Formula Sheet: Gas Phase

Table of Contents
1. Gas Laws and Fundamental Relationships
2. Dalton's Law and Partial Pressures
3. Kinetic Molecular Theory
4. Graham's Law of Effusion and Diffusion
5. Real Gases and Van der Waals Equation
6. Density and Molar Mass Relationships
7. Gas Stoichiometry
8. Pressure Conversions
9. Temperature Conversions
10. Kinetic Molecular Theory Postulates
11. Vapor Pressure and Phase Equilibrium
12. Special Gas Processes
13. Additional Important Relationships
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Gas Laws and Fundamental Relationships

Ideal Gas Law

• Ideal Gas Law: \[PV = nRT\]
  • \(P\) = pressure (atm, Pa, mmHg, torr)
  • \(V\) = volume (L, m³)
  • \(n\) = number of moles (mol)
  • \(R\) = universal gas constant
  • \(T\) = absolute temperature (K)
• Gas Constant Values:
  • \(R = 8.314\) J/(mol·K)
  • \(R = 0.0821\) L·atm/(mol·K)
  • \(R = 62.4\) L·mmHg/(mol·K)
• Alternative Form (using density): \[PM = \rho RT\]
  • \(M\) = molar mass (g/mol)
  • \(\rho\) = density (g/L)
• Alternative Form (using number of molecules): \[PV = NkT\]
  • \(N\) = number of molecules
  • \(k\) = Boltzmann constant = \(1.38 × 10^{-23}\) J/K
  • Relationship: \(k = R/N_A\)

Combined Gas Law

• Combined Gas Law: \[\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\]
  • For a fixed amount of gas (\(n\) constant)
  • All temperatures must be in Kelvin

Individual Gas Laws (Special Cases)

• Boyle's Law (constant \(n\) and \(T\)): \[P_1V_1 = P_2V_2\]
  • Pressure and volume are inversely proportional
  • Temperature remains constant (isothermal process)
• Charles's Law (constant \(n\) and \(P\)): \[\frac{V_1}{T_1} = \frac{V_2}{T_2}\]
  • Volume and temperature are directly proportional
  • Pressure remains constant (isobaric process)
• Gay-Lussac's Law (constant \(n\) and \(V\)): \[\frac{P_1}{T_1} = \frac{P_2}{T_2}\]
  • Pressure and temperature are directly proportional
  • Volume remains constant (isochoric process)
• Avogadro's Law (constant \(P\) and \(T\)): \[\frac{V_1}{n_1} = \frac{V_2}{n_2}\]
  • Volume and number of moles are directly proportional

Standard Temperature and Pressure (STP)

• STP Conditions:
  • Temperature: \(T = 273\) K (0°C)
  • Pressure: \(P = 1\) atm = 760 mmHg = 760 torr = 101.325 kPa
• Molar Volume at STP: \[V_m = 22.4 \text{ L/mol}\]
  • One mole of any ideal gas occupies 22.4 L at STP

Dalton's Law and Partial Pressures

• Dalton's Law of Partial Pressures: \[P_{total} = P_1 + P_2 + P_3 + ... + P_n\]
  • Total pressure equals sum of partial pressures
  • Applies to mixtures of non-reacting gases
• Partial Pressure from Mole Fraction: \[P_i = X_i P_{total}\]
  • \(P_i\) = partial pressure of gas \(i\)
  • \(X_i\) = mole fraction of gas \(i\)
• Mole Fraction: \[X_i = \frac{n_i}{n_{total}}\]
  • \(n_i\) = moles of gas \(i\)
  • \(n_{total}\) = total moles of all gases
  • Note: \(\sum X_i = 1\)
• Partial Pressure using Ideal Gas Law: \[P_i = \frac{n_i RT}{V}\]
  • For gas \(i\) in a mixture at temperature \(T\) and volume \(V\)

Kinetic Molecular Theory

Root-Mean-Square Speed

• Root-Mean-Square Speed: \[v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}\]
  • \(v_{rms}\) = root-mean-square speed (m/s)
  • \(R\) = 8.314 J/(mol·K)
  • \(T\) = absolute temperature (K)
  • \(M\) = molar mass (kg/mol)
  • \(k\) = Boltzmann constant
  • \(m\) = mass of one molecule (kg)
• Speed Relationships:
  • Higher temperature → higher speed
  • Lower molar mass → higher speed
  • At same temperature, lighter molecules move faster

Average Kinetic Energy

• Average Kinetic Energy (per molecule): \[KE_{avg} = \frac{3}{2}kT\]
  • \(k\) = Boltzmann constant = \(1.38 × 10^{-23}\) J/K
  • \(T\) = absolute temperature (K)
  • Independent of molecular mass
  • Only depends on temperature
• Average Kinetic Energy (per mole): \[KE_{avg} = \frac{3}{2}RT\]
  • \(R\) = 8.314 J/(mol·K)
• Relationship between KE and Speed: \[KE_{avg} = \frac{1}{2}mv_{rms}^2\]
  • \(m\) = mass of one molecule

Graham's Law of Effusion and Diffusion

• Graham's Law (rate comparison): \[\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} = \sqrt{\frac{\rho_2}{\rho_1}}\]
  • \(r_1, r_2\) = rates of effusion or diffusion
  • \(M_1, M_2\) = molar masses
  • \(\rho_1, \rho_2\) = densities
  • Lighter gases effuse/diffuse faster
• Graham's Law (time comparison): \[\frac{t_1}{t_2} = \sqrt{\frac{M_1}{M_2}}\]
  • \(t_1, t_2\) = times for effusion/diffusion
  • Heavier gases take longer
  • Time is inversely proportional to rate
• Definitions:
  • Effusion: passage of gas through a tiny opening into vacuum
  • Diffusion: spreading of gas molecules through space or another gas

Real Gases and Van der Waals Equation

Deviations from Ideal Behavior

• Conditions for Ideal Behavior:
  • High temperature (increases kinetic energy, reduces intermolecular forces)
  • Low pressure (increases distance between molecules)
  • Low density
  • Small molecular size
  • Weak intermolecular forces
• Conditions for Non-Ideal Behavior:
  • Low temperature
  • High pressure
  • Strong intermolecular forces
  • Large molecular size

Van der Waals Equation

• Van der Waals Equation: \[\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT\]
  • \(P\) = measured pressure
  • \(V\) = measured volume
  • \(n\) = number of moles
  • \(a\) = correction for intermolecular attractions (L²·atm/mol²)
  • \(b\) = correction for molecular volume (L/mol)
• Pressure Correction Term: \[\frac{an^2}{V^2}\]
  • Accounts for intermolecular attractive forces
  • Real pressure is less than ideal due to attractions
  • Larger \(a\) values indicate stronger intermolecular forces
• Volume Correction Term: \[nb\]
  • Accounts for the volume occupied by gas molecules themselves
  • Real available volume is less than container volume
  • Larger \(b\) values indicate larger molecular size

Density and Molar Mass Relationships

• Density from Ideal Gas Law: \[\rho = \frac{PM}{RT}\]
  • \(\rho\) = density (g/L)
  • \(P\) = pressure
  • \(M\) = molar mass (g/mol)
  • \(R\) = gas constant
  • \(T\) = temperature (K)
• Molar Mass from Density: \[M = \frac{\rho RT}{P}\]
  • Useful for determining molar mass of unknown gases
• Density Relationship: \[\rho = \frac{m}{V} = \frac{nM}{V}\]
  • \(m\) = mass of gas
  • \(n\) = moles of gas
  • \(M\) = molar mass

Gas Stoichiometry

• Moles from Ideal Gas Law: \[n = \frac{PV}{RT}\]
  • Use to find moles when P, V, and T are known
• Volume at STP: \[V = n × 22.4 \text{ L}\]
  • Only valid at STP (273 K, 1 atm)
• Mass-Volume Relationship: \[n = \frac{m}{M} = \frac{PV}{RT}\]
  • \(m\) = mass (g)
  • \(M\) = molar mass (g/mol)
• Stoichiometric Calculations:
  • Use ideal gas law to convert between volume and moles
  • Use mole ratios from balanced equations
  • Remember: equal volumes of gases at same T and P contain equal moles

Pressure Conversions

• Common Pressure Units:
  • 1 atm = 760 mmHg = 760 torr
  • 1 atm = 101,325 Pa = 101.325 kPa
  • 1 atm = 14.7 psi
  • 1 bar = 100 kPa ≈ 0.987 atm
• Gauge Pressure vs. Absolute Pressure: \[P_{absolute} = P_{gauge} + P_{atmospheric}\]
  • Absolute pressure includes atmospheric pressure
  • Gauge pressure measures pressure above atmospheric
  • Always use absolute pressure in gas law calculations

Temperature Conversions

• Celsius to Kelvin: \[T(K) = T(°C) + 273.15\]
  • Or approximately: \(T(K) = T(°C) + 273\)
• Kelvin to Celsius: \[T(°C) = T(K) - 273.15\]
• Fahrenheit to Celsius: \[T(°C) = \frac{5}{9}[T(°F) - 32]\]
• Celsius to Fahrenheit: \[T(°F) = \frac{9}{5}T(°C) + 32\]
• Important Note:
  • Always use Kelvin for gas law calculations
  • Absolute zero = 0 K = -273.15°C

Kinetic Molecular Theory Postulates

• Key Assumptions:
  • Gas molecules are in constant, random motion
  • Volume of gas molecules is negligible compared to container volume
  • No intermolecular forces (attractions or repulsions)
  • Collisions are perfectly elastic (no energy lost)
  • Average kinetic energy is proportional to absolute temperature

Vapor Pressure and Phase Equilibrium

• Vapor Pressure:
  • Pressure exerted by vapor in equilibrium with its liquid/solid phase
  • Increases with temperature
  • Independent of container volume at equilibrium
• Boiling Point:
  • Temperature at which vapor pressure equals external pressure
  • Normal boiling point: when \(P_{vapor} = 1\) atm
• Collecting Gases Over Water: \[P_{total} = P_{gas} + P_{water}\]
  • \(P_{total}\) = atmospheric pressure
  • \(P_{gas}\) = partial pressure of collected gas
  • \(P_{water}\) = vapor pressure of water at given temperature
  • Must subtract water vapor pressure to find gas pressure

Special Gas Processes

Isothermal Process

• Constant Temperature (ΔT = 0):
  • Use Boyle's Law: \(P_1V_1 = P_2V_2\)
  • Internal energy change: \(\Delta U = 0\) for ideal gas

Isobaric Process

• Constant Pressure (ΔP = 0):
  • Use Charles's Law: \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\)

Isochoric (Isovolumetric) Process

• Constant Volume (ΔV = 0):
  • Use Gay-Lussac's Law: \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\)

Adiabatic Process

• No Heat Transfer (Q = 0):
  • Temperature, pressure, and volume all change
  • Process occurs too quickly for heat exchange

Additional Important Relationships

• Number of Molecules: \[N = n × N_A\]
  • \(N\) = number of molecules
  • \(n\) = number of moles
  • \(N_A\) = Avogadro's number = \(6.022 × 10^{23}\) mol^-1
• Relationship between R and k: \[R = N_A × k\]
  • \(R\) = universal gas constant
  • \(k\) = Boltzmann constant
  • \(N_A\) = Avogadro's number
• Concentration (for gases): \[C = \frac{n}{V} = \frac{P}{RT}\]
  • \(C\) = molar concentration (mol/L or M)
  • Derived from ideal gas law