Deriving the ideal gas law

L. AllenStonington HSHolt Modern Chemistry, Chapter 11

Avogadro’s Law (1811)

Equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.

Amadeo Avogadro, the ugliest of the dead chemists

Significance of Avogadro’s hypothesis

If we can fix, for example, the temperature and pressure at STP, we can then compare how many molecules are present by simply comparing volumes. We find that one mole of any gas – ANY gas – has a volume of 22.41410 liters.

What does this mean?

For starters, if we convert any amount of gas at any temperature and pressure to STP using the combined gas law, we can convert the resulting volume to moles using this conversion factor.

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Example:

How many moles of gas are present in 46.5 liters of gas at 28 ° C and 2.75 atm of pressure?

Strategy? Use the 4-Step method to solve for V2, then use dimensional analysis to convert from liters to moles

4-Step method

Write the given Rearrange the

equation Plug in numbers

and units Solve

Write the givens Rearrange the equation

P1 = 2.75 atm V1 = 46.5 L T1 = 28 ° C + 273 =

301 K P2 = 1.00 atm V2 = ? T2 = 273 K 𝑉 2=

𝑃1𝑉 1𝑇 2𝑃 2𝑇 1

Don’t leave units out!

PLUG IN VALUES WITH LABELS

DOES THIS MAKE SENSE? Notice, BTW, that the

pressure goes down, which will make the volume go up. Also, the temperature is lower at STP; that makes the gas shrink. This is consistent with what we see as we disaggregate the parts of this equation.

Finish!

SOLVE

V2 = 116 Liters at STP (3 sig figs)

USE DIMENSIONAL ANALYSIS TO CONVERT FROM LITERS AT STP TO MOLES.

Shouldn’t this be easier?

If we know that 1 mole = 22.41 L at 273.15K and 1.000 atm, can’t we organize our work so that we don’t have to do this step by step? Can’t we come up with some kind of multiplier, something to include the 22.41, the 1.000 atm, the 273.15 K? Let’s fool around with the math…

Random mole from internet, lost url

Let’s combine these

P = 1.000 atm V = 22.41 L T = 273.15 K n = 1.000 mole

R = .0820578437 or 0.08206 (4 sf) or 0.0821 (3sf)

What is the unit on this?

We call this R, the ideal gas constant.

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Now let’s try the original problem again, and see how it comes out.

n= 5.18 moles

This is the same answer, with only one equation, which is simpler. Win!

Can you see where this is heading?

Gosh, if I can calculate moles, I can find mass, or number of molecules… wait a sec, isn’t this how the stoichiometry chapter got started?

Classwork:

End of chapter 11, page 392 #40-52 Do NOT ignore sig figs. We always use

one more sig fig in “book values” than in the givens, so that when we round back, we get the right answer. There is actually one correct answer for each question.

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