperature (T) and pressure (P).

Ideal Gas Equation and what it represents

PV = nRT; it represents the conservation of energy.

Combined Gas Equation

(P1V1)/(n1T1) = (P2V2)/(n2T2)

Connected System formula for final pressure (Pf) after opening stopcocks

Pf = (P1V1 + P2V2 + P3V3 …) / (V1 + V2 + V3 …)

Open Vessel equation (constant P and V)

niTi = nfTf

Conservation of Moles in an open vessel (ne = moles escaped)

ni = nf + ne

Pressure in terms of Height of liquid column

P = ρgh (where ρ is density, g is gravity, h is height).

Calculation of atmospheric pressure using Mercury column height

Patm = P(in cm)/76 = P(in mm)/760 in atm.

Pressure exerted at a given point in a column of components

Calculated by adding the pressures of all components present above it.

Pgas in an Open-end Manometer (liquid higher in open end)

Pgas = Patm + h(cm of liquid)

Pgas in a Close-end Manometer

Pgas = h(cm of liquid)

Dalton’s Law of Partial Pressure condition

Applicable only for non-reacting gaseous mixtures.

Definition of Partial Pressure

The pressure a gaseous component exerts when allowed to occupy the entire volume of the mixture at the same temperature.

Partial Pressure (Pi) in terms of Mole Fraction (Xi)

Pi = Ptotal · Xi

Relationship between Moist Gas and Dry Gas pressure

Pmoist gas = Pdry gas + P°H2O (where P°H2O is Aqueous Tension).

Diffusion vs. Effusion

Diffusion: Spontaneous mixing, large mass transfer, fast; Effusion: Movement through an orifice, small mass transfer, slow.

Graham’s Law of Effusion (Proportionality)

Rate (r) ∝ (PA)/√M, where A is area of orifice.

Rate of Effusion (r) measurable definitions

r = neffused/t; r = Veffused/t; r = Lengthcovered/t.

Fundamental Equation of KTG

PV = 1/3 mNc² or P = 1/3 ρc²

Root Mean Square Speed (Crms) formula

Crms = √(3RT/M) = √(3P/D)

Average Speed (Cav) formula

Cav = √(8RT/πM)

Most Probable Speed (Cmp) formula

Cmp = √(2RT/M)

Ratio of Cmp : Cav : Crms

1 : 1.128 : 1.225 (or √2 : √8/π : √3)

Average Kinetic Energy of one molecule

KEavg = 3/2 kT, where k is Boltzmann Constant (R/NA).

Total Kinetic Energy of 'n' moles of gas

KE = n(3/2 RT)

Relationship between Pressure (P) and Kinetic Energy per unit volume (E)

P = 2/3 E

Two Faulty Assumptions of KTG for ideal gases

1. No intermolecular attraction; 2. Molecules are point masses (volume is negligible).

Van der Waals Equation for 'n' moles of real gas

(P + an²/V²)(V - nb) = nRT

Significance and units of constant 'a'

Measures intermolecular attraction; Units: atm · L² · mol⁻².

Significance and units of constant 'b'

Measures excluded volume; b = 4 × Vmolecule × NA; Units: L · mol⁻¹.

Compressibility Factor (Z) definitions

Z = PV/nRT = Vreal/Videal

Deviation at Very High Pressure

Z = 1 + Pb/RT (Z > 1; repulsion dominates).

Deviation at Low/Moderate Pressure

Z = 1 - a/(VRT) (Z < 1; attraction dominates).

Conditions under which a Real Gas behaves Ideally

Low Pressure and High Temperature.

Critical Temperature (Tc) definition and formula

Max temperature to liquefy a gas; Tc = 8a/27Rb.

Critical Pressure (Pc) and Critical Molar Volume (Vc)

Pc = a/27b² and Vc = 3b.

Compressibility Factor at Critical Point (Zc)

Zc = 3/8 = 0.375.

Boyle’s Temperature (TB)

Temperature where a real gas behaves ideally; TB = a/Rb.

Inversion Temperature (Ti)

Ti = 2a/Rb = 2TB.

Virial Equation of State and the 2nd Virial Coefficient (B)

Z = 1 + B/V + C/V² …; where B = b - a/RT.

Mean Free Path (λ) definition and formula

Avg distance between 2 collisions; λ = 1 / (√2 π σ² N*).

Collision Number (Z1)

Z1 = √2 π σ² v_av N* (collisions per molecule per second).

Collision Frequency (Z11)

Z11 = [π σ² v_av (N*)²] / √2 (total collisions per unit volume per second).

Eudiometry Reagents: CO2/SO2, O2, and O3

CO2/SO2: KOH/NaOH; O2: Alkaline Pyrogallol; O3: Turpentine Oil.

General Hydrocarbon Combustion equation

CxHy + (x + y/4)O2 → xCO2 + (y/2)H2O

Pressure unit conversions (1 atm)

1 atm = 760 mmHg = 760 torr = 101325 Pa = 1.01325 bar.

Values of Universal Gas Constant (R)

0.0821 L·atm/mol·K; 8.314 J/mol·K; 1.987 (≈ 2) cal/mol·K.

Temperature Scale relationship (Celsius to Fahrenheit)

°C/5 = (°F - 32)/9

Molar Volume at NTP vs. STP

NTP (1 atm, 273K): 22.4 L/mol; STP (1 bar, 273K): 22.7 L/mol.