Thermodynamics Notes

Thermodynamics I

• 1st law of thermodynamics

• Internal energy

• State and path functions

• Work and heat

• Heat capacity

• Enthalpy

• Isothermal and adiabatic expansions

• Thermochemistry

What is Thermodynamics?

• Thermodynamics is the science of heat and temperature, focusing on the laws governing energy conversion (mechanical, electrical, chemical).

• Examples include combustion in car engines and cooling in refrigerators.

• Greek origin: "thérme-" (heat), "dy’namis" (power).

Energy (E)

• SI unit: Joule (J).

• Older unit: calorie (cal), where 1 cal = 4.184 J

• Potential energy: energy by virtue of position or composition.

• Kinetic energy: energy by virtue of motion.

System and Surroundings

• Universe = System + Surroundings

• System: includes gaseous H2 and O2.

• Surroundings: cylinder, piston, and everything else.

• Any change in system and/or surrounding affects the universe.

Open, Closed and Isolated System

• Open: exchange of matter and energy with surroundings.

• Closed: exchange of energy, but not matter.

• Isolated: no exchange of matter or energy.

1st Law of Thermodynamics

• The internal energy (U) of an isolated system is constant (energy conservation).

• \Delta U{universe} = \Delta U{system} + \Delta U_{surroundings} = 0

• \Delta U{system} = -\Delta U{surroundings}

• This conservation applies to U, H, and other thermodynamic functions.

What is Internal Energy (U)?

• Internal energy includes translational, rotational, and vibrational energies; bond energies; and potential energy between molecules.

• For a closed system, change in U is due to heat (q) or work (w).

• \Delta U = q + w

Work (w)

• Work is energy flow between system and surroundings due to a force (F) acting through a distance (x).

• Examples: inflating a balloon, moving ions, gas compression/expansion.

• Gas compression/expansion work: w = -P_{ex}\Delta V

• w = F \cdot x

• w = -P{ex}(Vf - V_i)

• Positive w: work done on the system.

• Negative w: work done by the system.

Heat (q)

• Heat is energy flow between system and surroundings due to temperature difference.

• Heat flows spontaneously from high T to low T.

• Positive q: heat flow into the system.

• Negative q: heat flow out of the system.

• \Delta U = q + w

• Work and heat are transitory; internal energy is associated with the state of the system.

State and Path Functions

• State function: change depends on initial and final states only (e.g., ΔU).

• \Delta U = \int{Ui}^{Uf} dU = Uf - U_i

• Path function: depends on how the change occurs (e.g., q and w).

• We do not write Δq, qf, qi or Δw, wf, wi.

• q and w are inexact differentials unless the path is specified.

Enthalpy is Heat Change at Constant Pressure

• Constant volume: \Delta U = q + w = q_V

• Constant pressure: \Delta U = q + w = qP - P{ex}\Delta V

• Enthalpy: H \equiv U + PV; \Delta H = q_P

Enthalpy (H)

• Enthalpy accounts for heat flow in constant pressure chemical processes.

• H = U + PV

• \Delta H = Hf - Hi = \Delta U + P\Delta V

• ΔH is a state function.

• Endothermic: \Delta H > 0, \Delta U > 0

• Exothermic: \Delta H < 0, \Delta U < 0

Heat Capacity

• Amount of energy to raise the temperature of a substance by 1 K.

• Expressed at constant P (CP) or V (CV).

• CP = \frac{qP}{\Delta T} = \frac{\Delta H}{\Delta T}

• CV = \frac{qV}{\Delta T} = \frac{\Delta U}{\Delta T}

How do H2O Molecules store energy?!

• Molecules store energy in translation, rotation, and vibration.

• Higher T means faster movement, quicker rotation, and more rigorous vibration.

• E{trans} = \frac{3}{2}kT E{rot} = \frac{3}{2}kT E_{vib} = kT

• At room temperature, molecules store energy in translation and rotation.

CP and Cv

• Constant P heating: some heat used for gas expansion.

• CP \Delta T (const. P) = CV \Delta T (const. V)

• \Delta T (const. P) < \Delta T (const. V)

• CP > CV

• CP - CV = nR

CP,M and CV,M of Mono- and Di-atomic Gases

• Monoatomic: energy deposited into translations only. C{V,M} = \frac{3}{2}R C{P,M} = \frac{5}{2}R

• Diatomic: energy deposited into translation and rotation. C{V,M} = \frac{5}{2}R C{P,M} = \frac{7}{2}R

• C{P,M} - C{V,M} = R

• \gamma = \frac{CP}{CV} = \frac{C{P,M}}{C{V,M}} > 1

Isothermal Expansion

• Constant temperature expansion (ΔT = 0).

• Irreversible: w{irrev} = -Pf(Vf - Vi)

• Reversible: w{rev} = -\int{Vi}^{Vf} \frac{nRT}{V} dV = -nRT \ln{\frac{Vf}{Vi}}

• \Delta U = q + w = 0 \implies -q = w

Irreversible Path – Work Done is from Surr.

• Irreversible Compression: w{irrev comp} = -Pi(Vi - Vf)

• \Delta U{cycle} = 0 \implies -q{net} = w_{net}

• Reversible process: system and surroundings revert to original states by reversing the process.

Isothermal Reversible Expansion

• Isothermal reversible expansion gives the maximum work done.

Adiabatic Reversible Expansion

• Adiabatic: no heat change (q = 0).

• \Delta U = q + w = w

• \Delta U = w = C_V \Delta T

• CV \Delta T = w = -P{ex} \Delta V

Adiabatic Reversible Expansion Equations

• \frac{Tf}{Ti} = \left(\frac{Vf}{Vi}\right)^{1-\gamma}

• \frac{Tf}{Ti} = \frac{PfVf}{PiVi}

• PiVi^{\gamma} = PfVf^{\gamma}

• Gas expansion leads to cooling; gas compression to heating.

Isothermal vs Adiabatic Reversible Expansion

• Adiabatic expansion curve is steeper than isothermal.

• Adiabatic expansion reaches lower T because q = 0.

• Isothermal: PV = constant

• Adiabatic: PV^{\gamma} = constant

Isenthalpic Expansion through a Porous Hole

• Gas moves from high to low pressure through a porous plug (q=0).

• \Delta U = Uf - Ui = w

• w{left} = PiVi w{right} = -PfVf w = PiVi - PfVf

• Hf = Hi \implies \Delta H = 0

Joule−Thomson Effect

• Cooling effect observed in isenthalpic expansion.

• Joule-Thomson coefficient: \mu{JT} = \left(\frac{\Delta T}{\Delta P}\right)H

• For ideal gas, \mu_{JT} = 0.

• Positive \mu_{JT}: gas cools on expansion.

• Negative \mu_{JT}: gas heats up on expansion.

Properties of Enthalpy

• Enthalpy is an extensive property.

• ΔH for forward reaction is equal and opposite to ΔH for reverse reaction.

• ΔH depends on the state of products and reactants.

Standard Enthalpy of Reaction (ΔHo r)

• ΔHo r: enthalpy of products minus reactants at standard conditions (1 atm, 298.15 K).

• Hess's Law: if a reaction is carried out in steps, ΔH is the sum of enthalpy changes in individual steps.

Standard Enthalpy of Formation (ΔHo f)

• ΔHo f: heat change for formation of one mole of a compound from its elements in standard states.

• ΔHo f for the most stable form of an element is zero.

Relationship between ΔHo r and ΔHo f

• \Delta Hr^o = \sum n \Delta Hf^o(products) - \sum m \Delta H_f^o(reactants)

Dependence of a function on variables

• If the function now becomes f(x,y), the partial derivatives are:
  \frac{df}{dx} = \frac{\partial f}{\partial x}y \frac{df}{dy} = \frac{\partial f}{\partial y}x

  The total differential is the sum
  of partial derivatives:

  Describes the total changes in f when x → x + dx and y → y + dy.
  df =\frac{\partial f}{\partial x}y dx +\frac{\partial f}{\partial y}x dy