Question

Hi everyone! Can someone look at this posted answer and tell me what they are doing where I marked in red? Thanks! :o)

Answer 1

Hey @Kainui, want to try this one?

Answer 2

Hi Kainnui!

Answer 3

Hey, I can't say unless I know what differential equation this is intended to solve. On its own it just seems completely random I agree haha.

Answer 4

it's y"=y' using power series

Answer 5

I did the work, and I got the answer where my arrow agrees...no idea what they are doing in the other places unless they are doing some weird technique to prepare to put the answer in a power series form

Answer 6

do you want me to post the entire solution from Chegg?

Answer 7

Nah this is fine I was just away for a second reading about insect brains lol

Answer 8

insect brains...ewww lol

Answer 9

I'm kinda leaning towards them just doing some weird technique to prepare to put the answer in a power series form

Answer 10

Since my teacher doesn't want power series form as the final answer, I think the middle solution is fine don't you?

Answer 11

Hmmm yeah I'm sort of lost in figuring this one out, I'll just go ahead and solve it on my own by power series and see if I can work out why they'd do something weird like this

Answer 12

ok...also, didn't you say your degree was in chem?

Answer 13

Yeah haha

Answer 14

In that case I have a strange question...

Answer 15

Sure go for it, also if I were to solve this I wouldn't even use power series, it just seems so strange to do it that way when you can make a substitution u=y'

Answer 16

I was reading some strange article about when a water molecule touches an certain mineral, the bonding actually splits the water and a hydrogen floats away...kind of like electrolysis but without electricity :O)
does that seem weird or surprising to you?

Answer 17

oh the question tells me to solve in both a conventional way and a power series that's why...it only takes 30 seconds to solve normally lol :o)

Answer 18

Yeah, that mineral is just a form of catalyst, something that doesn't get used up in a reaction.

Answer 19

so is that pretty normal to happen given the right pressure and temperature?

Answer 20

omg I just got my 101st fan! YAY

Answer 21

Like in cars, they have these special metal plated things at the exhaust that helps convert unburned hydrocarbons and carbon monoxide into water and CO2, really common! In fact it's not that different from electrolysis, it's just instead of there being a full current of electrons flowing there's just a few electrons being exchanged locally, and this is called a reduction/oxidation reaction, pretty much the bread and butter of chemistry haha.

Answer 22

redox! i remember that! lol

Answer 23

Part of what makes biology so interesting is that we have the ability to make proteins that are shaped and influenced in cool ways so that they can catalyze reactions, like burning hydrocarbons, at a much lower temperature. Quite a few proteins act as catalysts, I think they're specifically called enzymes, but proteins can be used to do other things like provide structure for the cell or whatever.

Yeah congrats on the fan haha, I guess I didn't really answer the math question though, I guess in some way that \(\ c_0-c_1 \) end up combining together to form a constant on its own though

Answer 24

you either fell asleep with your face on the keyboard are you are really typing away! :o)

Answer 25

actually a little of both lol

Answer 26

I'm gonna make some coffee haha brb.

Answer 27

I don't care about the math question anymore lol
I wanna pick your chemistry brain for a few minutes!

Answer 28

so in that reaction where they split the water and the hydrogen flew away...
the mineral was Mg2SiO4 + H2O
How can you mathematically prove that the splitting will actually take place?

Answer 29

is simply balancing the equation enough to actually prove what will happen or is math involved somehow?

Answer 30

Yeah awesome! Not enough people are interested in chemistry... and I admit that includes me some times haha, but I'm in the mood.

Well as far as that reaction goes, you really aren't looking at anything on such absolute terms. I'm sure you've heard of a mole and how it's such a large number, so often times the reality is that you actually have multiple reactions all happening at the same time that are all going in all different directions. But overall what you end up with is the most likely reaction, which is influenced by temperature, presence of a catalyst, among other things.

The main way you can find out if a reaction will take place is to just do the reaction, there are a LOT of things that you learn that give you chemical intuition however to predict reactions, and they're somewhat mathematical in nature but it's just sort of hard to explain.

When it comes to balancing a chemical equation, that's definitely enough to say that theoretically this can happen. It's the same thing as saying I can build two separate models out of the same set of legos, it doesn't necessarily mean that it will have a high yield. The two sides of it are really like in competition, and you end up having an equilibrium where the rate of turning one into the other matches. You can shift this dynamic equilibrium by adjusting temperature but you may also add a completely new possible thing that you didn't have before that can compete with your goal... It's definitely a juggling act!

I hope that's not too long winded haha I can go into more detail since I know that's kinda vague. XD

Answer 31

I forgot to add, currently there's a branch of chemistry called computational chemistry that basically deals with using computers to do quantum mechanics to predict reactions, chemical properties, and protein folding, that kind of thing. That's based around solving the Schrodinger equation, which is a differential equation coincidentally. Solving this in spherical coordinates for hydrogen gives you the orbital shapes you probably learned about in highschool chem.

Answer 32

no, I get it...just coming from a physics perspective, I don't like the idea of a lump of water touching an area of mineral...I actually would prefer to break it down to one molecule of water and one molecule of the mineral...know what I mean? moles are dirty little things anyway! lol

Answer 33

a LOT of modern chemistry relies on the Schrodinger equation, for a fun example, the stability of TNT.

Answer 34

I read that power series have to be used to actually solve the Schrodinger equation when doing quantum mechanics! Funny you should mention that! :o)

Answer 35

Hahaha yeah, well you can definitely treat it that way if you like, I don't know the mechanism though but I could probably guess at it and draw curved arrows and things like that. Since all molecules have some kinetic energy it's really just the temperature that would probably affect the rate of reaction, it would still happen on an individual molecule basis like you describe.

I think a common pitfall is that people imagine molecules needing "energy" to do something, but the energy is already in the system being constantly shifted around and you always have a population of molecules with high and low energy in the same container always!

Answer 36

Yes, there are a lot of interesting and nifty methods to solving the Schrodinger equation since it's not very easy to solve. Have you heard of the 3 body problem? That's basically what makes it so difficult since after Hydrogen you end up with all these electrons and protons in a molecule all interacting simultaneously.

Answer 37

okay well I have a really interesting thought then...
Let's say you have the Mg2SiO4 + H2O reaction which equals "X"
But what if you had HDO or D2O (semi-heavy water and heavy water) instead of just normal water?
Mg2SiO4 + DHO = ? or
Mg2SiO4 + D2O = ?

If deuterium has a different bonding strength, how could you calculate whether the hydrogen or the deuterium would pop off?

Answer 38

Well since Deuterium is twice the mass but the same charge it will have some effect, but this is usually pretty negligible because of how small Hydrogen and Deuterium are, but you can model the bond as a spring with a spring constant for instance. (Or perhaps we should think the other way around and say springs can be modeled as a bunch of atoms bonded together? =P ) So hooke's law says F=-kx and so to measure the difference we can do Infrared Spectroscopy.

What that amounts to is taking H2O and D2O gas and shooting IR light through it and seeing if it was absorbed with a sensor on the other side. IR is of a certain energy (frequency) that will be absorbed only by the stretching motions of the H-O and D-O bonds, and not much else so you can look at a graph of peaks to determine if the energy absorbed is of a resonant energy with the light. This probably sounds super weird, and it's sort of hard to explain well haha, but this is the condensed version. So we'd be able to see quite clearly what the bond strengths are that need to be overcome to break/make bonds.

There's a bunch of other methods, and spectroscopy is quite a broad area that lets us use radio waves and magnets in a process called Nuclear Magnetic Resonance to determine molecular structure among other things (also found in MRI machines) to how microwaves wiggle water to heat it up at the right resonant frequency to warm your food, to doing x-ray diffraction and other neat stuff. Uhhh did I answer the question I kinda went on a tangent haha.

Answer 39

The main thing to consider is if you have some reaction like A --> B then you need to consider the entropy and free energy of the two states to see if the reaction is favorable or not.

I think a problem I see in kind of explaining this is that there's a difference between thermodynamics and kinetics. Thermodynamics can only tell you if a process is favorable or not to occur. Kinetics tell you how fast that it happens. For example, diamonds turning into graphite is actually a favorable reaction, it's just that it happens very slowly!

Also, don't forget that just because you put the mineral and water together at room temperature, that doesn't mean that this is some kind of default where nothing should happen. Room temperature is about 300 Kelvin above absolute zero, so there's a lot of energy in the system to push the reaction forward if it is favorable, which it is.

Answer 40

Yeah, you answered it the best you could.
Is there a quick way to just estimate what would happen between H2O and HDO? Not D2O...I am just curious if the OH- would stick to the mineral or the OD- would stick...
Is there a table that says what energy is needed to split a hydrogen bond from water or a single deuterium bond from semi-heavy water (HDO) ?

Answer 41

One has to be stronger than the other you would think

Answer 42

Yeah definitely! Doing a substitution of a single H with D is a very common thing in chemistry, I know there's some math behind it, but I don't know it off hand but it's really not too complicated maybe I could find it in a few minutes in my book. There is a table that we can look up the bond strengths on the internet though. The thing is, you have inductive effect from other atoms on the molecule that affect the bond strengths.

When it comes down to it, mathematically speaking a bond is an area of higher electron probability density, which comes from solving the Schrodinger equation. So no table is really accurate when it comes to describing O-H to O-D bonds, they're more like useful approximations in general that are most likely determined from experimental evidence and only seen to match the mathematics.

There are all sorts of ways to get better approximations and numbers but it really depends on what or why you're trying to determine it. I feel like I'm leaving out a lot of information here, but I don't really know a quick way to explaining any of it.

Answer 43

No...that's interesting...it has always bothered me that you can balance an equation and say "this happens" but never was there math to prove it in my classes...
but now realizing that the quantum probabilities of electron densities is what dictates the reactions in the first place, the mathematics to prove the probability of such reactions is wrapped up in the quantum mechanical Schrodinger equation!
neat! :o)

Answer 44

Yeah, I guess a fun thing I should mention is you can derive the periodic table from solving the Schrodinger equation.

Answer 45

no way!

Answer 46

.
[PS awesome thread, thanks folks]

Answer 47

Are you typing? If not, I have another question regarding the SE and the periodic table predictions

Answer 48

So to explain some quantum mechanics, you have orbitals which are just where electrons are likely to be found. Orbitals are actually defined throughout all space, from the center of the atom all the way out to the milky way, it's just that the pictures you see of them describe where you're most likely to find the electron about 95% of the time, it can easily be inside or outside that.

(realize by the way that there are some necessary assumptions and simplifications here, never take anyone's statements about QM without many grains of salt) So for a given energy you basically have these orbitals, each can "hold" two electrons of opposite "spin". As you move up, you gain more sets of orbitals and these orbitals come in increasing pattern of odd numbers (this all comes from the Schrodinger equation!) which is 1,3, 5, 7,...

So think of each row on the periodic table as representing increasing energy as you go down, with Hydrogen the smallest. We can have an atom now with one electron or two electron before our single orbital is filled. That's the s-orbital, which corresponds to our 1 orbital holding 2 electrons.

Let's increase our energy to the next row down, we gained a new set of orbitals to fill, so in addition to a new, higher energy s-orbital we also gained 3 new orbitals, called p-orbitals. So since each of these can hold 2 electrons, we now have 2*1+2*3=8 electrons before we fill them up and have to click down to the next row. Wait, 8, that's where the octet rule is coming from! =P

Leaving out many important details, a couple rows down we add 5 more orbitals, we call these d-orbitals and so we end up with 10 possible electrons to pile into those along with our s and p orbitals.

There's a lot more to say, since I've barely skimmed the surface of "atomic orbitals" because by doing something called "linear combination of atomic orbitals" which is literally because solutions to the schrodinger equation can be added together linearly to create new solutions, called superposition, we can create "molecular orbitals" which are really important to chemists in simplifying and making assumptions about bonding and when there's a single, double, triple, or higher order bonds as well as non-integer order bonds and all sorts of weird stuff like resonance and aromaticity, which is essentially what makes benzene and other molecules with hexagonal shapes so stable as long as they have a special pattern of atoms (4n+2) where n is any natural number. This is why graphene is so stable and conducts electricity despite being made out of carbon, a traditionally poor conductor haha.

Answer 49

Yeah ask away you don't have to wait for me to finish typing haha I tend to sorta get into it and I'm kind of in a haze right now. More coffee BRB!

Answer 50

Well I was curious to know that since you can derive the entire periodic table by using the SE, has anyone ever calculated and predicted an "island" of stable super-heavy elements past the 110's?

Answer 51

Yeah, that's exactly what this is: http://en.wikipedia.org/wiki/Island_of_stability

Answer 52

wow...so if I'm reading that correctly, once we stuff together 150 proton and neutron give or take, we might have elements we could use for who knows what!? amazing!

Answer 53

Another interesting application that I learned about this summer was that of Neutron stars. Apparently these are stars that got so large that the electric force keeping the molecules apart failed and gravity forced it together creating what is essentially a single atom with an atomic number that's billions and billions of atoms large. And using the schrodinger equation they have successfully predicted some things modelling it as an atom! Just sort of mind boggling hahaha.

Answer 54

an atom the size of a neutron star?

Answer 55

The neutron star IS an atom.

Answer 56

an atom with no electrons or protons?

Answer 57

just degenerate neutrons?

Answer 58

Well ok I guess I exaggerated a bit, I guess I should just say we can successfully model a neutron star as an atom, but I don't know that much about it. I just thought it sounded interesting but I really don't know much about what goes on with cosmology type stuff. http://en.wikipedia.org/wiki/Neutron_star#Giant_nucleus

Answer 59

okay...that was interesting though...you can't get much by me lol :o)

Answer 60

well if it is a single atom, the quantum physics should be able to predict the strength of its magnetic field so if those predictions then matched observed magnetic strengths, then they might have something there!

Answer 61

okay well I have to get back to math! thanks for all the info, i saved it in a word doc, I might revisit it with you later...should we delete all this stuff that didn't pertain to my original question?

Answer 62

Yeah, I honestly don't know. There's sort of a difference between what a chemistry major and physics major study about the schrodinger equation. Chemists are much more concerned with electrons, and everything else is sort of just extra haha.

Answer 63

I don't see why it should be deleted, it doesn't really matter, it's not harming anything, the point of the site is to learn right? I mean I'm a mod too so... hahaha

Answer 64

okay! :o) thanks Kainui! here's a medal for good measure!

Answer 65

Yeah thanks for the interest, I think I'll try to direct myself back into tensors so I can understand more of Dirac's Relativistic Quantum mechanics! XD

Answer 66

sounds interesting...wish I had time...I want to derive Einsteins field equations sometime to truly understand how he did it...now I laso want to derive the SE as well! ugh!

Answer 67

okay I will close this out now! Thanks again!

Answer 68

Same here! Those both sound like fun, and are goals of mine too. X_X