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).

• $\Delta U<em>{universe} = \Delta U</em>{system} + \Delta U_{surroundings} = 0$

• $\Delta U<em>{system} = -\Delta U</em>{surroundings}$

• 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: $w = -P_{ex}\Delta V$

• $w = F \cdot x$

• $w = -P<em>{ex}(V</em>f - V_i)$

• 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).

• $\Delta U = \int<em>{U</em>i}^{U<em>f} dU = U</em>f - U_i$

• 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: $\Delta U = q + w = q<em>P - P</em>{ex}\Delta V$

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

Enthalpy (H)

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

• $H = U + PV$

• $\Delta H = H<em>f - H</em>i = \Delta U + P\Delta V$

• 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).

• $C<em>P = \frac{q</em>P}{\Delta T} = \frac{\Delta H}{\Delta T}$

• $C<em>V = \frac{q</em>V}{\Delta T} = \frac{\Delta U}{\Delta T}$

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.

• $E<em>{trans} = \frac{3}{2}kT$ $E</em>{rot} = \frac{3}{2}kT$ $E_{vib} = kT$

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

CP and Cv

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

• $C<em>P \Delta T (const. P) = C</em>V \Delta T (const. V)$

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

• CP > CV

• $C<em>P - C</em>V = nR$

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

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

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

• $C<em>{P,M} - C</em>{V,M} = R$

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

Isothermal Expansion

• Constant temperature expansion (ΔT = 0).

• Irreversible: $w<em>{irrev} = -P</em>f(V<em>f - V</em>i)$

• Reversible: $w<em>{rev} = -\int</em>{V<em>i}^{V</em>f} \frac{nRT}{V} dV = -nRT \ln{\frac{V<em>f}{V</em>i}}$

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

Irreversible Path – Work Done is from Surr.

• Irreversible Compression: $w<em>{irrev comp} = -P</em>i(V<em>i - V</em>f)$

• $\Delta U<em>{cycle} = 0 \implies -q</em>{net} = w_{net}$

• 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).

• $\Delta U = q + w = w$

• $\Delta U = w = C_V \Delta T$

• $C<em>V \Delta T = w = -P</em>{ex} \Delta V$

Adiabatic Reversible Expansion Equations

• $\frac{T<em>f}{T</em>i} = \left(\frac{V<em>f}{V</em>i}\right)^{1-\gamma}$

• $\frac{T<em>f}{T</em>i} = \frac{P<em>fV</em>f}{P<em>iV</em>i}$

• $P<em>iV</em>i^{\gamma} = P<em>fV</em>f^{\gamma}$

• 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).

• $\Delta U = U<em>f - U</em>i = w$

• $w<em>{left} = P</em>iV<em>i$ $w</em>{right} = -P<em>fV</em>f$ $w = P<em>iV</em>i - P<em>fV</em>f$

• $H<em>f = H</em>i \implies \Delta H = 0$

Joule−Thomson Effect

• Cooling effect observed in isenthalpic expansion.

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

• 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

• $\Delta H<em>r^o = \sum n \Delta H</em>f^o(products) - \sum m \Delta H_f^o(reactants)$

Dependence of a function on variables

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

  The total differential is the sum
  of partial derivatives:

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