Key Thoughts to Date• Summary-the four equations-3 variable systems• a/b=c• b/a=c• 1/(a*b)=c• a*b=c• Applications:

It is known that a & c are directly related and that a & b are inversely related. If a=9 and b=8, what is c?

There are four mathematically possible results:72 1.125 0.889 0.0139Which is it?a=c (from a & c being direct)a*b=c (from a & b being inverse which also requires that b & c

be direct)Thus, the answer is 72.

• Binary Changes of State• Direct: • state1= a1/b1

– Assume you are going to be solving for a2 => a1/b1 = a2 (this is not strictly true)

– Since the relationship between a and b is direct, b2 must be written on the same level as a2, yielding: (a1/b1)*b2 = a2

• In general one can describe the solution of all changes of state involving directly related variables as fitting the following general algorithm:

• (Fully defined state written as a ratio with the term identical to the missing term written as the numerator)*given variable=missing value

• Inverse• state1= a1*b1

• a1*b1=b2

• (a1*b1)/a2=b2

• In general:

(fully defined state written as a product)/given variable=missing variable

What about dimensions?

• The text in sections 1.5 and 1.9 discusses SI units, both fundamental and derived. In gen chem, we are most interested in mass, volume, moles and T, as well as a number of derived units.

• By next Tues (10/7),The student needs to be comfortable and be able to readily (as in instantaneously) convert between the following:– mass: mg,g,kg – vol: L,mL,cm3

– If practice is needed, there are web pages available.

• Other dimensions will be introduced as appropriate

Mass, volume and density-a study in intrinsic and extrinsic properties

• Intrinsic-not dependent on quantity, source or method of preparation• Extrinsic-not intrinsic• Key thought-most of what we measure/observe is sample dependent-or

at least we don’t know that it isn’t.• The value of intrinsic properties lies in their transferability to any

sample. Pick up any reference text and it contains only intrinsic information

• The key process then is the conversion of measured properties(extrinsic most often) into intrinsic ones

• An obvious example of this transition is the relationship between mass, volume and density.

– No one is likely to argue against mass and volume being sample dependent– density is then defined as mass per unit volume => d=m/V where m is

usually in grams and V can be cm3,mL or L.– density is said to be intrinsic-why? What does that say about the rules of

the universe?

• Is “mass per unit volume” a good definition of density?

Looking at d=m/V• What does d=m/V say about the relationships between the

variable pairs? Is it what you would logically expect?

• d=1.00g/mL is roughly the density of water at RT. It’s often useful to frame discussions with that as a reference

• if d>1.00 what quick relationship can you see between m and V-assuming that all of the dimensions are consistent?

• What about d<1.00?

• What is really significant about d=m/V is that it is the classic three variable equation. Master this and you’ll be able to handle the majority of “problems” you’ll encounter in the course.

Density Problems• Since this is a three variable system, it clearly provides

three basic questions-given any two what’s the third. Hardly rocket science.

• Working problems in lecture is one way to fill 100mins, but it has several shortcomings. In particular, problem solving is not a spectator sport. What is useful, is the discussion of common pitfalls or error traps which the instructor will lay for you.

• The student should realize that a major role of the instructor in a testing environment is to cause the student to make a mistake. It’s a game. The instructor is trying to make you err and you are trying to avoid those errors. To be frank, the instructor likes to lose.

Avoiding the Pitfalls• What is the volume in liters of a sample of a liquid with a density of 0.732g/mL

and a mass of 0.670kg?• Line up your ducks

– m=0.670kg=670.g– V=? (in liters)– d=0.732g/ml

• Recall that when d<1.00 V>m so V>670. If you are going to apply this, be certain that the dimension of the given variable matches that of the density.

• now you can play the algebra game d=m/V to V=m/d or you can simply realize that the only combination of 670 and 0.732 that yields an answer greater than 670 is 670/0.732

• either yields V=915.300546448087431693398907103825• hopefully you’ll round this to the allowed three sig figs =915mL• note:students commonly write out the correct calculation and then perform it

incorrectly. The most common errors being switching multiplication and division or inverting a ratio. More than one student would multiply instead of divide the above and get 490. That answer would be accepted and put in one of those “magic boxes”.

• you see your initial note that the answer is to be in liters and convert it to 0.915L

A few density facts• water(l)=1.00g/mL

• gas densities tend to be about 1/1000 or less than that of liquids and are often presented as g/L.

• metals tend to be in the range of

10-20g/cm3

Questions without solutions• What is the volume of 345g of a material

with d=1.17g/cm3

• A liquid has a volume of .465L and a mass of .756kg. What is its density?

Question of the Day

Why are you having us calculate densities in dimensions like kg/mL? Seems like a total waste of time. I’ve got lots of things going on besides this class, you know!!

It’s all about your state of mind

• There are really two key thoughts that the student must be comfortable with:

1. I will never ask you a question that is not fully solvable based solely upon previously covered/discussed material

2. There is nothing I can ask that you can’t do.

Tuesday’s Quiz- A Change of State?• How many grams of hydrogen are present in a sample of

ammonium carbonate that contains 0.13moles of oxygen?• yes=42 No=107 Who cares=98• (NH4)2CO3

• State 1 is fully defined by the formula which contains 8.00g of H and 3 moles of O

• State 2 contains 0.13moles of O• mass of H and moles of O are directly related• You should be able to see readily that the answer must be less

than 0.80g of H but more than 0.08g. Why?• setting it up as a direct change of state (a1/b1)*b2=a2

• (8.00g of H/3moles of O)*0.13moles=0.347g of H• what are these #s : 0.0487 185• 3/8*.13 8/.13*3 • notice that both fail our test

Getting ahead of the game-n and T

• For purposes of this exercise, ignore any issues involving temperature of which you may be aware.

• A gas system is kept at constant P and V• Initially, T=400K and n=0.45moles.• If constant P and V are maintained-what will T be

when n=0.057 moles?