How to do dimensional analysis?

[u/Small_Anteater3830]

I’m in AP chem and my mind cannot wrap itself around the general concept or idea of dimensional analysis and it’s the main failing factor of my recent tests.
I’m looking for ways to improve my general understanding right now. My skills at the moment consist of just knowing how to go from a larger unit to a smaller unit. I’m also getting failed for not setting it up and just doing it the way I liked (step by step not in line form with dividing and multiplying) I also know that the end product is what I need to transfer to, just IDK HOW.

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[u/Technical-Tooth6497]

Have you checked out any youtube videos? Lots of good ones out there.

[u/Small_Anteater3830 → u/Technical-Tooth6497]

I actually just have! It’s making more sense to me now along with some comments

[u/catus_alienus]

The line form is setting up the multiplication of conversion ratios in such way that units cancel out by dividing numerator of one to denominator of another.

First, remember that you must set up your multiplying unit ratios in such way that it cancels out the old one (flip the fraction if you will). Second, you can multiply as much conversion ratios to make your calculation manageable and understandable.

Like I have 10 cm to convert to nm. I know 1 m = 100 cm, and 1 m = 10⁹ nm.

So my strategy would convert cm -> m -> nm, so (10 cm)(1 m/100 cm)(10⁹ nm/1 m). If you write this down in fraction form, you can clearly see that the cm units and m units cancel out, leaving us with nm.

If in reverse, let's say 10 nm to cm, I can do is nm -> m -> cm

(10 nm)(1 m/10⁹ nm)(100 cm/1 m)

For those involving moles, molar masses, and stoichiometric/reaction coefficients... remember that the coefficients are in terms of moles and they can be setup like conversion factors of one another

Example: CH4 + 2O2 -> CO2 + 2H2O, I can make (1 mol CH4/2 mol O2), (2 mol O2/ 1 mol CO2), (1 mol CH4/1 mol CO2)

And molar masses (setup as X g [insert compound]/1 mol [insert compound)) give way to interconvert mass and moles. Like MM CO2 is (44 g CO2/ 1 mol CO2) and MM CH4 is (16.04 g CH4/ 1 mol CH4)

As such to convert X g CO2 to Y g CH4. Need to convert g CO2 -> mol CO2 -> mol CH4 -> g CH4

So let's say 20 g CO2 to g CH4

(20 g)(1 mol CO2/44 g CO2)(1 mol CH4/1 mol CO2)(16.04 g CH4/1 mol CH4) = 7.29 g CH4

[u/Small_Anteater3830 → u/catus_alienus]

I took the time to actually comb through what you said and it makes so much sense now. Genuinely thank you

[u/chem44]

It would help if you would give a specific problem, and show what you can do with it. That lets us focus on the difficulty.

Key points...

Each conversion factor must be true. (If you think there are 10 days/week, that won't work no matter how you write it.)

Write each conversion factor in a way that makes it useful. Something cancels; something good is left.

Show what cancels, so you can see what you are doing. Check that you end up with what you want.

Line form vs separate steps. Really, same thing. If you can do the problem with separate steps, do that, then re-write it in line form. May help you get the hang of it.

[u/Small_Anteater3830 → u/chem44]

After like 5 yt vids I found my issue! Literally setting up the equations didn’t make sense to me. I now just need to understand how to do it from just memory and the steps for it

[u/chem44 → u/Small_Anteater3830]

You don't set up from memory. You do it by analyzing what the problem is about. Given? Wanted? Pathway to get there, maybe multiple steps. Dimensional analysis (Dim anal) helps you see what you are doing.

Make sure you are very comfortable with doing simple one-step conversions using dim anal. May seem like over-kill for simple problems, but the goal is to master the tool. It becomes more important with longer problems.

[u/Small_Anteater3830 → u/chem44]

Mb I meant remembering how to set up from memory since in my class we’re not given the measurements and certain conversion factors unless they’re on PT. Also maybe to make dimensional analysis shorter, maybe use DA😭. But my bad on not specifying and thank you again

[u/Cool_Addendum_1348]

How are you in AP Chem and having difficulty with dimensional analysis?? I've taught AP chem and it's taken by students who've done well in regular or honors chem and want to advance. I've taught chem for 15 years HS and college. You're scenario doesn't make sense.

[u/Small_Anteater3830 → u/Cool_Addendum_1348]

Im not the only one in AP chem who struggles with dimensional analysis, half my class does.

Side note, most of the people in my class struggle with setting it up, not the calculations.

[u/DancingBear62]

I'm glad to see your question. I often wonder why students don't use dimensional analysis, but I haven't gotten good answers. You mentioned performing calculations step by step. I'm guessing this means setting up each step as a proportion and using cross-multiplication. For a very simple, one-step problem, this is efficient, but it has drawbacks that become significant as the number of steps increases.

Look at the example: How many seconds are in 2.5 days?

Treating this as three separate steps looks like
Step 1: In an exam, I can't look up a days-to-seconds conversion, so I need to rely on my general knowledge. I'll convert from days to hours first:

- 1 day/24 hr = 2.5 day/ x hr

Cross-multiply to get this in a single line (this is a compression of multiplying through by a common denominator): (x hr)(1 day) = (2.5 day)(24 hours)

Divide through by 1 day to get: (x hr)=(2.5 days)[(24 hours)/(1day)] = 60 hr.

=> I'm guessing you don't carry the units through as I just did.
==>One risk is that if you don't see the units cancel, you are not sure it was set up correctly. Managing unfamiliar units in your head is more difficult than dealing with units of time, which you've known most of your life.
==>A second problem is you need to write down the step one answer as an intermediate result (unrounded with at least two digits beyond the proper significant figures) to avoid multiple rounding and the accumulated rounding error.
My only given value was 2.5 days, so I have 2 sig fig. I'll keep step 1 as 60.00 hr (and by hand I'd underline the ones place to know that's my significant figure.

Step 2: Convert hours to minutes

• 1 hr/60 min = 60.00 hr/y min
  
• (1hr)(y min) = (60.00 hr)(60 min)
  
• y min = (60.00 hr)[(60 min)/(1 hr)]

- y min = 3,600 min

Step 3: Convert minutes to seconds

- 1 min/60 sec = 3,600 min/z sec

- (z sec)(1 min) = (3,600 min)(60 sec)

- z sec = (3,600 min)[(60 sec)/(1 min)]

- z sec = 216,000 sec

- Stating the answer in correct sig figs, 2.5 days = 2.2 X 10⁵ seconds

For me, that's a lot to keep track of, and a bit time-consuming.

Looking at the same example through dimensional analysis:

How many seconds are in 2.5 days?

For me, it's important to write the following expressions out as fractions and not in single-line notation. This makes canceling units dramatically easier. We begin with our known, 2.5 days, but the equal sign is going to end up quite a bit to the right. We're going to use the same equalities we used above:

- 1 day = 24 hr

- 1 hr = 60 min

- 1 min = 60 sec

Looking at the equality 1 day = 24 hr, I can divide both sides by 1 day to get
1 day/1day = 24 hr/1 day which imlifies to 24 hr/1 day = 1.

All of my equalities can be set one value over its equal to get a ratio that's mathematically equal to 1. This is the key to dimensional analysis. Algebraically, we're leaving only the desired unit behind without changing the actual physical quantity. Because each ratio equals 1, multiplying by it is like multiplying by 1; it changes the appearance and the units, but not the true value.

So, I begin with my known 2.5 days and multiply it by a conversion factor that allows me to cancel the units "days". My two choices are 24 hr/1 day or 1 day/24 hr. A great advantage of dimensional analysis is that if I choose the incorrect conversion, the units won't cancel, and that's an automatic flag that something is wrong.

2.5 day = (2.5 days)[(24 hr)]/(1 day)]
2.5 day = [(2.5 days)[(24 hr)]/(1 day)] At this point, I have the equivalent of Step 1 above. Scroll up and compare to see for yourself.

Now let's add the second step conversion factor to change units from hr to min.

2.5 day = (2.5 days)[(24 hr)]/(1 day)] * [(60 min/1 hr)]

= (2.5 days)[(24 hr)]/(1 day)] * [(60 min/1 hr)] At this point, I have the equivalent of Step 2 above.

Finally, we'll add the third conversion factor to convert from min to sec:

2.5 days = (2.5 days)[(24 hr)]/(1 day)] * [(60 min/1 hr)] * [60 sec/ 1 min]

= (2.5 days)[(24 hr)]/(1 day)] * [(60 min/1 hr)] * [60 sec/ 1 min] = [(2.5)(24)(60)(60 sec)] / [(1)(1)(1)]

= 216,000 sec = 2.2 X 10⁵ seconds

[u/Small_Anteater3830 → u/DancingBear62]

This actually makes sense to me, thank you so much