The Thing

The square root law [math] \sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}} [/math] only applies for real positive values of x and y. It should actually be [math] \sqrt{\frac{x}{y}}=\pm\frac{\sqrt{x}}{\sqrt{y}} [/math]

I don't get it. What exactly is it? My questions: 1. Why is it that the greater the vapor pressure the more volatile something is? 2. Why is it that the boiling point of something is when its vapor pressure equals that of the atmospheric pressure? Thanks!

Yes you can. A couple of things to keep in mind: 1. The central atom of the tetrahedral molecule is the geometric center of the tetrahedron. 2. Assume the bond lengths are the same. (Like in CH4) 3. The tetrahedron is a regular tetrahedron. (Again, like in CH4.) So each face is an equilateral triangle. Sorry. I couldn't draw a tetrahedron on paint (died horribly:D ). But visualize a tetrahedron ABCD, with its center at O. Then the "angles" you're trying to find is the angle between any 2 vertices and O (such as Angle AOB, or Angle BOC, etc...) All of these angles are the same. So let's find, say, angle AOB. Try the following: 1. Give the tetrahedron a side length, like 2 or 3 or 4, any number will do. 2. Now, the center of the tetrahedron will be directly above the center of any of the faces. So it will be directly above the center of triangle BCD (let's use this as the base). 3. So knowing that, find the distance between B and the center of BCD. Try it! 4. Now, let's name the center of BCD, err, P. Consider the triangle ABP. 5. You have just found the distance BP in step 3. You know AB, so use Pythagoras to find AP. 6. In ABP, remember that the bond lengths are the same. So AO=BO. Now, you have to find out the distance of AO (or BO). Try this. Hint: Set up an equation that uses Pythagoras! 7. After you have the distance of AO and BO, you can use the cosine law on Angle AOB. 8. And you're done. The result should be [math]cos^{-1}(-\frac{1}{3})[/math] Good luck and try it! Tell me if you need any steps clarified.

Hi I have a weird question that is from a biology lab but has to do with chemistry. We were testing the effect of different pH on bacteria, and placed tablets soaked in different concentration of HCl on an agar plate and couple of days later we measured the circle of inhibition around the tablet. My question is, how far can the HCl spread to? Like one circle of inhibition was about 2 cm in diameter (with 2.0M of HCl). Going away from the center, how does the concentration of the HCl drop? I vaguely remember reading somewhere that the concentration of HCl decreases logarithmically when moving away from the center of the circle. Is that right? Thank you!

Hi everyone. I want to do an experiment on the growth of a certain kind of bacteria in acidic and basic environments. I have 2 questions: 1. How do I obtain a pure culture of just one kind of bacteria, without any contamination (preferably a non-pathogenic bacteria)? Do I need a sterilized lab for this? 2. How do I go about setting up the acidic and basic environments? I have pre-prepared agar plates at school that I will use. Do I just dump acid or base all over it? Or do I use some other way (like using the little discs of filter paper)? Thanks and excuse my ignorance. I know NOTHING about this.

Indirect proofs are proofs by contradiction. So you list out the possibilities, which in this case there are 2: square of an odd number is odd, or it's even. Then, you assume that the square of an odd number is EVEN, bring about a contradiction, which means that this possibility is impossible, then the remaining one possibility, that the square of an odd number is odd, must be true. Judging by your "indirect proof", I'd say your direct proof looked like this probably: Let n=2k+1. Then n^2=4k^2+4k+1=2(2k^2+2k)+1. That's in the form of 2m+1 where m is an integer, which means the expression is odd. Turn it around. Assume that 4k^2+4k+1 is even. Then raise a contradiction.

Hi I have a couple of questions about cis-trans isomers. 1. When a molecule has one double bond and exhibits one instance of cis-trans isomerism, I put the cis or trans at the very beginning of the name right? 2. When a molecule has two or more double bonds and has 2 or more places of cis-trans isomerism, then where do I put the extra cis or trans? 3. When there are no identical groups around the double bond, do I write cis or trans depending on the orientation of the parent chain? Or do I not write cis or trans at all? Thank you so much.

Hey, that's quite nice! Put the series in pairs like this: [math]1+n-1+2+n-2+...+n-1+1+n[/math] [math]=n+n+n...+n[/math]. There are [math]n[/math] n's adding together. Which is the square that John drew above. So: [math]n^2[/math] is the series's sum.

Arithmetic Series. Get the value of [math]1+2+3+...+n[/math], add it to [math](n-1)+(n-2)+...+1[/math]: [math]\frac{n(n+1)}{2}+\frac{n(n-1)}{2}[/math] Simplify, gives the series as [math]n^2[/math].

Swansont just said I shouldn't fit something like a 23rd degree polynomial to it just because it fits well... Well, thanks. But can you give me an example of what method WOULD be appropriate in determining whether a relationship is linear?

Never mind. Thanks a lot though. By the way, the other question. Is comparing the coefficient of determination for different fits a valid method of deciding on whether the relationship is linear?

Really? Hooke's Law can be derived? Can you show me how? Also is comparing the coefficient of determination for different fits a valid method of deciding on whether the relationship is linear? Thanks.

Okay, let's make the scenario into deciding whether there is a linear relationship. However, after you plot your points there are usually some high-degree polynomial curves (with some extremely strange coefficients) that fit the points better than a line. Of course a line looks pretty close to the data, but then the higher-degree polynomials are even closer. Say we know that the relationship is indeed linear, and the higher-degree polynomials are caused by inevitable random errors in the sample. It looks linear, but a high-degree polynomial can fit more points than the line. In this case, how can we show that the relationship is indeed linear despite the fact that a high-degree polynomial curve fit the data points better?

Not sure if this belongs in the Physics or Math forum. I'm leaning towards Math more. Say, when scientists do experiments, record data, and come up wtih a formula, how do they prove that the formula is correct? I'm talking about those formulae that cannot be mathematically derived. One example might by Hooke's Law.There will inevitably be some error so how do they prove, besides just looking at the graph, that the formula should still stand in spite of the errors affecting the data? I know that they can't prove that the formulae is 100% true, but they can show, using statistical analysis that the formula is highly unlikely to be untrue. My question is, HOW? Say I've been experimenting on Hooke's Law F=-kx with a spring, a ruler, and some masses. There will be imprecision and random errors from the ruler. And if I collect the data, plot it, it will not assume the shape of a perfect line. So my question is: how do I know that I should fit a linear regression to it? How can I prove that it is very likely to be a linear relationship and not some other, like an exponential relationship? I look at the graph, yes, it looks linear, but how do I prove on paper or with a software mathematically that it is highly likely that this is a linear relationship? Thanks a lot.

Let y = a negative number with a huge absolute value.