So the question becomes. For a given high and low temperatures (Thigh and Tlow), what is the optimal choice of ΔH and ΔS, such that the system is most stable between those to temperatures?
Stability is not something you've decleared very well. I'm going to interpret the notion of "stability" as implying that the reaction would absorb (or release) as much heat as possible to counter a temperature change in the room.
Let's just spend a moment to see that this is a sensible approach:
Diagram 2: How the room loses / gains thermal energy Thermal Energy change
in the Room, cΔT ----> heat lost to the outside world, ΔE (due to... poor insulation etc.)
/\
|
|
Heat released by chemical equillibrium system, ΔU
Under the situation shown in the above diagram we have
Change in thermal energy of the room = c.ΔT = ΔU - ΔE
[eqn 10]
with c = heat capacity of the room; ΔE = energy transferred out of the room; ΔU = heat released by the chemical system into the room.
Note the sign conventions we have established: Room cooling: ΔT<0 ; ΔU >0 the chemical system will release heat (thermal energy) into the room ; ΔE > 0 heat is lost to the outside world. Let's not spend too long discussing this, it should be apparent that ΔE can be considered as the sum of natural heat exchange to the world outside the room + energy exchanged due to heating or cooling systems like radiators or air-conditioning units that a person has inside the room. A negative value indicates a net heat input to the room (e.g. from the house central heating system).
Note also that the chemical system is considered as part of the contents of the room. In the most degenerate situation the room is only the chemical system but usually the room is the chemical system + the rest of the room (like walls and furniture) such that the chemical system is just a minor contribution to the mass and heat capacity for the room. (LATE EDITING: Stuff removed - the above is just conservation of energy ideas and doesn't need further comment).
The following results follow from [Eqn 10]
(i) ΔU ≠ ΔE iff the room changes temperature.
(ii) .. result removed in late editing... serious error here, sorry. The following result still holds but for different reasons...
Therefore, |ΔU| ≤ |ΔE| always and equality holds if and only if ΔT =0.
[Result 11]
We can now see that no chemcal equillibrium system A↔Z exists which would perfectly maintain a fixed temperature in the room. Assume some energy ΔE≠0 is removed from (or added to) the room. For ΔT = 0, we must have ΔU = ΔE but the chemical equillibrium would not change at all if the temperature didn't change, so we would have ΔU = 0. Therefore, when some energy enters or leaves the room we are always in the situation with |ΔU| < |ΔE| and the best we can hope for is that the chemical equillibrium opposes ΔE as best it can (so that the temperature change will be less severe than if the chemical equillibrium system wasn't there).
I would suggest that the best interpretation of "maximum stability" for our temperature buffering system will be trying to keep |ΔT| as low as possible for a given amount of ΔE (energy transferred to/from something outside the room). For our purposes, the quantity ΔE is beyond our control or influence, that requires someone to insulate the room and turning on/off the central heating system of the house etc.
From [Eqn 10] ; [Results 11] and recognising that under our sign conventions ΔU opposes the sign of ΔT we have,
|ΔT| =
[Equation 12]
So, that
minimising the size of the temperature change for the room due to a given Energy transfer into / out of the room, requires the size of |ΔU| to be MAXIMISED.
CLAIM that ΔU = heat released from the chemical system = function of the current Temperature T, the parameters ΔH° and ΔS° of the reaction A→B and, of course, of the temperature change ΔT that occurrs. PROOF: We have the following,
ΔU = ΔN
A . ΔH° / (Avogadro)
[Equation 13]
Where, ΔN
A = the change in N
A (change in the number of molecules of A); ΔH° is the specific enthalpy of the reaction change (heat change for 1 mol of A being converted to Z); (Avogadro) = Avogadros number. The signs work (I hope), recall ΔN
A is negative, N
A decreases, when molecules change from A to Z and ΔH° = negative when A→Z is exothermic, making ΔU = heat released to the room = positive.
This is taking too long.... let's just speed this up.... we can obtain an expression for ΔN
A from [Equation 4] in terms of the change in the equillibrium constant K. We have a thermodynamic expression for K from [equation 5]. That establishes the "CLAIM" that ΔU is a function of the things mentioned above. The dependance on ΔT (the change in temperature) came from ΔN
A but that will be clearer in a moment...
During the actual operation of the chemical equillibrium device, you cannot change the parameters ΔH° or ΔS° of the reaction or add extra molecules of A, or do very much to change the way the device can abosrb or release heat. The ony changes that occurr are temperature changes in the room. So in calculating the quantity ΔN
A during the actual operation of the device, we can treat all other parameters as fixed constants. This saves a mess writing derivatives as partial derivatives, for example K = K(T) = a function only of Temperature during actual operation, even though K has dependance on ΔH and ΔS looking at [eqn 5]. This should also help to explain why finding partial derivatives w.r.t. variables like ΔH° is not something we will do. Just to be clear on this: It might seem that since we are trying to maximise ΔU (amount of heat released by the chemical system) with a carefull selection of parameters like ΔH° for the reaction, then we would want to find derivatives w.r.t. ΔH° and treat the problem as a typical maximisation problem. However, that is not what we will do. Instead, we treat the reaction parameters as a fixed constant (don't specify the actual value - just recognise that it is a constant) and see how the system will naturally behave in operation. After that, we are free to see how those results change or what they will mean when the reaction parameters like ΔH° and ΔS° are given a particular numerical value. This method seems to work better than just directly trying to find maxima from derivatives w.r.t. reaction parameters. (For example, there doesn't seem to be any local maximum. This might be discussed later - the larger ΔH° can be the better, the progression is endless and no local maxima exists BUT also you just don't get much useful information about the system anyway).
Let's see [Eqn 13] again:
δU = δN
A . ΔH° / (Avogadro)
[Equation 13' ]
We'll drop the Avogadros number (as if ΔH° was stated per molecule not per mole), it only scales the value of ΔU, the maximum won't be affected without it anyway. Note that we are now considering only small changes, as marked with δ symbols - the quantity ΔH° remains with a big delta because it is just a reaction parameter, a constant and not something that changes during the operation of the device. We'll also just write ΔH instead of ΔH° because it's shorter and should be clear enough that this is a fixed constant, a reaction parameter not something that changes dynamically during operation.
From [Eqn 4] we can differentiate, noting that temperature is the only variable that changes during operation of the device:
δN
A =
K and the derivative dK/dT can be found from [Eqn 5] and [Eqn 7]
=
[Check the signs work here: If ΔH >0, then A→Z is endothermic. K = [Z]/[A] and we expect K to increase when T increases. dK/dT >0 is what we have. Next up δN
A has a negative sign out at the front, so it's negative and yes we have A→Z so the number of molecules of A would decrease. This all looks OK to me. ]
Substitute this into [Eqn 13' ] and we have:
δU =
[Eqn 14]
where ζ = e
ΔS/R = a convenient constant to hold all the entropy information. Note that a lot of the quantities appearing in {Eqn 14] are constants, T is the only variable.
Recall that we wish to maximise δU given a fixed transfer of energy δE from the room. Straight away from [Eqn 14] we can see that if N (the total number of molecules of A and Z) is increased then δU is increased. There is also a term on the numerator ~ (ΔH)
2 and although ΔH does appear elsewhere its effect is most noticeable here. *LATE EDITING: Reasonably true for small to medium ΔH. However, when ΔH becomes very large, then the exponential function e
-ΔH will dominate and bring the numerator towards 0. This effect is more important than I first thought because ΔH usually is large - hundreds of thousands in standard units of Joules/mol. *
We can tidy up our expression a little. We are trying to maximise δU for a fixed value δE of energy transferred out of (or into) the room. However, we can imagine the maximisation requirement differently. Assume that a small change in temperature (δT) of the room has happened. We would like to have this cause a release (or absorption) of energy by the chemical system (δU) that is as large as possible because that would have forced δE to be as large as possible. [Eqn 12] can be used to formalise this condition and establish that it is equivalent. The post is too long, I'm skipping that. We will just re-state the maximisation problem as required: To buffer the temperature of the room as much as possible, we wish to maximise δU for a given fixed change in temperature δT. Anyway, with that modification we can divide by δT and obtain the following result (where δT→0 to form the derivative on the R.H.S.)
[Eqn 15]
Our objective is to maximise this quantity.
..... I stopped writing here.... there are some derivatives and some graphical plots to show what is happening that i would have inlcuded.... but by now no-one is reading this and the whole set of expressions have become sufficiently messy, no-one wants to see them anymore. They belong on a computer system like SymbolLab or Mathematica... or whatever your preferred mathematical manipulation software might be. They do not belong on a text document for people to read or differentiate by hand.
Minor Notes: We were really maximising |δU| , the size of a change. As it happens δU will oppose δT in sign, which means that we are often actually trying to minimise δU because its negative and a lower negative has a greater size. I started modifying everything to |δU| in the early part of the post... but time limits cut in... I never carried it through to the end of the post.
Here are some plots of what we get from [Eqn 15].
Parameters N=1 (arbitrary scalar anyway). ΔH = 10 000 J/mol ΔS = 33.33 J/mol/K (deliberately chosen so that ΔH/ΔS ≈ 300 kelvin ≈ approx. room temperature and ΔG would become 0 at that temp).
Plot-300.JPG (66.42 kB . 1363x418 - viewed 2329 times) Note that although I would have expected the minima (maxima of |dU/dT| ) to be around T = 300 k, it was actually a bit below that.
Paremeters as before, ΔH raised to 20 000 J/mol. ΔS = 33.33. Expected T = 600 kelvin but the maxima is still slightly off.
plot-600.JPG (82.3 kB . 1401x535 - viewed 2340 times)Anyway, the general behaviour of the chemical system seems to be reasonable..... it always has maximum temperature buffering ability at a particular value of temperature. Changing the values of ΔH and ΔS move that maxima up or down. You can probably also see why you can't get clear results by taking something like [Eqn 15] and directly differentiating w.r.t. reaction parameters like ΔH..... You can have a system that is actually tuned to buffer around a temperature of 100 kelvin * instead of 300 kelvin BUT if the value of N (the number of molecules in the system) is large enough and also ΔH is large eneough then it would still be a numerically better buffer at room temperature than a smaller and more sensibly ΔH valued system. * LATE EDITING in the previous sentence: It's more noticeable where the system is tuned to buffer below room temperature not above. See the actual plots with given parameters. It is, in my opinion, only by analysing the situation with ΔH and ΔS as fixed parameters and recognising that T is the only thing that can chnage in operation of the device where the natural behaviour of the device is observed. For a realistic situation, where you can't have an infintely sized system or infinitely high values of ΔH, you would want to work with the natural behaviour of the system and set it up with ΔH and ΔS parameters that put it's maximum buffering ability right at the room temperature you wish to try and maintain.