Tuesday, October 30, 2007
351 q3 moved to Friday
I won't make a habit of succumbing to the mob but I am moving the quiz to Friday. Hopefully this change doesn't screw anyone over. Tomorrow, Chapter 6, in which we will take thermo to a whole new level.
The Plight of Boltzmann+ Four Laws
By the time the first lecture of Week 7 had finished, we had discussed all four laws of thermodynamics, in mathematical and verbal forms, established the probabilistic nature of entropy and worked through ten types of entropy calculations. In fact, nearly all of the framework has now been constructed for Chapter 6, in which everything comes to fruition as we discuss spontaneity and equilibrium.Backtracking, last Friday we explored the work of [my idol] Ludwig Boltzmann, who beat his head against the scientific establishment as he spearheaded the development of statistical mechanics. Bypassing all the entropy is disorder nonsense, here we can see the physical underpinnings of entropy are probabilistic and that entropy is a measure of the number of accessible microstates to a system [which may arise as translational, rotational, vibrational, electronic, nuclear, configurational, etc]. In other words, entropy is a metric related to the number of ways that energy can be dispersed (into these microstates). Now that you have been equipped with this interpretation of absolute entropy, you can successfully point-and-laugh at all those who persist in utilizing the now-debunked disorder interpretation.
On Monday, we finished [finally] our set of ten processes:
01. cyclic process
02. reversible adiabatic
03. reversible isothermal
04. reversible phase change (at constant T, P)
05. reversible change of state [ideal gas]
06. irreversible change of state [ideal gas]
07. change of state [general] (two versions: T,V and T,P)
08. mixing of ideal gases A and B (also ideal solutions)
09. irreversible phase change (at constant T, P)
10. chemical reactions
These 10 processes cover nearly every situation of interest in chemistry.
Finally we elucidated the Third Law of Thermodynamics, that S → 0 as T → 0. As indicated in class, this is a restatement of what we saw in the Carnot engine, that absolute zero cannot be attained (although we've gotten way way down there, to 450 pK, where matter acts truly bizarre because of the dominance of quantum over thermal effects).
The absolute entropy of real matter, incidentally, usually approaches a nonzero S0, the residual entropy, which is a loose measurement of the strength of low-temperature intermolecular forces.
I hope it is clear by now that the Laws of Thermodynamics, in essence, establish a logical code from which nearly all energy transfer (and, hence, all phenomena) can be described. Turning this framework into usable results is not always easy, however.
Interestingly, a Fourth Law of Thermodynamics is often proposed, the Onsager reciprocal relations which we will not cover until pchem 2.
And now, the Four Laws of Thermodynamics, translated for Sanitation Engineers:
0th: There is shit.
1st: You can't get rid of it.
2nd: It gets deeper.
3rd: A nice empty trashcan is wishful thinking.
Thursday, October 25, 2007
The Second Law Is Better Than The Zeroth Law By Two Units
On Wednesday, we finally made it to possibly the most powerful and darkly beautiful statement in all of thermodynamics: the Second Law. But, before that, we examined how we might calculate some entropy changes using the Clausius relation [dS = dqrev/T], underscoring how this equation only works while tracing reversible paths. The irreversible heat transfer from a hot to cold reservoir can, for example, be broken into three reversible -- and calculable -- steps. It is from this simple system that leads us to the 2nd Law conjecture:∆Suniv ≥ 0 [= for equil/reversible, > for spont/irreversible]Unfortunately, abuses of this statement are many and takes but a moment's googling to find them. For many scientists, it holds a special place:
The law that entropy always increases-the second law of thermodynamics-holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations-then so much the worse for Maxwell's equations. If it is found to be contradicted by observation-well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation." – Sir Arthur Eddington [1928]And to reiterate some general statements from class, which will hopefully help you more fully grasp spontaneity and reversibility:
All spontaneous processes are irreversible.Some common misstatements about the Second Law:
For all irreversible processes, ∆Suniv > 0.
All reversible processes are at equilibrium.
For all reversible processes, ∆Suniv = 0.
All systems tend to greater disorder.Before moving onto specific applications of the Clausius relation, we paused, for bookkeeping's sake, to mention the Zeroth Law: If systems A and B are in equilibrium, and systems B and C are in equilibrium, then systems A and C are in equilibrium. Not only does this establish that a state property common to them must be equal (the temperature), it also allows for the possibility that two systems could be in equilibrium without in direct contact.
False: Not only is "disorder" a poorly constructed idea and overly dependent on human interpretation, the underlying premise is wrong.
All systems tend towards greater entropy.
False: Some systems tend towards greater entropy while others don't. The restriction rests on ∆Suniv, not on ∆S.
Lastly we began our march of calculating entropy changes for ten processes, finishing four:
01. cyclic
02. reversible adiabatic
03. reversible isothermal
04. reversible phase transition [at const T, P]
On Friday, we will finish this list of ten, visit the revolutionary work of Boltzmann and, if time permits, elucidate the Third Law of Thermodynamics.
Tuesday, October 23, 2007
The Direction of Spontaneous Change
We began the first lecture of Week 6 with an examination of the inadequacy of the First Law to sufficiently describe thermodynamic events. For example, it does not preclude a penny, say, from absorbing thermal energy from a table and turning it into gravitational work (that is, springing up off the table). The Boltzmann formula will demonstrate that the probability of this event is nonzero but exceedingly small (so unimaginably improbable that perhaps we should call it impossible?) But the point is that, macroscopically, we see a definite directionality to energy transfer:
Through his theoretical work on heat engine efficiency, the French engineer Sadi Carnot was our first thermodynamicist. His memoirs, lost for twenty years and posthumously rescued by college friend Benoit Clapeyron, inspired the work of Clausius and Thomson [Kelvin], both of whom essentially triggered the thermodynamic revolution. Carnot's greatest achievement was to demonstrate that heat flow could be harnessed and transmogrified into usable work by engines, but with a maximum efficiency less than 100%. Indeed, this maximum efficiency is dependent only on the reservoir temperatures and not on the material used in the engine, nor on the actual steps of each cycle. It is a thermodynamic limit imposed on us by nature, who has decreed that heat is a form of energy rather than a transferred substance.
[Note: In today's lecture, I believe that I inadvertently flipped the subscripts for the temperatures in the adiabatic formulas. Consult Engel-Reid for consistency]
Looking further at the results of Carnot we see, as Clausius did, a hidden state function, one that sums to zero as we go around a cycle. From this fact we can back out the relationship dS = dq/T [the Clausius equation], introduced at the end of the hour but forming the basis of Wednesday's lecture to come.
First Law - limits the magnitude of energy transferBefore discussing how it was first discovered, we first needed to correct some misconceptions about entropy, one of the most thoroughly mangled concepts in all of science:
Second Law - limits the direction of energy transfer
Entropy is not equal to disorder, nor is it a measure of disorder (whatever that means scientifically).One of the most common [bad] examples demonstrating the alleged relationship between entropy and disorder is a deck of cards. When we shuffle an "ordered" deck of cards, we always see it become "disordered". The problem with this language is that there is no way to quantify order, especially since every outcome is equally probable. We have simply defined A,2,3,4 .. Q,K of each suit as being the ordered state -- but that definition is arbitrary. Nature should not -- and does not -- depend on such human definitions. Other [bad] examples include blaming messy desks and cluttered rooms on this "law of entropy."
When we connect it directly to the number of accessible microstates (ala Boltzmann) we will understand the probabilistic basis of entropy more fully.
Entropy is a measure of the tendency of energy to disperse, rather than being localized.
Through his theoretical work on heat engine efficiency, the French engineer Sadi Carnot was our first thermodynamicist. His memoirs, lost for twenty years and posthumously rescued by college friend Benoit Clapeyron, inspired the work of Clausius and Thomson [Kelvin], both of whom essentially triggered the thermodynamic revolution. Carnot's greatest achievement was to demonstrate that heat flow could be harnessed and transmogrified into usable work by engines, but with a maximum efficiency less than 100%. Indeed, this maximum efficiency is dependent only on the reservoir temperatures and not on the material used in the engine, nor on the actual steps of each cycle. It is a thermodynamic limit imposed on us by nature, who has decreed that heat is a form of energy rather than a transferred substance.
[Note: In today's lecture, I believe that I inadvertently flipped the subscripts for the temperatures in the adiabatic formulas. Consult Engel-Reid for consistency]
Looking further at the results of Carnot we see, as Clausius did, a hidden state function, one that sums to zero as we go around a cycle. From this fact we can back out the relationship dS = dq/T [the Clausius equation], introduced at the end of the hour but forming the basis of Wednesday's lecture to come.
Sunday, October 21, 2007
The Second Law
On Monday we will start Chapter 5 and move towards the Second Law of Thermodynamics, one of the most important statements in all of science. First I'll argue why the First Law is incomplete as a description of thermodynamics then dive headfirst into the Carnot cycle.
Wednesday, October 17, 2007
End of the First Law Era
How do we obtain reaction energies from reaction enthalpies, or vice versa? How do we determine a reaction enthalpy at a nonstandard temperature given a value at 298K? These two questions take us to the end of Chapter Four and the "First Law Era".
It is straightforward to derive the relation ∆rxnU°=∆rxnH°-RT∆rxnνgas which can be used to interconvert between reaction energy and enthalpy. When using this equation, we must keep in mind that (a) we are implicitly assuming all gases are ideal and (b) that, for liquids and solids, molar enthalpies and internal energies are approximately equal. In principle, we could jam in a real gas equation of state and create a ghastly version that is more general or, as is always done, use this version anyway and take the hit in accuracy. Assumption (b) is rather good because, unless we encounter extreme pressures, the heat capacities CP,m and CV,m are nearly equal for condensed phases.
Adjusting to a nonstandard temperature is rather important since many (most?) reactions do not actually occur at 25°C and the difference is often quite significant. Once the heat capacity CP,m is known as a function of temperature, we can calculate ∆∆rxnH°=∫∆rxnC°PdT.
To make integration life easier, the heat capacities are fit to simple polynomials of T:
CP,m=a + bT +cT2+dT3+eT4 [Shomate] or
CP,m=a + bT +cT-2
Exam 1 on Thursday. If you are nervous, just remember that if you got through organic chemistry, calculus and physics, you can do this. I have posted solutions to the two questions I assigned from Chapter 4. I also updated the study sheet that had some equations missing and an error in one of the thermodynamic equations of state.
It is straightforward to derive the relation ∆rxnU°=∆rxnH°-RT∆rxnνgas which can be used to interconvert between reaction energy and enthalpy. When using this equation, we must keep in mind that (a) we are implicitly assuming all gases are ideal and (b) that, for liquids and solids, molar enthalpies and internal energies are approximately equal. In principle, we could jam in a real gas equation of state and create a ghastly version that is more general or, as is always done, use this version anyway and take the hit in accuracy. Assumption (b) is rather good because, unless we encounter extreme pressures, the heat capacities CP,m and CV,m are nearly equal for condensed phases.
Adjusting to a nonstandard temperature is rather important since many (most?) reactions do not actually occur at 25°C and the difference is often quite significant. Once the heat capacity CP,m is known as a function of temperature, we can calculate ∆∆rxnH°=∫∆rxnC°PdT.
To make integration life easier, the heat capacities are fit to simple polynomials of T:
CP,m=a + bT +cT2+dT3+eT4 [Shomate] or
CP,m=a + bT +cT-2
Exam 1 on Thursday. If you are nervous, just remember that if you got through organic chemistry, calculus and physics, you can do this. I have posted solutions to the two questions I assigned from Chapter 4. I also updated the study sheet that had some equations missing and an error in one of the thermodynamic equations of state.
Tuesday, October 16, 2007
Thermochemistry and Hess' Law
The first lecture of Week 5 found us tackling thermochemistry, that subset of thermodynamics describing heat transfer that accompanies chemical reactions. In constant-volume calorimeters (often closed vessels), the heat transfer q we measure is ∆rxnU, whereas in constant-pressure calorimeters (open vessels), q will be ∆rxnH. (On Wednesday, we will see a simple method to interconvert them). Since reactions are typically performed at constant temperature and pressure, any results give us important information about the energy stored in chemical bonds.
Using the properties of state functions, we can predict the heat transfer under these two conditions using Hess' Law and, since constant-pressure conditions are more common in chemical systems, we tend to focus on ∆rxnH rather than ∆rxnU. The norm is to cast all reactions as simple sums of formation reactions, each of which represents the formation of 1 mole of a substance from constituent elements in their standard states/phases. Hess' Law is particularly powerful in thermochemistry because it applies equally well for any extensive state property. Note that this is our second usage of the ∆ symbol (the first being the familiar ∆Y = Yfinal - Yinitial). Whenever the subscript appears on the ∆ itself, as in ∆combY, we are calculating the sum of the products minus the sum of the reactants, each multiplied by appropriate stoichiometric coefficients. Hopefully this operation is still familiar from general chemistry.
Next lecture, we will finish thermochemistry and begin to tackle entropy, one of the most important and poorly understood concepts in all of science. If your intrepid instructor has the backbone to trudge through the quiz 2 carnage, he may be able to return them on Wednesday. I will admit that the uncharacteristic dearth of questions/comments/emails/office visits so far this quarter (quickly approaching the 50%-done mark) had lulled me to mistakenly believe that this class was further along the thermodynamic path than it actually was. Hopefully quiz 2 will be a valuable learning experience for many as we lumber towards Thursday (and remember, no class on Friday).
Using the properties of state functions, we can predict the heat transfer under these two conditions using Hess' Law and, since constant-pressure conditions are more common in chemical systems, we tend to focus on ∆rxnH rather than ∆rxnU. The norm is to cast all reactions as simple sums of formation reactions, each of which represents the formation of 1 mole of a substance from constituent elements in their standard states/phases. Hess' Law is particularly powerful in thermochemistry because it applies equally well for any extensive state property. Note that this is our second usage of the ∆ symbol (the first being the familiar ∆Y = Yfinal - Yinitial). Whenever the subscript appears on the ∆ itself, as in ∆combY, we are calculating the sum of the products minus the sum of the reactants, each multiplied by appropriate stoichiometric coefficients. Hopefully this operation is still familiar from general chemistry.
Next lecture, we will finish thermochemistry and begin to tackle entropy, one of the most important and poorly understood concepts in all of science. If your intrepid instructor has the backbone to trudge through the quiz 2 carnage, he may be able to return them on Wednesday. I will admit that the uncharacteristic dearth of questions/comments/emails/office visits so far this quarter (quickly approaching the 50%-done mark) had lulled me to mistakenly believe that this class was further along the thermodynamic path than it actually was. Hopefully quiz 2 will be a valuable learning experience for many as we lumber towards Thursday (and remember, no class on Friday).
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