TBR Gen Chem Ch. 6 Gases - Example problem 6.9 - Math problem

[inasensegone]

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Here is the problem (Concerning Avogadro's law):

When 0.15 moles of helium gas are added to a piston containing 0.82 moles of another gas, by what percent does the total volume increase?

A. 6.4%
B. 14.4%
C. 18.3%
D. 25.0%

My issue isn't with figuring out how to do this problem, it is with the explanation provided in the book. My method was just to compare the relative change in number of moles, and knowing that Volume was directly proportional, determine the percent increase. (In other words, 0.82/0.15 ~ .18) I of course approximated this value from the choices given and from the fact that 0.15 * 5 = .75, therefore 0.15 must be multiplied by a little over 5 and the actual value would be a little less than 20% (because the 5 would indicate an increase of 20%, a slightly larger factor would indicate slightly less than this 20%).

Anyway, the method in the book says this:

V1/n1 = V2/n2 *therefore* V2/V1 = n2/n1 ---> V2/V1 = 0.97/0.82,
So, n2/n1 = 0.97/0.82

[so far so good, but then:]

0.97/0.82 = (0.97 + 0.194)/(0.82 + 0.164) = 1.164/0.984 ~ 1.18

(The answer, again, is ~18% so they achieved the correct answer)

Where the hell did the +0.194/+0.164 come from?

I noticed this guy had the same issue, but didn't get any answers, so I decided to make a new thread with the whole problem. I'm sure it's a simple explanation but I couldn't figure it out and it was bothering me. Anyone know what's going on here?

Thanks in advance!

 

[Odi]

OP this thread killed so much of my time trying to figure this out

I'm 99.99% sure that I'm 80% they are using a math trick where they can get the denominator closest to 1 making the calculation very easy--and they are doing this by increasing the numerator and the denominator by 20% of their respective value.

0.194 is 20% of 0.97 and 0.164 is 20% of 0.82.

This way the ratio doesn't change because they are increasing by the same factor of their initial value. 20% was probably chosen because it gets the denominator very close to one. I have no idea how this would make it any easier.

Don't stress explanations as long as you understand how to find the answers and can do so for all similar questions.

 

[inasensegone]

Hahaha... I know the feeling Odi... Trying to figure their math out killed like 30 minutes of my time. That much time is precious in MCAT study minutes. Anyway, thanks for the help! I agree that it may have been an attempt to make the denominator 1, and it does appear to be a 20% factor, but I still don't see how it can be applied quickly to facilitate the math. Oh well... most of TBR isn't this frustrating anyway. Just time consuming. Thanks again!

 

[patrickmbyrne]

V2/V1 = 0.97/0.82 is easy enough to do in your head. .97/.82=97/82 1 + 15/82=1.1+68/82 -> =1.18

 

[inasensegone]

V2/V1 = 0.97/0.82 is easy enough to do in your head. .97/.82=97/82 1 + 15/82=1.1+68/82 -> =1.18

Thanks for the response Patrick, but my question was concerning TBR's methodology. I know how to do the problem, I just wanted to figure out their math approach.

By the way, I don't think your math is correct, because 1.1 + 68/82 does not equal 1.18

 

[patrickmbyrne]

68/82 *10^-1.

As for the math part the second guy seems right, but it still takes longer for me:
They are multiplying and dividing by 1+20% because 1+20% is the complement of 80% so this will bring the denominator to unity.
0.97/0.82 = (0.97 + 0.194)/(0.82 + 0.164)~(.97+.194)/(1) ~1+.194~1.19= 1.164/0.984 ~ 1.18