[Racket]

Hey, I'm taking an introduction to organic chemistry course.  I haven't had this stuff for a while so I'm a little rusty.  Anyway we just had a lab and now have some questions to answer, and I was hoping to get some help with this one I've been stuck on.  So, here it goes:

"A graduated cylinder, measured with a metric ruler, is found to be 12.7 cm high.  The cylinder is filled with mercury to its full height of 12.7 cm; the mass of mercury required to completely fill the cylinder is 105.5 grams.  The density of mercury is 13.6 g/ml.  Regarding the mercury in the graduated cylinder as a "metal cylinder", determine the diameter of the "cylinder" of mercury contained within the graduated cylinder."

Now I found the question worded oddly, but from what I get of it I start by getting the volume of the mercury.

Density of mercury = 13.6 g/ml
Mass of mercury = 105.5 g
V = M/D
So,
V = 105.5/13.6

From there I get 7.76 ml and then work backwards using the volume formula V = pi * r^2 * height (12.7 cm)


But I feel like I'm doing this wrong and highly overthinking this.  Am I on the right track or what?

[Dan]

Method looks sensible to me

[Racket]

Alright, awesome.  So does .14 cm for a diameter sound good?  Because it seems a little small to me, maybe I was messing up with my math after all.

[fledarmus]

Looks like a math error to me - can you set up the rest of the problem so we can see your calculation?

[Racket]

V = 105.5/13.6 = 7.76

V = pi * (d/2)^2 * h

7.76 = pi * (d/2)^2 * 12.7 - square both sides

2.79 = pi * d/2 * 12.7 - divide both sides by 3.14

.889 = d/2 * 12.7 - divide both sides by 12.7

.07 = d/2 - multiply both sides by 2

.14 = Diameter

Right now I see I didn't actually square the whole right side but I still feel as if that number would come out too small even with that fixed.

[Borek]

0.14 is wrong, but you already know it. I agree the correct final result doesn't look too convincing, but it doesn't look completely nonsensical either. And in general you are on the right track.

[DrCMS]

V = 105.5/13.6 = 7.76

V = pi * (d/2)^2 * h

So far so good but the next step is completely wrong.

7.76 = pi * (d/2)^2 * 12.7 - square both sides

2.79 = pi * d/2 * 12.7

Right now I see I didn't actually square the whole right side but I still feel as if that number would come out too small even with that fixed.

Well doing the maths wrong is never going to give the right answer so think again.  Try to divide by π and h before you think about the square part.

[Racket]

Thanks for all the help so far everyone.  I'm trying to understand what I'm doing here, I tend to second guess myself when doing equations like this a lot.

I thought exponents were always done first?

Doing it out I get 7.76/pi = 2.47

2.47/12.7 = .195

square root of .195 = .441

.441 * 2 = .882

So I must still be doing something wrong.

[Dan]

I thought exponents were always done first?

It doesn't matter what order you do it in, you just have to do it correctly. If you take the square root of both sides you have to take the square root of everything on both sides:

[tex]V = \pi r^2 h[/tex]

root both sides:

[tex]\sqrt V = \sqrt {\pi r^2 h} = \sqrt\pi\sqrt r^2 \sqrt h = r \sqrt \pi \sqrt h[/tex]

[Borek]

.441 * 2 = .882

So I must still be doing something wrong.

If memory serves me well, that's what I got yesterday.

[DrCMS]

Doing it out I get 7.76/pi = 2.47

2.47/12.7 = .195

square root of .195 = .441

.441 * 2 = .882

That looks correct to me, and now Borek.

[fledarmus]

Number looks good to me too, and that's about the size of my thinnest 10 mL graduated cylinder (reads to 0.02 mL). Not a bad piece of equipment for measuring out 7.76 mL.