Tuesday, September 16, 2008

Carboshift!

I remember reading in this article, where there was a study made on carbocations in solution; apparently they managed to stabilise the formation of carbocations in superacidic solution, and therefore observed the characteristics of carbocations, in particular their fluxional character. And they found that when they prepared the carbocation that had the structure:




After NMR analysis, they found that they had two other carbocations:



Can you fashion a guess? Haha.


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And if you still haven't figured it out yet, it's due to carbocationic rearrangements as such:


Quite cool huh? Haha.

Sunday, September 14, 2008

Circular Argument I

Here's something that caught my attention when I was tutoring one of my students in 'E' Math at the Secondary School level: the angle subtended by a chord of a circle at the centre of the circle is twice that subtended by the same chord at the circumference.

Perhaps a diagram might make things much easier:


The chord is the line AB, and the angle subtended at the centre is the angle AOB, and the angle subtended at the circumference is the angle ACB, as indicated in Greek above, haha. So this property of a circle says that angle AOB is twice that of angle ACB. But why?

Easy enough, I shall use the method that Shaun used during our lecture, which is a rather neat and easy proof; so kudos to Shaun! First of all, you divide the triangle ABC into half, down the centre as shown below with a line CD, and then notice I've coloured an isoceles triangle in green:

And then you should notice that this line should bisect the angles AOB and ACB, and we can conclude first that:


And from this, the following statements should therefore make sense:


Quite a neat proof huh? Shaun came up with it all by himself, pro!

Thursday, September 11, 2008

Speeding Towards Light

So, someone tagged that she wants to know why light always travels at the speed of light huh? Haha. Well here’s some information to get you started:



This is a typical wavefunction, which is a function of both position (x) and time (t). If you’re a good JC student in Singapore taking Mathematics at the H2 level, then you should be able to figure out that this function travels in the direction of positive x (if you can’t tell you can come ask me, heh) and it also travels in this direction with increasing time.

Now, we see that there’s an amplitude (A), and there’s an angular frequency (ω), and there’s a wavenumber (k). These quantities should, hopefully, be familiar to you, or at least I hope you’ve heard of them! Haha. These are all A Level knowledge, so hopefully you have!

I’d like to say that although what I’m about to say applies to all waves, not all waves have such a nice looking function like the one above! I just chose the easiest one of all so that it’s easy to digest what I’m going to say.

Going on, I’d like you to see that:


What I’ve done here is to take the second partial derivative, meaning I differentiate the wavefunction with respect to x only, twice, each time keeping t constant.

Aright, now, let’s try to take the partial derivative again, now with respect to t:


Cool! To sum it all up so that you follow me better:


It should be obvious to you then, that:


Does anyone want to fashion a guess what this means? Heh. Well, for waves we know that (from basic JC level Physics):


And therefore, we can say that:


But hey! We learnt in Secondary School that the speed v of any wave is related to the frequency f and wavelength λ by:


And therefore:


And thus our previous equation shows that:


That is, if we differentiate the wavefunction with respect to position twice, we obtain the derivative of the wavefunction with respect to time divided by the speed of the wave squared! What an interesting inherent symmetry! And this holds for all waves!

I’d really like to go on, but well, I don’t think most of you are acquainted with electromagnetic waves, are you? For electromagnetic waves, the equation turns out such that:


If you don’t know, ε is the permittivity of vacuum, and μ is the permeability of vacuum. Interesting enough, we note that:


Since all quantities on the left are constants, then we conclude that the speed of light (or rather, electromagnetic waves) must also be a constant regardless of anything, and therefore we now have:


Well well, I hope this is good enough! Because really, I can’t explain why light has a constant speed unless one deals with more advanced electromagnetic theory and wave mechanics. But haha, this is good enough I think, for all of you to chew upon! :p

Thursday, September 4, 2008

Double Angle Madness III

And I might as well show the last identity, right? Haha, and so here it is:


And that's all for now. :)

Double Angle Madness II

And continuing with the streak, I might as well show how to prove a very commonly used identity; for convenience, I've reused the diagram from the previous post:



Trigonometry isn't that hard right? :)

Double Angle Madness

Well, I gave one of my tuition students this identity to prove:

sin 2x = 2 sinx cosx

And is it really that hard? Well, actually all it needs is a single diagram and two lines of working, as I have illustrated below:


Cheers! :)