Thermodynamics

Thermodynamics is the study of heat and energy transformations based on a small set of fundamental principles and equations of state. It predicts whether processes and chemical reactions are energetically favorable and quantifies the direction and extent of spontaneous change. These principles govern phase transitions such as melting, boiling, and vaporization of water, which are critical in many biological and medical contexts. For gases, the ideal gas law often provides a simple, effective approximation of macroscopic behavior under moderate conditions.

Energy

Thermodynamics deals with the conversion of energy and the ability to perform work. Several forms of energy exist.

• Definition: energy is the capacity to do work
• Unit: J (joule), where 1 J = 1 N·m
• Forms
  • Thermal energy: the energy associated with the disordered motion of particles in a substance
  • Potential energy: energy an object possesses due to its position in a force field (e.g., gravitational, electric) or its configuration (e.g., a compressed spring)
  • Kinetic energy: the energy an object possesses due to its motion
  • Chemical energy: energy stored in the bonds of chemical compounds, which can be released during a chemical reaction (e.g., glucose in muscle cells)

Law of conservation of energy: Energy can be converted from one form to another and transferred between systems, but it cannot be created or destroyed. In an isolated system, the total energy remains constant.

Entropy

Entropy (S) is a measure of the disorder or randomness of a system at the particle level. The greater the entropy, the greater the disorder.

Formula: ΔS = Q_rev/T for a reversible process, and ΔS > Q_irrev/T for an irreversible process

• Unit: J/K
• ΔS = entropy change (J/K), Q_rev = heat transferred in a reversible process (J), Q_irrev = heat transferred in an irreversible process (J), T = absolute temperature (K)
• Relative entropy: the entropy of a substance depends on its physical state
  • S_gas > S_liquid > S_solid
  • Gases have the highest entropy due to the high degree of random motion and large volume occupied by their particles.
  • Crystalline solids have the lowest entropy because their particles are fixed in an ordered lattice structure with limited vibrational motion.

In an isolated system (which exchanges neither energy nor matter with its surroundings), the total entropy can only increase or remain the same for any spontaneous process.

Calorimetry is the science of measuring heat changes associated with chemical reactions or physical processes.

• Calorimeter: an insulated device used to measure the amount of heat absorbed or released
  • Constant-pressure calorimetry
    • Measures the enthalpy change (ΔH) of a reaction at constant atmospheric pressure
    • E.g., a coffee-cup calorimeter
  • Constant-volume calorimetry
    • Measures the change in internal energy (ΔU) of a reaction in a sealed container
    • E.g., a bomb calorimeter

• Thermodynamic system: the specific part of the universe being studied, separated from the rest (the surroundings) by a boundary
  • Isolated system: cannot exchange energy or matter with the surroundings
  • Closed system: can exchange energy but not matter
  • Open system: can exchange both energy and matter
• State function: a property of a system that depends only on its current equilibrium state, not on the path taken to reach that state
  • Examples include pressure (p), volume (V), temperature (T), internal energy (U), enthalpy (H), entropy (S), and Gibbs free energy (G).
  • Changes in state functions (e.g., ΔU, ΔH) are independent of the process or pathway.
  • Work (W) and heat (Q) are not state functions; they are path-dependent.

The four laws of thermodynamics provide the fundamental principles from which all other relationships in the field are derived. Understanding these laws is key to answering most questions in thermodynamics.

Zeroth law of thermodynamics

The zeroth law describes thermal equilibrium, the state where systems have the same temperature.

Zeroth law: If system A is in thermal equilibrium with system B, and system B is in thermal equilibrium with system C, then system A is also in thermal equilibrium with system C.

Heat transfer

A consequence of the zeroth law is that when two systems with different temperatures are in thermal contact, heat flows from the warmer system to the cooler one until they reach thermal equilibrium. The rate of this heat transfer depends on the thermal conductivity of the materials involved.

Mechanisms of heat transfer

• Conduction: the transfer of heat through direct physical contact
  • Occurs via molecular collisions, where more energetic particles transfer kinetic energy to less energetic ones
  • Predominant in solids
• Convection: the transfer of heat through the movement of fluids (liquids or gases)
  • Warmer, less dense fluid rises, and cooler, denser fluid sinks, creating convection currents that distribute heat.
  • Example: boiling water or the circulation of warm air in a room
• Radiation: the transfer of heat through electromagnetic waves
  • Does not require a medium for transfer
  • Example: the heat felt from the sun or a campfire

Temperature

• Definition
  • A measure of the average kinetic energy of the particles in a substance, which determines the direction of heat flow
  • Temperature indicates the thermal state of an object relative to a standard.
• Temperature scales: in everyday life, temperature is often measured in degrees Celsius (°C), but the standard scientific unit is the kelvin (K)
  • Conversion: T (K) = T (°C) + 273.15
    • The Celsius and Kelvin scales are shifted relative to each other but have the same magnitude.

A temperature change of one degree Celsius is equivalent to a temperature change of one Kelvin.

Thermal conductivity

This material property describes the rate at which heat is transferred through a material.

Formula: P = λ · A · ΔT/l

• Unit: W (watt)
• P = heat flow rate (W), λ = thermal conductivity (W/(m·K)), A = cross-sectional area (m²), ΔT = temperature difference (K), l = length or thickness of the material (m)

First and second laws of thermodynamics

The first law of thermodynamics and the second law of thermodynamics govern which processes are possible based on energy and entropy considerations.

First law: The change in a system's internal energy (ΔU) is equal to the heat (Q) added to the system plus the work (W) done on the system (ΔU = Q + W). This is a restatement of the law of conservation of energy.

PV diagrams and work

• PV diagram: a graph of pressure (P) versus volume (V) for a thermodynamic system
• Work calculation: the work done during a process can be determined from the area under the curve on a PV diagram
  • For an expansion process (increasing V), the work done by the system is positive (W < 0 for the system's energy change).
  • For a compression process (decreasing V), the work done on the system is positive (W > 0 for the system's energy change).
  • For a cyclic process, the net work done is the area enclosed by the curve.

Second law: The total entropy of an isolated system always increases over time for a spontaneous process.

Applying the first two laws to thermodynamic processes often requires the following concepts:

• Internal energy (U): the sum of all kinetic and potential energies of the particles within a system
  • Formula: ΔU = Q + W
    • Unit: J (joule)
    • ΔU = change in internal energy (J), Q = heat added to the system (J), W = work done on the system (J)
• Specific heat capacity: a material constant indicating the amount of heat needed to raise the temperature of a unit mass of a substance by one degree
  • Formula: c = Q/(m × ΔT)
    • Unit: J/(kg·K)
    • c = specific heat capacity (J/(kg·K)), Q = heat transferred (J), m = mass (kg), ΔT = change in temperature (K)
    • Temperature change: the formula can be rearranged to find the temperature change: ΔT = Q/(m·c)
• Thermal energy: the energy transferred as heat, which can be related to power and time
  • Formula: Q = P × t
    • Unit: J (joule)
    • Q = thermal energy (J), P = power (W), t = time (s)

Sample calculation

A person with a body mass of 65 kg feels cold and sits by a heater, warming their entire body by 1°C. Assuming the specific heat capacity of the human body is 3,500 J/(kg·K), calculate the amount of heat the person absorbed.

• Find: amount of heat Q
• Given: temperature difference ΔT, specific heat capacity c, mass m
  • c = Q/(m × ΔT) → Q = c × m × ΔT
  • Note: A change of 1°C is equal to a change of 1 K.
  • → 3,500 J/(kg·K) × 65 kg × 1 K = 227,500 J

Third law of thermodynamics

The third law relates to the behavior of systems as they approach absolute zero temperature.

Third law: The entropy of a perfect crystal at absolute zero (0 K) is zero. A consequence of this law is that absolute zero is unattainable.

The third law implies that perfect order is impossible because some thermal motion persists at any temperature above absolute zero. Creating order in a system always requires an input of energy to counteract the natural tendency toward disorder (entropy).

Chemical energy is the potential energy stored in the chemical bonds of compounds, which can be converted into other forms of energy during chemical reactions. The amount of stored energy depends on the strength and type of bonds between atoms.

Energy content in different bond types

The bond energy (or bond enthalpy) is the energy required to break a mole of a particular bond in the gaseous state. It quantifies the strength of the bond.

• Ionic bond: ∼100–1000 kJ/mol
• Covalent bond: ∼100–1000 kJ/mol
  • Bond energy is inversely related to bond length: shorter bonds are typically stronger.
  • Triple bonds are stronger (have higher bond energy) than double bonds, which are stronger than single bonds.
• Hydrogen bond: ∼5–50 kJ/mol
• Hydrophobic interactions: < 20 kJ/mol

The greater the bond energy, the stronger the chemical bond.

Absolute energy turnover

Chemical reactions involve the breaking of existing bonds and the formation of new ones, resulting in an overall energy change.

Enthalpy (H)

Enthalpy (H) is a measure of the total heat content of a system; the change in enthalpy (ΔH) represents the heat absorbed or released in a process at constant pressure

• Formula: H = U + p × V
  • Unit: J (or kJ)
    • H = enthalpy (J), U = internal energy (J), p = pressure (Pa), V = volume (m³)
• Reaction enthalpy (ΔH): ΔH_reaction = H_products – H_reactants
  • ΔH < 0: exothermic
    • H_products < H_reactants
    • A process that releases heat into the surroundings
  • ΔH > 0: endothermic
    • H_products > H_reactants
    • A process that absorbs heat from the surroundings
    • Example: evaporation
      • When liquid evaporates, it draws heat energy from its surroundings (e.g., the skin), causing a cooling effect. This required heat is the latent heat of vaporization (L_v) and can be calculated using the formula: L_v = heat energy (Q)/mass (m).

Calculating ΔH

Method 1: standard heat of formation

• Standard heat of formation (ΔH°_f): the enthalpy change when one mole of a compound is formed from its constituent elements in their standard states
  • Standard state: typically 298 K (25°C) and 1 atm pressure.
  • By definition, the ΔH°_f of an element in its most stable form (e.g., O₂, C(graphite)) is zero.
• Standard heat of reaction (ΔH°_rxn)
  • Formula: ΔH°_rxn = Σ(ΔH°_{f, products}) - Σ(ΔH°_{f, reactants})

Method 2: bond dissociation energies

• Bond dissociation energy: the energy required to break a specific covalent bond homolytically (one electron to each atom)
  • Formula: ΔH_rxn ≈ Σ(bond energies of bonds broken) - Σ(bond energies of bonds formed)
  • This method is an approximation as it uses average bond energies, which may differ slightly from the specific energies in a particular molecule.

Method 3: Hess's Law

• Hess's law: The total enthalpy change (ΔH) for a reaction is constant, no matter how many steps or what pathway the reaction takes.
  • Consequence: The final enthalpy change is always the sum of the enthalpies of the products minus the sum of the enthalpies of the reactants, regardless of any intermediate steps.
  • Application: The Born-Haber cycle uses Hess's law to calculate lattice energies of ionic compounds, which are difficult to measure directly.
    • The formation of NaCl from solid Na and gaseous Cl₂ involves these steps:
      • Sublimation of Na
      • Dissociation of Cl₂
      • Ionization of Na
      • Electron attachment to Cl
      • Formation of the NaCl lattice
    • Calculation: The lattice energy can be found by summing the other known enthalpy changes (sublimation, ionization, etc.) and the standard heat of formation (ΔH°_f).
    • Using typical values: lattice energy = -411 kJ/mol – 108 kJ/mol – 122 kJ/mol – 496 kJ/mol – (-349 kJ/mol) = -788 kJ/mol

Gibbs free energy (G)

Gibbs free energy (G) is a thermodynamic potential that combines enthalpy and entropy to predict the spontaneity of a process at constant temperature and pressure.

• Formula: ΔG = ΔH – T × ΔS
  • Unit: J/mol (or kJ/mol)
    • ΔG = change in Gibbs free energy (J/mol), ΔH = change in enthalpy (J/mol), T = absolute temperature (K), ΔS = change in entropy (J/(mol·K))
• Spontaneity
  • ΔG < 0: exergonic
    • A process that occurs spontaneously
    • A process that releases free energy, which can be used to do work
    • The equilibrium favors the products.
    • Example: ATP + H₂O ⟶ ADP + Pᵢ → ΔG ≈ -30.5 kJ/mol
  • ΔG > 0: endergonic
    • A process that is non-spontaneous
    • A process that requires an input of free energy to proceed.
    • The equilibrium favors the reactants.
  • ΔG = 0: The system is at equilibrium.

ΔG and chemical equilibrium

The standard free energy change of a reaction is related to its equilibrium constant (K) from the law of mass action.

• Formula: ΔG° = -R × T × ln(K)
  • Unit: J/mol (or kJ/mol)
    • ΔG° = standard Gibbs free energy change (J/mol), R = ideal gas constant (8.314 J/(mol·K)), T = absolute temperature (K), K = equilibrium constant (unitless)

ΔG and redox reactions

The standard free energy change for a redox reaction is related to its standard cell potential (E°_cell).

• Free energy for redox reactions: The free energy change for a redox reaction is related to the cell potential.
  • Formula: ΔG° = -n × F × E°_cell
    • Unit: J/mol (or kJ/mol)
      • ΔG° = standard Gibbs free energy change (J/mol), n = moles of electrons transferred (mol), F = Faraday constant (96,485 C/mol), E°_cell = standard cell potential (V)

Sample calculation: Gibbs free energy

A chemical reaction at 25°C has a standard entropy change (ΔS°) of 0.2 kJ/(mol·K). The enthalpy of the products is 400 kJ/mol, and the enthalpy of the reactants is 200 kJ/mol. Determine if the reaction is spontaneous under standard conditions.

• Find: Gibbs free energy change (ΔG)
• Given: H_reactants, H_products, reaction entropy ΔS, temperature T
  • First, calculate the reaction enthalpy: ΔH_reaction = H_products - H_reactants → ΔH_reaction = 400 kJ/mol - 200 kJ/mol = 200 kJ/mol
  • Next, use the Gibbs equation: ΔG = ΔH - T × ΔS → ΔG = 200 kJ/mol - (298 K × 0.2 kJ/(mol·K)) = 200 - 59.6 = 140.4 kJ/mol
  • Since ΔG > 0, the reaction is non-spontaneous under these conditions.

Sample calculation: chemical equilibrium

A reaction at 25°C has an equilibrium constant (K) of 5 × 10^-9. Determine the standard Gibbs free energy change for this reaction.

• Find: Gibbs free energy change (ΔG)
• Given: temperature T, equilibrium constant K
  • ΔG = - R × T × ln(K) → ΔG = -8.314 J/(mol·K) × 298 K × ln(5 × 10^-9) = -47,356 J/mol ≈ -47.4 kJ/mol

Sample calculation: electrochemical reaction

Consider the electrochemical reaction 2 Na + Cu²⁺ → 2 Na⁺ + Cu. The standard reduction potentials are E°_Na = -2.71 V and E°_Cu = +0.34 V. What is the standard Gibbs free energy change of the reaction?

• Find: Gibbs free energy change ΔG
• Given: standard potentials E°, reaction equation
  • Determine half-reactions and electrons transferred: 2 Na → 2 Na⁺ + 2 e^- (oxidation) and Cu²⁺ + 2e^- → Cu (reduction). The number of electrons transferred (n) is 2.
  • Calculate standard cell potential: ΔE° = E°_cathode - E°_anode = 0.34 V - (-2.71 V) = 3.05 V.
  • Calculate ΔG: ΔG = -n × F × ΔE° → ΔG = -2 × 96,485 C/mol × 3.05 V ≈ -588,500 J/mol ≈ -588.5 kJ/mol

Energy profile of chemical reactions

The energy changes during a reaction can be visualized in an energy profile diagram, which plots the free energy (G) on the y-axis against the reaction progress on the x-axis.

• Activation energy (E_a): the minimum energy that must be supplied for a reaction to start, represented by the peak of the "energy hill" in the diagram
• Intermediates and transition states: Transition states are high-energy, unstable configurations of atoms that exist for a fleeting moment at the peak of the activation energy barrier. Intermediates are short-lived species that are formed and consumed during the reaction, corresponding to local energy minima between transition states.
  • Appearance in the energy profile: intermediates appear as valleys between two energy peaks
  • Stability
    • The lower the energy of an intermediate, the more stable it is.
    • The step with the highest activation energy is the rate-determining step of the reaction.
  • Alternative reaction paths: if multiple reaction pathways are possible, the reaction will preferentially proceed through the pathway with the lowest activation energy

Reaction type	Free energy change	Energy profile
Exergonic	ΔG < 0
Endergonic	ΔG > 0

Catalysis

Catalysis provides an alternative reaction pathway with a lower activation energy. This is useful for reactions that are thermodynamically favorable but kinetically slow.

• Catalyst: a substance that increases the rate of a chemical reaction without being consumed in the process
  • Properties
    • Increases the reaction rate, allowing equilibrium to be reached faster, but does not change the position of the chemical equilibrium
    • Can be effective in very small amounts because it is regenerated during the reaction cycle
  • Heterogeneous catalyst: exists in a different phase from the reactants (e.g., a solid catalyst with liquid reactants)
  • Homogeneous catalyst: exists in the same phase as the reactants (e.g., both are dissolved in the same liquid)
  • Enzymatic catalysis: the catalyst is a biological enzyme
    • Special feature: enzymes are typically highly substrate-specific and stereoselective
• Energetic principle: By lowering the activation energy, the catalyst allows the reaction to proceed more quickly under given conditions, as shown in the energy profile.
• Molecular principle: catalysts often interact with a reactant to form a less stable intermediate, providing an alternative, lower-energy pathway to the products

A catalyst participates in the reaction but emerges unchanged at the end. It can then participate in another reaction cycle. This process repeats until the reactants are depleted.

In science, a "phase" is a region of space where the physical and chemical properties of a substance are uniform. The three primary phases are the states of matter: solid, liquid, and gas.

The states of matter

The three states of matter are distinguished by the arrangement, motion, and spacing of their constituent particles. Substances with larger molecules and stronger intermolecular forces generally have higher melting and boiling points.

Phase transitions

A phase transition is the conversion of a substance from one phase to another.

• Melting: transition from solid → liquid
  • Freezing: transition from liquid → solid
• Vaporization: transition from liquid → gas
  • Condensation: transition from gas → liquid
• Sublimation: transition from solid → gas
  • Deposition: transition from gas → solid

Enthalpy of phase changes

• General principle: During a phase transition, the temperature of the substance remains constant while heat is absorbed or released. This energy is used to change the potential energy and arrangement of the molecules, not their kinetic energy.
• Heat of fusion (ΔH_fus): the heat absorbed to convert a solid into a liquid at its melting point
• Heat of vaporization (ΔH_vap): the heat absorbed to convert a liquid into a gas at its boiling point

Phase diagrams

A phase diagram is a graph of pressure versus temperature that shows the conditions under which a substance exists in different phases.

Phase diagram

• X-axis: temperature (T)
• Y-axis: pressure (p)
  • The SI unit for pressure is pascal (Pa).
    • 1 atm ≈ 1.013 bar ≈ 1.013 × 10⁵ Pa
• Regions: the areas between the lines represent the conditions for the solid, liquid, and gaseous phases
• Lines: the lines represent the conditions of phase equilibrium where phase transitions occur
  • Fusion curve: line between solid and liquid phases
  • Vaporization curve: line between liquid and gaseous phases
  • Sublimation curve: line between solid and gaseous phases
• Special points
  • Triple point
    • The intersection of the three-phase boundary lines
    • The unique combination of temperature and pressure where all three phases coexist in equilibrium
  • Critical point
    • The endpoint of the vaporization curve
    • Beyond this point, the liquid and gaseous phases are indistinguishable, forming a supercritical fluid.
  • Normal boiling point
    • The temperature on the vaporization curve corresponding to standard atmospheric pressure (1 atm or 1.013 bar)
  • Normal melting point
    • The temperature on the fusion curve corresponding to standard atmospheric pressure

The phase diagram of water is unusual because its fusion curve has a negative slope. This means that at higher pressures, the melting point of ice decreases. This behavior is responsible for the density anomaly of water: ice is less dense than liquid water, which is why it floats.

Colligative properties

Colligative properties are properties of solutions that depend on the concentration of solute particles, not on their identity. These include changes to phase transition points.

• van 't Hoff factor (i): represents the number of discrete particles (ions or molecules) a solute produces in solution for each formula unit
• Freezing point depression: The freezing point of a solution is lower than that of the pure solvent.
  • Cause: dissolving a non-volatile solute in a solvent
  • Reason: the solute particles increase the entropy of the liquid phase, making it more stable and lowering the temperature at which it freezes
  • Formula: ΔT_f = i × K_f × m
    • ΔT_f = change in freezing point (°C), i = van 't Hoff factor (unitless), K_f = freezing point depression constant (specific to the solvent, °C kg/mol), m = molality (mol/kg)
  • Example: spreading salt on icy roads to melt the ice
• Boiling point elevation: The boiling point of a solution is higher than that of the pure solvent.
  • Cause: dissolving a non-volatile solute in a solvent
  • Reason: the solute increases the entropy of the liquid phase and lowers the vapor pressure, requiring a higher temperature to reach the boiling point
  • Formula: ΔT_b = i × K_b × m
    • ΔT_b = change in boiling point (°C), i = van 't Hoff factor (unitless), K_b = boiling point elevation constant (specific to the solvent, °C kg/mol), m = molality of the solution (mol/kg)
  • Example: salt water boils at a slightly higher temperature than pure water
• Vapor pressure lowering (Raoult's law): The vapor pressure of a solvent is reduced by the addition of a non-volatile solute.
  • Cause: dissolving a non-volatile solute in a solvent
  • Reason: the solute particles occupy some of the surface area of the liquid, reducing the rate at which solvent molecules can escape into the gas phase
  • Formula: P_A = X_A × P°_A
    • P_A = vapor pressure of the solvent in solution (atm), X_A = mole fraction of the solvent (unitless), P°_A = vapor pressure of the pure solvent (atm)
• Osmotic pressure (Π): the minimum pressure required to prevent the inward flow of pure solvent across a semipermeable membrane
  • Formula: Π = iMRT
    • Π = osmotic pressure (atm), i = van 't Hoff factor (unitless), M = molarity of the solution (mol/L), R = ideal gas constant (L⋅atm), T = absolute temperature (K)

Thermal expansion

• Coefficient of thermal expansion: describes how the size of an object changes with a change in temperature
  • Most substances expand when heated and contract when cooled.
  • The expansion can be linear (change in length) or volumetric (change in volume).

The gaseous state of matter is characterized by particles that are far apart and move rapidly and randomly. The behavior of gases can be described using a simplified model.

Ideal gas

The ideal gas is a theoretical model that simplifies the behavior of real gases under certain conditions. While no real gas is truly ideal, this model provides a useful approximation.

• Assumptions
  • Gas particles are treated as point masses with negligible volume compared to the volume of the container.
  • There are no intermolecular attractive or repulsive forces between particles.
  • Particles are in continuous, random motion.
  • Collisions between particles and with the container walls are perfectly elastic (no net loss of kinetic energy).
  • The average kinetic energy of the gas particles is directly proportional to the absolute temperature (in Kelvin).
• Properties
  • Pressure: the force exerted by a gas per unit area on the walls of its container, resulting from the constant collisions of gas particles
    • Measurement: a simple mercury barometer is used to measure atmospheric pressure
      • It consists of an inverted glass tube filled with mercury, standing in a bath of mercury
      • The weight of the mercury column is balanced by the atmospheric pressure exerted on the surface of the mercury bath.
      • Standard atmospheric pressure supports a column of mercury 760 mm high.

Boyle's law

• At constant temperature, the volume of an ideal gas is inversely proportional to its pressure (V ∝ 1/P or P ×V = constant).
• If the volume increases, the pressure decreases, and vice versa.
• P × V = k or P₁V₁ = P₂V₂
  • P = pressure (Pa or atm), V = volume (m³ or L), k = constant
  • P₁ and P₂ (initial and final pressure in Pa or atm), V₁ and V₂ (initial and final volume in m³ or L)

Charles' law

• At constant pressure, the volume of an ideal gas is directly proportional to its absolute temperature (V ∝ T).
• If the temperature increases, the volume increases if the pressure remains constant.
• V/T = k or V₁/T₁ = V₂/T₂
  • V = volume (m³ or L), T = temperature (K), k = constant
  • V₁ and V₂ (initial and final volume in m³ or L), T₁ and T₂ (initial and final temperature in K)

Gay-Lussac’s law

• At constant volume, the pressure of an ideal gas is directly proportional to its absolute temperature (P ∝ T).
• If the temperature increases, the pressure increases, provided the volume does not change.
• P/T = k or P₁/T₁ = P₂/T₂
  • P = pressure (Pa or atm), T = temperature (K), k = constant
  • P₁ and P₂ (initial and final pressure in Pa or atm), T₁ and T₂ (initial and final temperature in K)

Avogadro's principle

• At the same temperature and pressure, equal volumes of ideal gases contain the same number of molecules (V ∝ n).
• n/V = k or V₁/n₁ = V₂/n₂
  • n = moles of gas (mol), V = volume (m³ or L), k = constant
  • V₁ and V₂ (initial and final volume in m³ or L), n₁ and n₂ (initial and final moles of gas in mol)

Ideal gas law

The ideal gas law combines the relationships above into a single equation describing the state of an ideal gas.

• Formula: P × V = n × R × T
  • P = pressure (atm), V = volume (L), n = amount of substance (mol), R = ideal gas constant (8.314 J/(mol·K)), T = absolute temperature (K)
    • The value of the ideal gas constant is R ≈ 8.314 J/(mol·K).
    • Pressure units: 1 atm ≈ 1.013 bar ≈ 101,325 Pa ≈ 760 mmHg.

At standard temperature and pressure (STP: 0°C and 1 atm of pressure), one mole of an ideal gas occupies a volume of 22.4 L.

Combined gas law

• Formula: P₁ × V₁/T₁ = P₂ × V₂/T₂
  • P₁ and P₂ (initial and final pressure of the gas in Pa or atm), V₁ and V₂ (initial and final volume in m³ or L), T₁ and T₂ (initial and final temperature in K)
• This equation shows how a gas changes when pressure, volume, or temperature are altered, provided the amount of gas (n) remains constant.

Diving medicine
In water, the ambient pressure increases by approximately 1 atm (or 1 bar) for every 10 meters of depth. According to Boyle's law (p × V = constant), this increased pressure causes the volume of air in a diver's lungs to decrease. It also increases the amount of gases, particularly nitrogen, that dissolve in the blood and tissues. At great depths, this can lead to nitrogen narcosis, a state of altered consciousness. If a diver ascends too quickly, the rapid decrease in ambient pressure causes the dissolved gases to come out of solution and form bubbles in the body, a dangerous condition known as decompression sickness ("the bends"). A slow, controlled ascent is necessary to allow these gases to be safely expelled through respiration.

Sample calculation 1: molar volume at STP

What is the volume of 1 mol of an ideal gas at standard temperature and pressure (STP)?

• Find: volume V
• Given: n = 1 mol, T = 273.15 K, p = 1.013 × 10⁵ Pa
  • Formula: p × V = n × R × T → V = n × R × T/p
  • V = (1 mol × 8.314 J/(mol·K) × 273.15 K)/(1.013 × 10⁵ Pa) = 0.0224 m³ = 22.4 L

Sample calculation 2: pressure change

A diver at a depth of 50 m inhales 3 L of air from their tank. If the diver ascends rapidly to the surface without exhaling, what volume would the air expand to? (Assume constant temperature and that pressure increases by 1 atm for every 10 m of depth.)

• Find: volume at the surface (V₂)
• Given: initial volume V₁ = 3 L; pressures p₁ (at 50 m) and p₂ (at surface)
  • The pressure at the surface (p₂) is 1 atm. The pressure at 50 m (p₁) is the surface pressure plus 1 atm for every 10 m, so p₁ = 1 atm + 5 atm = 6 atm.
  • Using Boyle's Law (since T and n are constant): p₁V₁ = p₂V₂
  • V₂ = (p₁ × V₁)/p₂
  • V₂= (6 atm × 3 L) / 1 atm = 18 L
  • Since the total lung capacity is about 6–7 L, this rapid expansion would cause catastrophic lung injury (pulmonary barotrauma).

Particle velocity in a gas

The particles in a gas are in constant, random motion, known as Brownian motion. The speed of this motion depends on two main factors:

• Molar mass: at a given temperature, particles with a smaller mass move faster, on average, than particles with a larger mass
• Temperature: the higher the temperature, the higher the average kinetic energy and thus the higher the average velocity of the gas particles
• Maxwell-Boltzmann distribution: The velocities of individual gas particles are not all the same but follow a statistical distribution (i.e., some molecules are much faster than average and some are much slower).
• Average kinetic energy: E_{kin, avg} = (3/2) × k_B × T
  • Unit: J (joule)
  • E_{kin, avg} = average translational kinetic energy per molecule (J), T = absolute temperature (K), k_B = Boltzmann constant (1.38 × 10⁻²³ J/K)

The Maxwell–Boltzmann distribution explains that molecules in a liquid have a range of kinetic energies. Even below the boiling point, a small fraction of faster molecules always has enough energy to overcome intermolecular forces and evaporate; this is why clothes dry at temperatures far below 100°C. Conversely, slower gas molecules can condense back into the liquid. In a closed system, these simultaneous processes establish a dynamic equilibrium, which determines the liquid's vapor pressure.

Heat capacity of gases

• Heat capacity at constant volume (C_v): the heat required to raise the temperature of a gas by one degree while holding the volume constant
• Heat capacity at constant pressure (C_p): the heat required to raise the temperature of a gas by one degree while holding the pressure constant
  • More heat is required than at constant volume because some energy is used for expansion work done by the gas.
  • For an ideal monatomic gas, C_p is always greater than C_v.

Partial pressure

Gases are often mixed, as is the air around us. Total air pressure equals the sum of the partial pressures of the individual gases. A partial pressure is defined as the pressure a gas would exert if it were the only gas present in the volume.

• Definition: the pressure that a single gas in a mixture would exert if it occupied the total volume alone
• Formula: p_i = X_i × p_total
  • p_i = partial pressure of a gas, X_i = mole fraction of gas i (unitless), p_total = total pressure
• Dalton's law: the total pressure of a gas mixture is the sum of the partial pressures of its individual components
• Formula: p_total = p₁ + p₂ + p₃ + …
  • p_total = total pressure (Pa), p_1,2,3 = partial pressure of a component (Pa)
• Example: air
  • Composition: air is approximately 78% nitrogen, 21% oxygen, 0.9% argon, and 0.04% carbon dioxide
  • Air pressure: standard atmospheric pressure at sea level is approximately 1 atm (101.3 kPa or 760 mmHg)
  • Partial pressure of O₂: p_O₂ = 0.21 × 760 mmHg ≈ 160 mmHg

Sample calculation

What is the partial pressure of nitrogen in air at standard atmospheric pressure?

• Find: partial pressure of N₂ (p_N₂)
• Given: total pressure p_total = 1 atm, mole fraction of N₂ = 0.78
  • p_N₂ = 0.78 × 1 atm = 0.78 atm
  • In mmHg: p_N₂ = 0.78 × 760 mmHg ≈ 593 mmHg

Real gases

Real gases deviate from ideal behavior, particularly at high pressures and low temperatures, because the assumptions of the kinetic molecular theory are no longer valid.

• Qualitative deviations
  • Intermolecular forces: at low temperatures and high pressures, attractive forces between gas particles become significant, causing the particles to strike the container walls with less force. This leads to a lower pressure than predicted by the ideal gas law.
  • Particle volume: At high pressures, the volume occupied by the gas particles themselves is no longer negligible compared to the container volume. This reduces the available volume for movement, leading to a higher pressure than predicted.
• Quantitative deviations (Van der Waals equation)
  • An equation of state that modifies the ideal gas law to account for the behavior of real gases.
  • Formula: (P + an²/V²)(V - nb) = nRT
    • P = pressure (Pa), V = volume (m³), n = amount of substance (mol), T = absolute temperature (K), R = ideal gas constant (8.314 J/(mol·K)), a,b = gas-specific constants
  • Correction for pressure (a): The term an²/V² corrects for intermolecular attractive forces. The constant 'a' is specific to each gas and is larger for gases with stronger intermolecular attractions.
  • Correction for volume (b): The term nb subtracts the volume occupied by the gas molecules themselves. The constant 'b' represents the volume per mole of gas particles.

Real gases behave most like ideal gases at high temperatures and low pressures.

Gases in liquids

The concentration of a gas dissolved in a liquid is proportional to its partial pressure above the liquid (Henry's law). This principle is medically important for understanding gas transport in the blood. Additionally, water vapor is a component of gas mixtures like air, which is relevant for respiratory physiology (see humidity, BTPS, and STPD).

Humidity

Humidity measures the amount of water vapor in the air.

• Absolute humidity: the mass of water vapor per unit volume of air (g/m³)
• Relative humidity: the ratio of the current partial pressure of water vapor in the air to the saturation vapor pressure of water at that temperature, expressed as a percentage
  • Formula: relative humidity (%) = (p_{water vapor}/p_saturation) × 100
    • p_{water vapor} = partial pressure of water vapor (Pa), p_saturation = saturation vapor pressure at a given temperature (Pa)
• When a gas is saturated, the water-vapor partial pressure needs to be subtracted from the total pressure before using the remaining (dry‑gas) pressure in gas‑law calculations.

BTPS and STPD

In medicine, gas volumes are often standardized to specific conditions:

• BTPS (Body Temperature and Pressure, Saturated): used for physiological gas volumes (e.g., tidal volume corrected to body conditions)
  • T = 37 °C (310.15 K)
  • Pressure = ambient atmospheric pressure
  • Air is saturated with water vapor (p_H₂O ≈ 47 mmHg or 6.3 kPa).
• STPD (Standard Temperature and Pressure, Dry): used for comparing gas volumes under standard lab conditions
  • T = 0 °C (273.15 K)
  • Pressure = 1 atm (101.3 kPa)
  • No water vapor is present (p_H₂O = 0 kPa).

When a gas is saturated with humidity (e.g., 'collected over water'), the total pressure P_total includes the partial pressure of water vapor P_{water vapor}. According to Dalton's Law, the pressure of the 'dry' gas is P_{dry gas} = P_total - P_{water vapor}. You must use this P_{dry gas} value in the combined gas law ((P₁ × V₁)/T1 = (P₂ × V₂)/T₂) or ideal gas law (P × V = n × R × T) for any calculations. All gas law calculations require temperature in Kelvin.

Sample calculation

Calculate the difference in water vapor pressure between inhaled and exhaled air. Assume inhaled air is at 15°C with 20% relative humidity (saturation vapor pressure at 15°C is 1.7 kPa) and exhaled air is at body temperature (37°C) and is fully saturated (saturation vapor pressure at 37°C is 6.3 kPa).

• Find: Δp_H₂O
• Given: temperatures, saturation vapor pressures, and relative humidity
  • First, find the partial pressure of water vapor in inhaled air: p_inhaled = 20% × 1.7 kPa = 0.34 kPa.
  • The partial pressure in exhaled air is the saturation pressure at 37°C, so p_exhaled = 6.3 kPa.
  • The difference is Δp_H₂O= 6.3 kPa - 0.34 kPa = 5.96 kPa. This represents the water added to the air during respiration.