Thermodynamics - Equations

internal energy / first law

internal energy:
ΔU = Uf - Ui
(Note that U, is also shown as E in many books and often on Quest)

First Law of Thermodynamics
ΔU = q + w
(this is a mathematical version of the first law)


Substances not changing phase:
q   =   m Cs ΔT   (per gram)
q   =   n Cm ΔT   (per mol)

Substances that are changing phase (transition):
q   =   m ΔHtrans   (per gram)
q   =   n ΔHtrans   (per mol)

"trans" will be one of the following: fusion, freezing, evaporation, condensation, sublimation, or deposition


w   =   -PextΔV    (constant external pressure)
w   =   -ΔngasRT    (also constant external pressure)
Δngas = (#mol gas prod) - (#mol gas react)

enthalpy to internal energy

H = U + PV

ΔH = ΔU + PΔV (constant pressure)


ΔU = ΔH - ΔnRT

conditional heat flow
ΔH = qP   (constant pressure)
ΔU = qV   (constant volume)


qcal   =   -qsys

qcal   =   CcalΔT

coffee-cup Calorimetry (constant pressure, qP)
qcal = mwater · Cs,water · ΔT

bomb Calorimetry (constant volume, qV)
qcal = mwater · Cs,water · ΔT + Chardware · ΔT

Hess' Law

ΔHrxn = ΔH1 + ΔH2 + ΔH3 + ...
flip and scale various reactions to match the target reaction

ΔHrxn° = ΣnΔHf° (products) - ΣnΔHf° (reactants)
look up heat of formation data from a table

ΔHrxn° ≈ ΣnΔHbond°(breaking) - ΣnΔHbond°(making)
look up bond energies from a table

entropy and the second law

entropy is a measure of energy dispersal and the Second Law states that universal entropy must increase for any spontaneous process:
ΔSuniv = ΔSsys + ΔSsurr

entropy (S)
S  =  k ln Ω   Boltzmann formula

k is the Boltzmann Constant with a value of 1.38 × 10−23 J/K which is the same as the universal gas constant, R, divided by Avogadro's number, NA.

entropy change via reversible heat flow
ΔS  =  qrev / T   (constant T)

entropy change with changing T
ΔS  =  n Cp ln(Tf/Ti)   (changing T, constant P)

entropy change for phase transition (change)
ΔStrans  =  ΔHtrans / Ttrans

third law - absolute entropy

third law: the entropy of a perfectly crystalline substance is zero

ΔSrxn° = ΣnS° (products) - ΣnS° (reactants)

Note: S° is standard absolute entropy and is value you find in a thermodynamic table. It is NOT a "formation" reaction - notice the fact that there is no Δ there or a subscript "f". All values for substances that are solids, liquids, or gases are positive - aka: "absolute"

free energy

G   =   H - TS    (definition)
ΔG = ΔH - TΔS (constant pressure)

using formation data from a table...

ΔGrxn° = ΣnΔGf° (products) - ΣnΔGf° (reactants)


equilibrium is a state in which ΔSuniv = 0. We generally "recast" that idea into the free energy of the system and say that equilibrium is achieved when the free energy of the system is at a minimum and ΔG = 0.

So IF you have equilibrium in play...

ΔH = TeqΔS
Teq is the temperature at which equilibrium occurs. It is also the transition temperature where a process/reaction goes from spontaneous to non-spontaneous and vice-versa.

Conceptual Summary

A chemical system will tend to change from one state to another via whatever means are available such that universal entropy is increased. Maximum universal entropy change will coincide with the final state that tends to have the lowest possible total free energy of all the components. If that state is reached, then at that exact composition, all the reactants have the exact total free energy as all the products. That also means that ∆G = 0 at this point. This is the point at which equilibrium has been reached. There is no longer any tendency for the reaction to go in a net forward direction or net reverse direction. Either way would actually increase the free energy of the system. The equilibrium condition is basically what all reactions are "trying" to achieve. The "drive" to get to that position/condition is the change in free energy for the reaction, ∆G. If that quantity is positive, then the reaction will be driven in reverse from the way it is written on the page - non-spontaneous forward direction means spontaneous reverse direction. If the quantity is negative (-∆G) then the reaction will be driven forward.

One last thing... as reactions proceed forward, the amounts of reactants and products are constantly changing. The reactants decrease and the products increase in amount for a reaction going forward. Realize that free energy is tied to the amounts of substance as well as what the substance is. When a compound is decreasing, its free energy is decreasing as well because there is less and less of it. If it all reacts, then the free energy of that substance is now zero because it is gone. Of course, on the other side of the arrow are the products. They were at zero free energy when the reaction started (because nothing was there), but then the products begin to appear and the free energy climbs. The reaction will effectively stop when the total of all the free energies of the reactants equals the total free energy of the products. If the free energies match, then ∆G must be equal to zero, and the system has reached equilibrium.

So what does that mean? Well, no matter what your ∆G is for a reaction - whether it is positive or negative, the absolute value of ∆G will decrease and continue to decrease as the reaction proceeds forward (or backwards). Once the reaction hits that spot where ∆G = 0, then equilibrium is now "in play". Lots and lots of wonderful relationships and equations become important once you have equilibrium. We save all that "wonderfulness" for you in CH302.

Read the the previous paragraphs over and over and TRY to understand what they are saying. Understanding this is the KEY to having a good understanding of just what drives everything around us. Thermodynamics rules the driving force of all things.