Physical Chemistry - Lecture Notes on Gases

Fundamental Concepts and States of Matter

• Definition of Gases:
  • Visibly distinct state of ordinary matter alongside liquids and solids.
  • Distinguished from solids and liquids by relatively low density and viscosity.
  • Exhibits relatively great expansion and contraction with changes in pressure and temperature.
  • Possesses the ability to diffuse readily and has a spontaneous tendency to become distributed uniformly throughout any container.
• Comparison of Matter States:
  • Solids: Characterized by well-defined shapes and high resistance to compression.
  • Liquids: Free-flowing and bounded by self-formed surfaces.
  • Gases: Expand freely to fill their containers completely and possess significantly lower densities than liquids and solids.
• State Transitions and Coexistence:
  • Most substances exist in gaseous, liquid, or solid states depending on external environmental conditions such as temperature and pressure.
  • Not all aggregate states are possible for certain substances.
  • Coexistence of states: A substance can exist simultaneously in two or three states under specific conditions.
  • Water Equilibrium Example: Water exists in stable equilibrium simultaneously across three states (ice, liquid water, and water vapor) near $0\,^\circ\text{C}$ and low pressure.
  • Sublimation: The phase transition directly from the solid state to the gaseous state.
  • Liquefaction or Condensation: The phase transition from the gaseous state to the liquid state.
• State Functions:
  • Definition: A state function (or function of state) is a property of a thermodynamic system that depends solely on the current equilibrium state of the system, regardless of the path or process taken to reach that state.
  • Key Examples: Pressure ($p$), volume ($V$), temperature ($T$), internal energy ($U$), enthalpy ($H$), entropy ($S$), and Gibbs free energy ($G$).
  • Path Independence: The change in value of a state function between two states (e.g., $\Delta U$, $\Delta H$) depends strictly on the initial and final states, unlike path functions such as heat ($q$) and work ($w$) which depend on the specific path taken.

The Ideal Gas Law and Macroscopic Relationships

• Equations of State:
  • Empirical gas laws correlate macroscopic variables of a system.
  • An equation of state is defined as a mathematical relationship between system variables.
  • Key macroscopic variables for common gases include pressure ($p$), volume ($v$), and temperature ($T$).
• The Clapeyron-Mendeleev Equation:
  • An ideal gas is defined as a gas whose physical state is described by the Clapeyron-Mendeleev equation:
    $p v = n R T$
  • Variables and Constants:
  • $p$: Pressure of the gas.
  • $v$: Total volume occupied by the gas.
  • $n$: Number of moles of gas.
  • $T$: Absolute temperature.
  • $R$: Universal gas constant, relating to one mole of gas.
• Characteristics of the Universal Gas Constant ($R$):
  • Independent of the chemical nature of the gas and external conditions.
  • For one mole of gas ($n = 1\,\text{mol}$), the equation reduces to:
    $p V = R T$
    where $V = \frac{v}{n}$ represents the molar volume.
  • Physical Interpretation: $R$ is equal to the work performed during the expansion of one mole of an ideal gas at constant pressure when its temperature is increased by $1\,\text{K}$.
• Associated Empirical Laws:
  • Boyle's Law: For a gas maintained at a constant temperature, the volume is inversely proportional to the pressure.
  • Gay-Lussac-Joule Law: The internal energy of an ideal gas depends solely on its absolute temperature and is entirely independent of volume and pressure.

Gas Mixtures and Dalton's Law of Partial Pressures

• Behavior of Gas Mixtures:
  • Under ordinary physical conditions, gases can mix completely and homogeneously in any proportion.
  • Dalton's Law Statement: Within the limits of applicability of ideal gas laws, the total pressure exerted by a mixture of gases occupying a definite volume equals the sum of the partial pressures of all constituent gases.
• Partial Pressure Definition:
  • The partial pressure of an individual gas component in a mixture is the pressure that the gas would exert if it alone occupied the entire volume of the mixture at the exact same temperature.
• Mathematical Formulation of Dalton's Law:
  • Total pressure equation:
    $p = p<em>1 + p</em>2 + p<em>3 + p</em>4 + p<em>5 + \dots = \sum p</em>i$
    where $p_i$ represents the partial pressure of the $i\text{-th}$ gas component.
  • Individual component partial pressure from the ideal gas law:
    $p<em>i = \frac{n</em>i R T}{v} = \frac{n_i p}{n}$
  • Expanded total pressure equation:
    $p = (n<em>1 + n</em>2 + n<em>3 + n</em>4 + n_5 + \dots) \frac{R T}{v} = \frac{n R T}{v}$
• Mole Fraction and Application Limits:
  • Mole fraction: The ratio of the number of moles of a given component ($n<em>i$) to the total number of moles of the mixture ($n$), expressed as $\frac{n</em>i}{n}$.
  • Relationships: The ideal gas equation of state applies to individual components separately, to individual gases existing in isolation, and to the gas mixture as a whole.
  • Calculating Partial Pressure: Partial pressure of any component equals total pressure multiplied by its mole fraction ($p<em>i = p \times \frac{n</em>i}{n}$).
  • Scope Restriction: This relation holds strictly and exclusively for mixtures of ideal gases.

The Kinetic Theory of Gases

• Microscopic Basis of Macroscopic Properties:
  • Connects classical Newtonian mechanics of individual molecules to macroscopic physical properties.
  • Volume ($v$): Reflects the positional distribution of molecules and represents the total available space within which molecules can freely move.
  • Pressure ($p$): Measures the average momentum change per unit time experienced by molecules as they collide with and rebound from container walls.
  • Temperature ($T$): Directly proportional to the average kinetic energy of the molecules, or to the square of their average velocity.
  • Scope: Predicts ideal gas equations of state, statistical velocity distributions, and transport phenomena (thermal conductivity, coefficient of diffusion, viscosity).
• Assumptions of Kinetic Theory:
  1. Gas molecules are in continuous chaotic movement in all directions, colliding with one another and with the walls of the containing vessel.
  2. Dimensions of molecules are negligible compared with the distance between them.
  3. Forces of attraction between molecules are negligible.
• Kinetic Theory Equations and Derivations:
  • Fundamental pressure relation:
    $p = \frac{1}{3} N m \bar{u}^2$
    where $N$ is the number of molecules in $1\,\text{m}^3$ of gas, $m$ is the mass of a single molecule, and $\bar{u}$ is the root mean square velocity.
  • Root mean square (RMS) velocity definition:
    $\bar{u} = \sqrt{\frac{\sum u<em>i^2}{N}}$ where $u</em>i$ represents the velocity of individual molecule i$.
  • Fundamental equation of kinetic theory:
    p V = \frac{1}{3} NA m \bar{u}^2 = \frac{1}{3} M \bar{u}^2 = \frac{2}{3} R T where NA = N V$is Avogadro's number and$M = N_A m is the molecular weight.
  • Mean kinetic energy of translational motion:
    \bar{E} = \frac{m \bar{u}^2}{2}
  • Combining kinetic energy and ideal gas relations:
    \bar{E} = \frac{3}{2} \frac{R T}{NA} = \frac{3}{2} k T where k = \frac{R}{NA} is Boltzmann's constant.
  • Absolute Temperature Meaning: Absolute temperature measures and is directly proportional to molecular kinetic energy; two gases at the same temperature possess identical average molecular kinetic energies.
  • Molecular RMS Velocity Equation:
    \bar{u} = \sqrt{\frac{3 R T}{M}}
  • Velocity Relationships and Dynamics:
  • RMS velocity is directly proportional to the square root of absolute temperature (\sqrt{T}).
  • RMS velocity is inversely proportional to the square root of molecular weight (\frac{1}{\sqrt{M}}).
  • Smaller molecular mass results in higher average velocity.
  • Energy redistribution: Molecular collisions redistribute kinetic energy among molecules while keeping total system kinetic energy constant.

Real Gases and the Van der Waals Equation

• Deviations from Ideal Gas Behavior:
  • The ideal gas equation (p V = R T) provides an approximation; real gases deviate significantly, especially under non-ideal conditions.
  • Ideal gases could never become liquids or solids regardless of temperature reduction or pressure increase.
• The Van der Waals Equation of State:
  • Formulated as:
    \left(p + \frac{a}{V^2}\right)(V - b) = R T
  • Parameters a$and$b:
  • Adjustable material parameters determined experimentally for specific real gases.
  • Vary from gas to gas and are not universal constants.
  • Term \frac{a}{V^2}: Corrects for mutual attraction of molecules, acting as internal pressure compressing the gas.
  • Parameter b$: Corrects for actual molecular volume and short-range repulsive forces. Approximately equals four times the volume of molecules calculated from their gas-kinetic radius ($b \approx 4 \times V_{\text{molecules}}).
• Temperature and Pressure Dependencies:
  • Theoretical assumptions treat a$and$b$as independent of temperature and pressure; experimental data show$a$depends on temperature and$b varies with volume.
• Microscopic Interpretation of Intermolecular Forces:
  • Interactions are strongly repulsive in close proximity, mildly attractive at intermediate distances, and vanish at long distances.
  • Repulsion creates a volume of exclusion (b$) around each molecule, reducing freely available space to$(V - b).
  • Limiting behavior: When volume V$is extremely large, correction terms become negligible and the equation coincides with the ideal gas equation ($p V = R T).
• Phase Transitions and Coexistence:
  • Occur at low temperatures (reduced molecular motion) and high pressures or reduced volumes (reduced intermolecular spacing).
  • System enters a high-density bound state with a bounding surface, marking the onset of the liquid state (phase transition).
  • Accounts for phase transitions and describes a two-phase coexistence region terminating at a critical point, above which no physical distinction exists between gas and liquid phases.

Heat Capacity of Gases

• Definition of Heat Capacity:
  • Quantity of heat required to raise system temperature by 1\,\text{K}$(or$1\,^\circ\text{C}) when no chemical reactions or changes of state occur.
  • Molar Heat Capacity (C): Heat capacity per mole of substance.
• Isochoric and Isobaric Heat Capacities:
  • Isochoric Heat Capacity (C_V): Heat capacity measured at constant volume.
  • Isobaric Heat Capacity (C_p): Heat capacity measured at constant pressure.
  • Relative Magnitude: Heat capacity at constant pressure is always greater than heat capacity at constant volume (Cp > CV).
• Heat Capacity Difference for Ideal Gases:
  • Mathematical relation:
    Cp - CV = n R
  • Explanation: The difference equals the work performed by the gas against external pressure when its temperature is increased by 1\,\text{K} at constant pressure:
    n R (T + 1) - n R T = n R$$