Thermodynamics Condensed
by Dr. McCord
if you want to see only the formulas, go hereThe First Law of Thermodynamics
Energy can neither be created nor destroyed - only changed in form.
The energy of the universe is constant.
internal energy:
\(\Delta U = U_{\rm f} - U_{\rm i}\)
(Note that U, is also shown as E in many books and often on Quest)
First Law of Thermodynamics
\(\Delta U = q + w\)
(this is a mathematical version of the first law - sort of)
Substances not changing phase:
\(q = m\cdot C_{\rm s}\cdot \Delta T\) (per gram)
\(q = n\cdot C_{\rm m}\cdot \Delta T\) (per mol)
Substances that are changing phase:
\(q = m\cdot\Delta H_{\rm trans}\) (per gram)
\(q = n\cdot\Delta H_{\rm trans}\) (per mol)
"trans" is for phase transition and will be one of the following: fusion, freezing, evaporation, condensation, sublimation, or deposition
\(w = - P_{\rm ext}\Delta V\) (constant external pressure)
or
\(w = -\Delta n_{\rm gas}RT\) (also constant external pressure)
Δngas = (#mol gas prod) - (#mol gas react)
\(H = U + PV\)
\(\Delta H = \Delta U + P\Delta V\) (constant pressure)
\(\Delta U = \Delta H - P\Delta V\)
\(\Delta U = \Delta H -\Delta nRT\)
conditional heat flow
\(\Delta H = q_{\rm P}\) (constant pressure)
\(\Delta U = q_{\rm V}\) (constant volume)
\(\Delta U = 0\) (constant T)
\(q = -w\)
\(w = \int_{\rm i}^{\rm f} {nRT\over V}{\rm d}V\)
\(w = -nRT \ln\left({V_{\rm f}\over V_{\rm i}}\right)\)
\(q = nRT \ln\left({V_{\rm f}\over V_{\rm i}}\right)\)
\(q_{\rm cal} = -q_{\rm sys}\)
\(q_{\rm cal} = C_{\rm cal}\Delta T\)
coffee-cup Calorimetry (constant pressure, directly getting qP which is ΔH)
\(q_{\rm cal} = m_{\rm water} C_{\rm s,water} \Delta T\)
bomb Calorimetry (constant volume, directly getting qV which is ΔU)
\(q_{\rm cal} = m_{\rm water} C_{\rm s,water} \Delta T\) + \(C_{\rm hardware}\Delta T\)
\(\Delta H_{\rm rxn} = \Delta H_1 + \Delta H_2 + \Delta H_3 + \cdots\)
flip and scale various reactions to match the target reaction
\(\Delta H^\circ_{\rm rxn} = \sum n\Delta H^\circ_{\rm f}{\rm (products)}\) \(- \sum n\Delta H^\circ_{\rm f}{\rm (reactants)} \)
look up heat of formation data from a table
\(\Delta H^\circ_{\rm rxn} = \sum n\Delta H_{\rm bond}{\rm (breaking)}\) \(- \sum n\Delta H_{\rm bond}{\rm (making)} \)
look up bond energies from a table
The Second Law of Thermodynamics
All spontaneous changes are accompanied by an increase in universal entropy.
\(\Delta S_{\rm univ} = \Delta S_{\rm sys} + \Delta S_{\rm surr}\)
(note that universal entropy is BOTH the system AND the surroundings)★★ Spontaneity ★★
"Spontaneity", in the chemical sense, is the tendency or propensity of a given reaction or process to proceed forward as shown. All spontaneous reactions are said to be "energetically favorable".
This means that the initial state and the final state must be communicated in order to establish the direction of change. Other factors may impede or even stop the tendency to proceed forward - this still does not change the direction of the spontaneous change.entropy is a measure of energy dispersal
entropy (S)
\(S = k \ln \Omega\) 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
\(\displaystyle\Delta S = {q_{\rm rev}\over T}\) (constant T)
entropy change with changing T
\(\displaystyle\Delta S = n\;C_{\rm m} \ln\left(T_{\rm f}\over T_{\rm i}\right)\) (changing T, constant P)
entropy change with changing V (isothermal expansion)
\(\displaystyle\Delta S = n\;R \ln\left(V_{\rm f}\over V_{\rm i}\right)\) (changing V, constant T)
entropy change for phase change
\(\displaystyle\Delta S = {\Delta H_{\rm trans}\over T_{\rm trans}}\)
The Third Law of Thermodynamics
The entropy of a perfectly crystalline substance at absolute zero is zero.
(this law is what allows us to have absolute entropy values, S)\(\Delta S^\circ_{\rm rxn} = \sum n S^\circ{\rm (products)}\)\(- \sum n S^\circ{\rm (reactants)} \)
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"
\(G = H - TS\) (definition)
\(\Delta G = \Delta H - T\Delta S\) (constant temperature)
using formation data from a table...
\(\Delta G^\circ_{\rm rxn} = \sum n\Delta G^\circ_{\rm f}{\rm (products)}\) \(- \sum n\Delta G^\circ_{\rm f}{\rm (reactants)} \)
equilibrium is a state in which \(\Delta S_{\rm univ}=0\). 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 \(\Delta G = 0\).
So IF you have equilibrium in play...
\(\Delta H = T_{\rm eq}\Delta S\)
\(T_{\rm eq}\) 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.
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.