Thank you.
Yes, my understanding is that total delta G is always non-positive at the particular temperature and pressure at which it occurs. That is a consequence of the 2nd Law of Thermodynamics. That is provided that you include all the relevant terms in the free energy, so that all sources...
Ah. Fuzzy. I can see that.
Okay, maybe I can make the question more precise. Outside of biology, are there reactions in which a positive ΔG process is driven by a negative ΔG process? For example, suppose that the reaction that produces some compound AB out of its constitutents A and B is...
Can you elaborate? How is it fussy? Maybe I can improve it.
Excellent! Thank you for those examples. I will do a google search on those. So these reactions where a -ΔG reaction drives a +ΔG reaction?
Okay, that does clarify your point for me somewhat. Thank you. Maybe I can clarify my question by putting it this way: are biological reactions, as in metabolism, entirely a thing to themselves--a completely self-contained area of study? Is there nothing in geology that shows a glimmer of a...
Thank you for the link.
Perhaps I should have specified catalyzed coupled reaction. But I get the point that there is more than one way to draw the box that includes the "system". I guess there may be, in principle, all sorts of coupled reactions in geology if you completely ignore reaction...
I'm just wondering if all known coupled reactions are biological, or if there are some that geological and could have preceded the emergence of life on earth. In other words, are there some compounds in geochemistry that are much more abundant than you would expect if the reactions that produce...
Nice article! Adrian Bejan talks about this kind of thing in his book Design in Nature. He even extends the reasoning to the evolution of technology, like cars and planes. His idea is that the world is organized by flow. That structures that increase the flow of matter are selected...
Both the scalar triple product and the determinant (in 3 dimensions) have the geometrical interpretation of volume. ##(\vec u \times \vec v) \cdot \vec w## is the volume of the prism formed by vectors u, v, and w. They are probably expecting you to make the identification
$$ \det A \quad =...
Okay, thanks.
You are right, I did not cite that as a rule of inference. In Hilbert and Ackermann's logic, it is a derived rule of inference. Their basic rule of inference is uniform replacement, where you have to replace all occurrences of one thing with another thing. So they show that if...
Nice. I like how you were able to prove it all in one sequence with equations. I don't know what all the abbreviations, like "Leib", stand for, but I follow what you did.
I broke it up into two steps, as you originally suggested. I used the weaker "one-way" version of that theorem you...
Cool. How did you do it?
Yeah, that's a good idea. Depending on how much you like logic, you might try to prove the axioms of one system as theorems in another system, and vice versa. That can help you develop a feel for what is going on.
Okay, if there are no occurrences of x in B, then some of the manipulations RUber did ought to do the trick. A couple more hints are...
To get 1), use the contrapositive and the fact that ∀x(F→G(x)) → (F→∀xG(x)) when x does not occur free in F.
To get 2), you'll need to know that...
You are doing fine right up to that point. The something goes wrong. Why not collect terms with a right there and then solve for a? Everything else in that last equation is a known quantity.
I do not believe using matrices would make this calculation any shorter. Notice that most of the...
Crap. I messed that up. That was supposed to say...
Obviously the integral is typically not zero for non-cyclic paths, even when they are reversible. In that case it is just the change in entropy.
While I'm at it, my claim that Carnot cycles are the logical basis of thermodynamics doesn't...