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### Tools in Quantitative Chemistry

• Dimensional analysis is the use of conversion factors to convert between different units, including the standard SI units of measurement.
• Percent error is related to accuracy, while standard deviation is related to precision.
• Significant figures are related to precision, and must be considered when performing calculations.
• Significant figures are a relate to a measurement’s uncertainty.

### Tools in Quantitative Chemistry

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Lesson Overview 0:07
• Units of Measurement 1:23
• The International System of Units (SI): Mass, Length, and Volume
• Percent Error 2:17
• Percent Error
• Example: Calculate the Percent Error
• Standard Deviation 3:48
• Standard Deviation Formula
• Standard Deviation cont'd 4:42
• Example: Calculate Your Standard Deviation
• Precisions vs. Accuracy 6:25
• Precision
• Accuracy
• Significant Figures and Uncertainty 7:50
• Consider the Following (2) Rulers
• Consider the Following Graduated Cylinder
• Identifying Significant Figures 12:43
• The Rules of Sig Figs Overview
• The Rules for Sig Figs: All Nonzero Digits Are Significant
• The Rules for Sig Figs: A Zero is Significant When It is In-Between Nonzero Digits
• The Rules for Sig Figs: A Zero is Significant When at the End of a Decimal Number
• The Rules for Sig Figs: A Zero is not significant When Starting a Decimal Number
• Using Sig Figs in Calculations 15:03
• Using Sig Figs for Multiplication and Division
• Using Sig Figs for Addition and Subtraction
• Using Sig Figs for Mixed Operations
• Dimensional Analysis 16:20
• Dimensional Analysis Overview
• General Format for Dimensional Analysis
• Example: How Many Miles are in 17 Laps?
• Example: How Many Grams are in 1.22 Pounds?
• Dimensional Analysis cont'd 19:43
• Example: How Much is Spent on Diapers in One Week?
• Dimensional Analysis cont'd 21:03
• SI Prefixes
• Dimensional Analysis cont'd 22:03
• 500 mg → ? kg
• 34.1 cm → ? um
• Summary 25:11
• Sample Problem 1: Dimensional Analysis 26:09

### Transcription: Tools in Quantitative Chemistry

Hi, welcome back to Educator.com.0000

Today's lesson in general chemistry is on the tools of quantitative chemistry.0002

We are going to get introduced into how chemists go ahead and make measurements0010

and the basic concepts and methods of performing everyday calculations.0015

Our first point in the lesson is going to be the SI units of measurement.0023

Once we become familiar with the units of measurement,0027

we will then get into explaining how exactly good or bad your measurement was.0030

This is done by something we call percent error and also standard deviation.0039

We will then get into a discussion of what we mean by precision versus accuracy.0044

We will then get into something a little more quantifiable which is called significant figures and uncertainty.0048

Followed by the following: identifying sig figs; then using these significant figures in calculations.0054

We will then get into a very important concept which is fundamental to all of the physical sciences.0063

This is called dimensional analysis which is your use of what we call conversion factors for problem solving.0069

Finally we will wrap up the lesson with a brief summary followed by some sample problems.0078

The SI units of measurement are what we call the international system of units.0085

They are included as part of the measuring system.0090

It is used in all parts of the world except the United States.0093

The basic SI unit for mass of course is going to be the kilogram.0098

For length, it is going to be the meter.0106

For volume, it is going to be the liter.0109

Kg again stands for kilogram; lowercase m is going to be abbreviating meter.0114

Finally capital L is going to be symbolized for liter.0122

Again mass is going to be kilogram; length is going to be meter; volume is going to be liter.0128

When you make a measurement, how good or poor is it?0139

How close was a measurement to an actual or already known value?0143

This is what we call percentage error.0148

Percentage error is equal to the actual value minus the experimental value divided by the actual value.0150

Actual value is the known value.0158

The experimental value is what you actually measure in an experiment; what you measured.0163

Of course we always like to get as close as possible to the actual.0172

To illustrate this very simple equation to use, suppose you weighed an object and recorded a mass of 86.2 grams.0177

The known value is 97.9 grams; calculate the percentage error.0185

Known value is going to be actual.0189

What you recorded, this is our experimental value.0194

The percent error... let's just plug everything into the equation.0202

It is going to be equal to the actual value 97.9 grams minus the experimental value of 86.2 grams.0206

Divided by the actual value which was 97.9 grams.0216

All of this is going be multiplied by 100 to get you your percentage error.0222

The next statistic that we can also calculate is something called standard deviation.0229

Sometimes when you repeat an experiment several times, it is useful to know how successful you were in reproducing your results.0235

If you did for example ten trials of the same measurement,0243

how well were you able to get the same measurement in each of the ten trials?0248

For our purposes we want of course the standard deviation to be as low as possible.0255

Standard deviation has the following equations.0261

It is equal to the square root of all of these terms where x is equal to basically each individual score.0263

X-bar is simply the average of all the x's.0271

N is the number of values or the number of trials.0274

Summation of course means we are going to add up across all the values.0278

Let's go ahead and illustrate the use of this equation.0283

It looks like a lot but it is really not too bad.0287

You weighed an object three separate times with masses of 32.6, 31.8, 34.1 milligrams.0291

All I am going to do is set it up for us right now.0300

The standard deviation is going to be equal the square root of everything underneath it; summation x minus x-bar.0302

X is each score; 32 minus 6, minus the average, squared.0311

I am going to put all of this in brackets showing that we are going to add what is next.0321

Plus, 31.8 minus x-bar, squared; then plus 34.1 minus x-bar, squared; then closed brackets.0326

That is what is meant by summation x minus x-bar, squared.0349

Then divided by n minus 1 where n is going to be equal to 3 trials.0353

All of that divided by 3 minus 1.0358

All we need then to finish this problem is x-bar.0362

X-bar is just your average of the three values.0368

That is just going to be 32.6 plus 31.8 plus 34.1.0371

All of that divided by 3, giving you your average score; that is standard deviation.0377

The next item is what we call precision and accuracy.0385

Precision simply represents how well you were able to reproduce your results over several trials.0388

High levels of precision tend to support accountability and reliability.0394

Precision is going to be quantifiable by standard deviation; measured by standard deviation.0399

Again you want as low a deviation as possible; low standard deviation desirable.0410

The word accuracy however represents how well you were able to measure a value0422

in relation to an already known or published value from the literature.0427

Of course this is going to be measured by percent error.0431

Once again we want as low a percentage error as possible; low percent error is also desirable.0443

A lot of people use the terms precision and accuracy interchangeably.0454

But of course as you have just seen they are not the same.0457

One represents your ability to reproduce a certain measurement.0461

The second represents how well you were able to get to a known value.0466

The next item is very important whenever you make measurements.0473

This is called significant figures and uncertainty.0476

Consider the following two rulers; let me go ahead and draw one ruler.0480

I am going to go ahead and draw a second ruler on the bottom.0486

Now the difference is going to be their tick marks; one, two, three.0489

Then the next one is going to be the following; one and two.0495

Let's go ahead and measure a piece of wood.0510

That is this length right here measured by green.0514

I am going to do the same thing with the other ruler; just like that.0517

How would you measure the green stick with the above ruler?0529

One person could say maybe 1.4 cm; or another person could say 1.5 cm.0533

It turns out that neither person is incorrect.0541

Why?--because the 1 is what we know for sure.0545

We know that green object is longer than 1 for sure; 1 cm.0549

Really this last digit here, the .4 and the .5, is really uncertain.0554

It is the digit of uncertainty.0560

The digit of uncertainty is strictly up to you the user; up to person making the measurement.0566

Because the ruler is so poor, the tick marks are so large,0578

there is no way we can determine if it is .4 or .5.0584

It is not provided by the tick marks on the ruler at all.0587

However when we go ahead and look at this ruler down here,0593

we finally get the tick marks that is representative of the appropriate digit.0596

So you hear, one person could say 1.4.0601

Because it looks like the green mark is right on the .4, one person could even say 1.40 cm.0606

The other person could say 1.41 cm; one could say 1.42cm.0613

Someone could even say it is a little less; 1.39 cm.0619

It turns out that because of the tick marks, we can go one digit more.0623

We can provide a better measurement; better measurement due to what we call a higher level of precision.0629

Again it is a better measurement due to higher level of precision.0651

The rule of thumb is the following.0656

How far or how many digits do you know how to record a measurement?0658

You are always going to go one digit past whatever is given to you or whatever the limit is on the ruler.0663

Go one digit past what is given by the ruler or instrument.0671

For example when we go ahead and read a graduated cylinder, you always want to look at the bottom of the curve.0690

You see how this water level is slightly curved.0698

That is what you call the meniscus.0701

You always want to look at the bottom of it.0705

What is provided to us here on this graduated cylinder?0707

50 milliliters is right here; 55 milliliters is right there.0712

We know that the meniscus is approximately 53 milliliters.0726

The graduated cylinder can tell us if it is 51, 52, or 53.0735

What the rule is is we are going to go one digit past this.0741

We can say something like 53.0; we can say something like 53.1.0745

We could even go under and say 52.9 milliliters or even 52.8 milliliters.0750

Once again you are going to one digit past the last digit of certainty provided by the instrument.0757

When you make a measurement, it is important to always write it down with the correct number of these significant figures.0765

What are their significant figures?--what are the rules for identifying them?0774

The rules for sig figs are the following; all nonzero digits are significant.0779

A zero is significant when it is in between nonzero digits.0786

A zero is significant when at the end of a decimal number.0791

A zero is not significant when starting a decimal number.0795

Let's go take a look at a couple examples; all nonzero digits are significant.0799

For example 562, we have a grand total of three sig figs because none of them are zeros.0804

A zero is significant when it is in between nonzero digits.0810

Something like 501; this is three sig figs.0814

The zero counts because it is in between nonzero digits.0818

5001; both zeros count because they are in between nonzero digits.0822

How about 50010?--only the two zeros count here because they are in between nonzero digits.0829

This last zero here does not count; we only have four sig figs here.0837

A zero is significant when at the end of a decimal number; for example 500. and 500.0.0843

It turns out that in 500. all of these are significant.0853

The zeros come at the end of a decimal number.0858

Here, 500.0, these are all significant because they come at the end of a decimal number.0860

A zero is not significant when starting a decimal number.0870

For example 0.00321, the two zeros do not count here because they start a decimal.0873

We only have three significant figures here.0882

However 0.003210, these do not count at all.0884

However you see the zero here, that comes at the end of a decimal number.0891

This definitely does count; we have a grand total of four significant figures.0895

When we use significant figures in calculations, we have to learn how to incorporate the rules now.0906

For multiplication and division, the answer will have the same number of sig figs as the fewest number of sig figs present.0913

For addition and subtraction, the answer will have the same number of decimal places as the fewest number of decimal places present.0921

For example. 0.321 times 0.57, we are going to get an answer that is only two sig figs.0929

Why?--because here in 0.321, you have three sig figs.0941

Here in 0.57, you have two sig figs.0946

When we do the addition and subtraction with the same numbers, 0.321 minus 0.57,0949

you see now we go by digits after the decimal places.0956

Here there is three digits after the decimal; here there is two.0960

Our answer is going to have two digits after the decimal.0963

Finally when you have mixed operations, you never want to round until the end.0971

You want to carry all digits through.0976

Now that we have talked about significant figures, let's go ahead and discuss what we mean by dimensional analysis.0982

Dimensional analysis utilizes the following.0989

It utilizes ratios of different units that we call conversion factors to convert from one unit to another.0991

If you want to go for example from unit A to unit B, how do we go ahead and do that?0999

The general format is the following; we are going to take unit A.1007

We are going to multiply by this ratio, something over something.1011

That is going to go ahead and give me unit B.1015

Unit A goes downstairs to get cancelled.1018

Unit B goes upstairs to get carried through to the final answer.1022

This ratio right here of unit B to unit A, that is what we call your conversion factor.1027

Let's go ahead and look at a couple of examples.1038

There are 4 laps in 1 mile; how many miles are in 17 laps?1040

We are going to say 17 laps times something over something.1045

That is going to give us our answer in units of miles which is what the question is asking for.1050

It is going to go downstairs to get cancelled.1059

Miles goes upstairs to get carried through to the final answer.1061

The conversion factor is actually given to us in the problem because they state that 4 laps is equal to 1 mile.1065

This is then 1 mile on top and 4 laps on the bottom.1071

This is going to round to 4.3 miles.1081

Let me tell you why: 17 laps, you have two sig figs.1085

However for conversion factors, we are assuming that a conversion factor are what we call exact numbers.1090

That is they have infinite precision.1097

They have infinite significant figures; there is no uncertainty.1100

You are going to ignore conversion factors for sig fig purposes.1104

Once again you are going to ignore conversion factors for sig fig purposes.1113

Let's go ahead and look at one last example here on dimensional analysis.1121

One pound is 454 grams; how many grams are in 1.22 pounds?1125

1.22 pounds times something over something is going to give us our answer in grams.1131

I want to cancel the pounds; that goes downstairs.1141

I want to keep grams; that is going to go upstairs.1144

You are told that 1 pound is 454 grams; 1 pound on the bottom and 454 grams on top.1147

That is going to give us an answer of 553.88 grams.1155

Here we have three sig figs; our answer is going to round to 554 grams.1162

That again, that is what we call dimensional analysis.1170

It is important to really master this because we are going to be using this incredibly heavily throughout all our lessons in general chemistry.1173

One last example then; suppose a diaper cost us 35 cents.1185

You have a newborn who goes through about 14 diapers a day.1189

How much is spent on diapers in one week?1192

Suppose you have a diaper costing 25 cents; that is actually a statement already.1196

We are told that 25 cents costs us each diaper; 25 cents per diaper.1203

We are trying to multiply through and cancel units.1211

The other item we see here that has diapers is 14 diapers a day; that is another ratio.1214

14 diapers goes upstairs to get cancelled; then 1 day on the bottom.1220

Finally I want to get my answer into week or dollars per week.1226

This is going to now be 7 days on top to get cancelled, divided by 1 week on the bottom.1233

When all is said and done, you are going to get an answer of1241

24 dollars and 50 cents per week to be spent on diapers.1243

As you can see, the reason why we are doing these examples1250

that are not chemistry yet is to show you that dimensional analysis1253

which we use in chemistry and the physical sciences can actually be very easily applied to everyday life.1256

The next type of dimensional analysis deals with unit conversion.1266

You have heard of terms like centi and milli and kilo before.1271

But how do you convert between the three?1276

We are going to learn that dimensional analysis is all behind this; converting between units.1278

You should ask your chemistry instructor which units you actually have to know.1284

But let me go ahead and just point out a few.1288

Mega is 106; kilo is 103; deci is 10-1; centi is 10-2.1291

Milli means 10-3; micro is 10-6; nano is 10-9.1301

if at all or if they are going to be given to you.1313

Now that we have been introduced to these prefixes and what they mean,1317

let's go ahead and see how we can use them in calculations.1322

For example, 500 milligrams is equal to how many kilograms?1326

What I always like to say is that sometimes it helps to convert to the base unit first.1330

What I mean by the base unit is that it has no prefixes.1335

In other words, let's get to the unit with no prefixes first.1341

First step is to go from milligrams to regular grams then onto kilograms.1347

This is going to be a two step process.1355

500 mg, just going to set it up, times something over something.1358

The first step is to get g; g goes upstairs; mg goes downstairs to get cancelled.1364

Once I am in g, now I can go to kg; times something over something.1369

That is going to give us our answer in units of kg.1375

You see that g is upstairs here; it is going to go downstairs to get cancelled.1378

Kg goes upstairs to get carried through to the final answer.1383

What numbers do we put in and where?--the rule is the following.1388

For the prefix multipliers like kilo and milli, you put the multiplier with the base unit; put multiplier with the base unit.1393

Once again you should put the multiplier with the base unit.1409

For milli, milli stands for 10-3.1414

That is going to go with the base unit here; 10-3 on top; 1 on the bottom.1419

Kilo stands for 103; that is going to go with the base unit.1424

1 goes to kg; 103 is going to go the g.1428

Once again you always put the multiplier with the base unit.1433

Let's go ahead and do a last one; this is centimeters to micrometers.1443

The first step is to go from centimeters to regular meters; then from regular meters on to micrometers.1447

34.1 centimeters times something over something is going to give me my answer in meters.1457

Cm goes downstairs to get cancelled.1465

M goes on top to get carried through to the final answer.1467

Once I am in meters, I can then go on and get micrometers; times something over something.1471

That is going to give me my answer in micrometers.1476

You see that m is on top; it has to go downstairs to get cancelled.1479

Then micrometers goes upstairs to get carried through to the final answer.1483

When we look up the prefix for centi, centi stands for 10-2.1488

That goes with the prefix-less unit; 10-2 on top; 1 on the bottom.1492

When we look up the multiplier for micro, it is 10-6.1497

That goes downstairs with meters; 1 on top.1503

That is going to get you your answer in units of micrometers.1507

To summarize, when you perform measurements, you want to gauge how well you are doing.1514

We can do this by percentage error calculation which again is going to tell us a little about your accuracy.1519

We can also calculate what is called the standard deviation.1527

That is going to tell us of how precise you were.1531

We also learned the concept of significant figures; significant figures is related to precision.1535

Finally we learned a very fundamental concept in all of the physical sciences which is dimensional analysis1545

which follows the same basic pattern where we can go from unit A to unit B using a conversion factor.1551

There is our summary; now let's go ahead and tackle some sample problems.1567

An intramuscular medication is given at a dosage of 5.00 milligrams per kilogram of body weight.1571

If you give 0.425 grams of medication to a patient, what is the patient's weight in pounds?1584

0.425 grams of medication is here.1592

Somehow we want to go from grams of medication to whatever the question is asking for which is pounds of body weight.1597

In addition we see in the first sentence that we have a statement here1611

that 5 milligrams of medication are given per kilogram of body weight.1615

That represents a very nice ratio.1619

5.00 mg of medication for every kilogram kg of body weight; that is our conversion factor.1622

This unit is 5 milligrams of medication; we are given 2.45 grams of medication.1637

We have to get the 0.425 grams of medication into milligrams first.1643

0.425 grams of medication times something over something is going to give us our answer in units of milligrams of medication.1648

Milligrams goes upstairs; g goes downstairs.1659

When we look up the prefix for milli, it is 10-3.1662

That goes with the prefix-less unit on the bottom; then 1 on top.1666

That is going to give us milligrams of medication which is going to be 425.1670

We can then take the 425 milligrams of medication, multiply it by something over something.1677

That is going to give us our answer in kilograms of body weight.1684

Mg is going to go on the bottom; kg is going to go on top.1698

We know that from the sentence, it is 1 kg for every 5.00 mg.1701

Finally we can then take our kilogram of body weight and we can go to pounds of body weight.1708

We do that from the conversion factor where 1 kilogram is equal to approximately 2.20 pounds.1717

We are going to take kilograms of weight, multiply it by something over something.1726

That is going to give us our answer in pounds.1732

Pounds goes upstairs; kg goes downstairs; just 2.20 divided by 1.1736

When all is said and done, we should get an answer of 187 pounds using the correct number of sig figs.1743

That is our lesson from general chemistry on quantitative tools.1752

I want to thank you for your attention.1758

I will see you next time on Educator.com.1760