CHEM 205

Ideal Gas Law

• Relationship between P, V, n, T for an ideal gas

• R = gas constant

• Equation forms:

  • PV = nRT

  • PV_m = RT (molar volume form)

    • V_m = V/n

To use the Ideal Gas Law:

1. Rearrange PV = nRT

2. Convert units

3. Sub values

Kinetic Model Assumptions

What assumptions define a perfect (ideal) gas?

1. Molecules in random motion

2. Point particles (no volume)

3. Move in straight lines between collisions

4. No interactions except elastic collisions

Pressure from Molecular Collisions

What causes gas pressure?

• Pressure = force from molecular impacts on container walls

• Derived from momentum change during collisions

• Leads to relation:

  • PV = 1/3 Nmv²

Translational Kinetic Energy

What is the avg translational kinetic energy of an ideal gas?

• Per mole: E_trans = 3/2 RT

• Per molecule: 3/2 k_BT

• Shows U depends only on T for ideal gases

• U = E

Equipartition Principle

What does equipartition say about energy contributions?

• Each translational or rotational degree of freedom → RT/2

• Each vibrational degree → RT

• Explains why diatomic gases have:

  • High T: U = 7/2 RT

  • Room T: U = 5/2 RT (vibration not excited)

Internal Energy U of Different Molecules

What is U for different ideal gases?

• Monatomic: U = 3/2 nRT

• Linear molecules: U = 5/2 nRT

• Nonlinear molecules: U = 3 nRT

• Only valid when vibrational modes not excited

Key insights of U:

• U = U(T) only

• Independent of volume or pressure

• Breaks down real gases due to interactions

Van der Waals Equation

Van der Waals introduces:

• Repulsion (finite size): replace V → (V - nb)

• Attraction: replace P → (P + an²/V²)

• Full equation:

  • (P + an²/V²) (V - nb) = nRT

Compressibility Factor Z

What is compressibility factor z?

• z = PV/nRT

• z = 1: ideal gas

• z ≠ 1: real gas deviations

• used to quantify non-ideality

Virial Equation:

• Expansion: z = PV/RT = 1 + B₂P/RT + B₃P²/RT + …

• Describes intermolecular forces

What Thermodynamics studies

• Transformation of energy in chemical systems

• Predicts direction of reactions + equilibrium

• Determines driving forces (ΔG, ΔS, ΔH)

0th, 1st, 2nd, 3rd Laws

The four laws of thermodynamics:

• Zeroth: Defines temperature via thermal equilibrium

• First: Energy conservation → ΔU = q + w

• Second: Direction of spontaneous change → ΔS_univ >= 0

• Third: Defines absolute entropy (S → 0 at 0K for perfect crystal)

System / Surroundings / Universe

• System = Part of universe being studied

• Surroundings = Everything else

• Universe = system + surroundings

Types of systems:

• Open: exchanges matter + energy

• Closed: exchanges energy only

• Isolated: exchanges neither

State Functions

What is a State function?

• Depends only on current state, not path

• Examples:

  • P, V, T, U, H, G, S

  • q and w are NOT state functions

Zeroth Law

• If A is in thermal equilibrium with B, and B with C, then A is with C

First Law

ΔU = q + w

• q > 0: heat into system

• w > 0: work done ON system

Sign conventions:

• w > 0: surroundings compress system

• w < 0: system expands (does work ON surroundings)

Heat vs Work

• Heat q: energy transfer due to temperature differences

• Work w: energy transfer due to force acting through distance

Reversible vs Irreversible Processes

• Reversible: System always in equilibrium; max work

  • ΔS_univ = 0

• Irreversible: Real processes; spontaneous; not at equilibrium

  • ΔS_univ > 0

Types of processes

• Isothermal: T constant

• Isobaric: P constant

• Isochoric: V constant

• Adiabatic: q = 0

• Exothermic: release heat into surroundings (q < 0)

• Endothermic: absorb heat from surroundings (q > 0)

Expansion Work

w = -P_ext ΔV

• Expansion (ΔV > 0): w < 0

• Compression (ΔV < 0): w > 0

• P_ext = constant

Free Expansion:

• Expansion into vacuum → P_ext = 0

• w = 0, q = 0, so ΔU = 0

Reversible Isothermal Expansion/Compression of an ideal gas:

• P_gas = nRT/V

• Reversible: P_ext = P_gas

• w = -nRT ln(V_f / V_i)

• If isothermal, ΔU = 0, so q = -w

Enthalpy Definition

H = U + PV

• State Function

• At constant V:

  • ΔU = q_v

• At constant P:

  • ΔH = q_p

• Determines which heat capacity to use (C_V or C_P)

ΔH: tracks heat at constant pressure → calorimetry

Heat Capacity

C = dq/dT

q = C ΔT

• C_V = heat capacity at constant volume

• C_P = heat capacity at constant pressure

• C_P > C_V

Specific vs Molar Heat Capacity

• Specific (C_s) = per gram

• Molar (C_m) = per mole

• C_{P, m} = C_P / n

Constant Volume Process

• V = constant → no PV work

• w = 0

• ΔU = q_V

• Heat at constant volume directly changes internal energy

Heat Capacity:

• C_V = n C_{V, m}

• ΔU = n C_{V, m} ΔT

Relationship between C_P and C_V

C_{P, m} = C_{V, m} + R

• Monatomic ideal gas:

  • C_{V, m} = 3/2 R, C_{P, m} = 5/2 R

Reaction Enthalpy (delta H_rxn)

• Δ_rH = sum[v_i H_m,i (products)] - sum[v_j H_m,j (reactants)]

Standard Reaction Enthalpy:

• Same formula as Δ_rH but using standard molar enthalpies

Kirchhoff’s Law

• Δ_rH⁰(T) = Δ_rH⁰(T₀) + \int^T_T₀ [sum v Cpm(prod) - sum v Cpm(react)] dT

Why the 2nd Law?

• 2nd Law introduces entropy S, criterion for spontaneity

Spontaneous vs Nonspontaneous Processes

Spontaneous:

• Occurs in 1 direction only (irreversible)

• Always involves loss of work ( w → q)

• can be slow

Nonspontaneous:

• Occurs only with external intervention (work input)

• First law may show ΔU = 0, but spontaneity still requires entropy

Entropy S

• Measure of disorder, randomness

• High S → high disorder

• Perfect crystal at 0K has S = 0 (third law)

Definition of entropy change:

• ΔS = \int^{state 2}_state1 [d q_rev / T]

• dS = dq_rev / T

• Only reversible heat matters

• If the process is irreversible, construct a reversible path

2nd Law Statements:

• Reversible: ΔS_univ = 0

• Irreversible (spontaneous): ΔS_univ > 0

• S_univ = system + surroundings

Cases of Entropy

Isolated System:

• No heat or matter exchange

• Entropy must increase for any spontaneous change

  • ΔS_sys > 0

Isothermal Expansion:

• ΔS = nR ln(V₂ / V₁) = nR ln(P₁ / P₂)

  • P1V1 = P2V2 = nRT

  • works for reversible and irreversible expansions

Constant V (Isochoric) or Constant P (Isobaric):

• dS = C_{V or P} dT / T

• If C is constant:

  • ΔS = C_{V or P} ln(T₂ / T₁)

• Applies to any Isochoric/Isobaric processes

Adiabatic Process:

• Reversible adiabatic:

  • q_rev = 0 → ΔS = 0

• Irreversible adiabatic:

  • ΔS_univ > 0 → S_sys > 0

Phase Changes:

• for reversible phase transitions:

  • ΔS = ΔH / T

Standard Entropy of Reaction

delta_r S^deg = sum[v_iS^deg(prod)] - sum[v_jS^deg(react)]

Temp Dependence of reaction entropy:

• Δ_rS⁰(T) = ΔS⁰(T) + \int^T_T₀ [sum(prod) - sum(react)] dT / T

Gibbs Free Energy

• Because spontaneity requires evaluating ΔS_sys + ΔS_surr → inconvenient

• Gibbs shows at constant T and P, S can be determined using:

  • G = H - TS

  • A state function with units of J

  • Molar Gibbs energy: G_m = G / n

• At constant T:

  • ΔG = ΔH - TΔS

Spontaneity Criterion (Constant T and P):

• ΔG < 0 → spontaneous (ΔS_univ > 0)

• ΔG = 0 → equilibrium (ΔS_univ = 0)

• ΔG > 0 → nonspontaneous

Standard Gibbs Energy of Reaction:

• ΔrG^deg = sum[v ΔrG∘(prod)] - sum[v ΔrG∘(react)]

Gibbs vs 2nd Law

When can we replace the 2nd law with ΔG?

• Only when T and P are constant

• ΔS_univ > 0 ←> ΔG < 0

Chemical Potential Definition

μ_i = (dG / dn_i)_{P, T, n}

• Partial molar Gibbs energy → measures how much G changes when adding species i

3rd Law of Thermodynamics

• A perfect crystal at 0K has 0 entropy:

  • S(0) = 0

• Defines the absolute entropy scale

Boltzmann Entropy:

• Statistical definition of entropy

  • S = k ln(W)

  • W = # of microstates

  • Perfect crystal: W = 1 → S = 0

Phase Diagram

What does a P vs T phase diagram show?

• Shows which phase (solid, liquid, gas) is most stable at each (P, T)

• Phase boundaries = equilibrium between two phases

• Triple point = all 3 phases coexist

• Critical point = end of liquid-gas boundary (supercritical fluid)

What do these three phase boundaries represent?

• Fusion curve: equilibrium ⇌ liquid

• Vaporization curve: liquid ⇌ gas

• Sublimation curve: solid ⇌ gas

• Each point on a boundary = unique (P, T) where two phases coexist

Vapor Pressure:

• Pressure of vapor in equilibrium with its liquid

• function of T onle: P_vap(T)

Triple Point:

• Unique (P, T) where s, l, g coexist

• Degrees of freedom F = 0 → neither P nor T can vary

Critical Point:

• Liquid-gas boundary ends

• Above T_c: no liquids exists regardless of pressure

• supercritical fluid forms → gas-like expansion, liquid-like density

The Gibbs Phase Rule

F = C - P + 2

• For one-component (C = 1) systems: F = 3 - P

  • P = 1 → F = 2 (T and P vary freely)

  • P = 2 → F = 1 (boundary line)

  • P = 3 → F = 0 (triple point)

What condition defines phase equilibrium?

• μ_A = μ_B

• G_A = G_B

• ΔG = 0 along phase boundaries

Clausius-Clapeyron Equation (general)

• dp/dT = ΔS/ΔV = ΔH / TΔV

• Vaporization form

  • ln(P₂ / P₁) = - ΔH_vap / R (1/T₂ - 1/T₁)

Standard State

What is activity “a” and why do we use it?

• Activity a corrects for non-ideal behavior so that

• G - G⁰ = RT ln(a)

• remains valid for all systems

• standard state: a = 1

  • solids: a = 1

  • liquids: a = 1

  • ideal gas: a = P/P⁰

  • solute: a = y(c/c⁰)

  • solvent: a = yx

Reaction Gibbs Energy and Q

How is ΔG related to Q?

• ΔG = ΔG⁰ + RT ln(Q)

• Forward spontaneous if ΔG < 0

• Reverse spontaneous if ΔG > 0

At equilibrium:

• ΔG = 0, so:

  • ΔG⁰ = -RT ln(K)

  • Q = K

• So can say:

  • ΔG = RT ln(Q / K)

What does magnitude of K tell us?

• K > 1: products favored

• K < 1: reactants favored

• K = 1: neither favored

• Even if ΔG⁰ > 0, K > 0 always (some forward reaction must occur)

Q vs K:

• Q < K: forward spontaneous

• Q > K: reverse spontaneous

• Q = K: equilibrium

Van’t Hoff Equation

How does K change with temperature?

• ln(K₂ / K₁) = - ΔH⁰ / R (1/T₂ - 1/T₁)

• Assumes ΔH⁰ and ΔS⁰ are constant with T

Free Energy and Work

What is the physical meaning of ΔG?

• At constant T and P:

  • -ΔG = W_max

  • real processes are irreversible → actual work < max work

Temperature and Spontaneity

• ΔH < 0, ΔS > 0 → spontaneous at all T

• ΔH < 0, ΔS < 0 → spontaneous at low T

• ΔH > 0, ΔS > 0 → spontaneous at high T

• ΔH > 0, ΔS < 0 → never spontaneous

Electrochemistry, Cell Reaction Construction

How do you construct a full redox reaction from half-reactions?

• Write both half-reactions as reductions

• Cell reaction = (rhs) - (lhs)

• Electrons must cancel → defines n

• Ex:

  • Cu²⁺ + 2e^- → Cu

  • Zn²⁺ + 2e^- → Zn

  • Total: Cu²⁺ + Zn → Cu + Zn²⁺

Nernst Equation

• E = E⁰ - RT/nF ln(Q)

• at equilibrium: E = 0 → Q = K

Boiling Point Elevation

Why does adding solute raise the boiling point>

• Solute lowers solvent vapor pressure → requires higher T to reach P_vap = P_ext

• Boiling point elevation:

  • ΔT_b = K_bb_B

  • K_b = solvent-dependent

  • b_B = molality

• Depends only on # of solute particles, not identity

Freezing Point Depression

Why does adding solute lower the freezing point?

• Solute disrupts crystal formation → requires lower T for solid to form

• ΔT_f = -K_fb_B

• b_B = n / m = m_B / M_B / m_solvent

• Depends only on # of solute particles

Osmosis

• Flow of pure solvent → more concentrated solution through a membrane

• Osmotic Pressure:

  • Π = c_BRT

  • c_B = n/V

Donnan Membrane Equilibrium

• Donnan Potential:

  • E = -RT/z_iF ln( [ion]₂ / [ion]₁)

  • z_i: charge of ion

Why quantum mechanics?

Could not explain:

1. Blackbody radiation

2. Photoelectric effect

3. Line spectra

Planck’s Quantization

• Energy is emitted/absorbed in discrete packets (photons)

  • E = hv = hc/λ

  • v = freq

Particle in a Box

• Wavefunctions: chi(x) = sqrt(2/L) sin(n pi x / L)

• Energies: E_n = n²h²/8mL²

  • Energy spacing increases with n

  • ΔE = hv gives absorption wavelength

Absorption vs Emission

• hv = E₂ - E₁

• Absorption: E₁ → E₂

• Emission: E₂ → E₁

Beer-Lambert Law

• A = εcl

• A: absorbance

• ε: molar extinction coefficient

• c: concentration

• l: path length

Transmittance Relation

• A = -log₁₀(T) = log₁₀(I₀ / I)

• I: intensity

• T = I / I₀

Conjugation and λ_max

More conjugation → smaller ΔE → longer lambda (red-shift)

Molecular Orbitals

• Electronic transitions occur between MO’s

• Linear combi of atomic orbitals → bonding (sigma, pi) + antibonding (sigma\, pi\) orbitals

• Nonbonding (n) orbitals from lone pairs also participate

Rules MO electrons must follow:

• Max 2 electrons per MO

• Opposite spins (Pauli exclusion principle)

• Fill lowest energy first (Aufbau)

Term Symbols and Spin multiplicity

Multiplicity = 2S + 1

• S = total electron spin

• Singlet (S=0), Triplet (S=1)

Frank-Condon Principle

• Electronic transitions occur faster than nuclei can move

• Geometry fixed during excitation (vertical transition)

• Intensity depends on vibration wavefunction overlap

When is an electronic transition intense?

• When initial and final vibrational wavefunctions have large overlaps → high probability density

Fluorescence

1. Absorption: S₀ → S₁ (vertical FC transition)

2. Vibrational relaxation (nonradiative)

3. Fluorescence: S₁(v=0) → S₀(v>0)

4. Relaxation to S₀(v=0)

Stokes Shift

Fluorescence occurs at longer lambda (lower energy) than absorption due to vibrational relaxation before emission

IR Spectroscopy

What does IR spectroscopy probe?

• vibrational transitions within the same electronic state

What is the IR unit wavenumber?

• ν~ = 1/lambda (cm-1)

• higher v~ → higher energy

Harmonic Oscillator Model

Bond modeled as spring:

• F = -kx

• energy levels equally spaced

• selection rule: Δv = +-1

IR Activity Rule

When is a vibration IR active?

• Dipole moment must change during vibration

  • Dipole moment: size of an electric dipole

• Homonuclear diatomics → IR inactive

• Heteronuclear → IR active

Morse Oscillator

Real bonds are anharmonic → allows:

• Overtones (Δv = +-2, +-3, …)

• Hot bands

• Bond Dissociation

Number of Vibrational Modes

How many fundamental vibrational modes?

• Linear: F = 3N - 5

• Nonlinear: F = 3N - 6

Which nuclei are NMR-active?

• Odd mass, odd atomic #, or both → nuclear spin I ≠ 0 → magnetic moment

• ¹³C and ¹H

Spin states for C and H:

• I = ½ → 2I + 1 = 2 states

• m_i = +1/2 (alpha, low E)

• m_i = -1/2 (beta, high E)

Energy Gap in NMR

Expression for ΔE between alpha and beta states?

• ΔE = hv = yhB₀

• B₀ increase → ΔE increase → v increase

Larmor Frequency

v = yB₀/(2pi)

• Precession freq of nuclear magnetic moment around B₀

Resonance Condition

When does NMR absorption occur?

• When RF photon matches ΔE:

  • hv = yhB₀

• spin flip alpha → beta

Chemical Shift (dirac)

dirac = v - v_ref / v_ref * 10^6 ppm

How do shielding effects change dirac?

• Shielding: omega B < 0 → B_loc < B₀ → dirac decreases (upfield)

• Deshielding: omega B > 0 → B_loc > B₀ → dirac increases (downfield)

What controls chemical shift?

• Local electron density → shielding

• electronegativity, hybridization, etc

¹H NMR (chemical shift vs intensity)

¹H NMR provides:

1. Location (chemical shift dirac)

2. Area (integration → # of H)

3. Fine structure (J-coupling → neighbors)

J-coupling Basics

What causes spin-spin splitting

• Through bond coupling (1-5 bonds)

• Neighboring spins (alpha/beta) shift resonance → multiplets

• J (Hz) = constant, independent of B₀

n+1 Rule

Splitting pattern for n neighboring nonequivalent protons?

• Multiplicity = n + 1

• Intensitiies follow Pascal’s triangle

AX Systems

Splitting pattern for an AX system:

• A split by X → doublet

• X split by A → doublet

• 2 doublets, same J

AX₂ Systems

• A split by 2 equivalent X → triplet (1:2:1)

• X split by A → doublet

A₃X₂ Systems

• CH₂ has 3 neighbors → quartet (1:3:3:1)

• CH₃ has 2 neighbors → triplet (1:2:1)

• OH usually a singlet (rapid exchange)