Gas Laws

Normally, I'd just do all the separate "laws" and then combine them to make the big equation of state for gas - the Ideal Gas Law (IDL). Here it is right up front:

\[PV = nRT\]

Typically, we have our pressure (\(P\)) in units of atmospheres (atm), and volume (\(V\)) in units of liters (L). Temperature (\(T\)) HAS to be in absolute units of kelvin (K), and of course \(n\) is our standard amount of substance, the mole (mol). Putting all that together will get you the most used version of \(R\) for gas law problems:

\[R = 0.08206\;{\rm L\;atm\over mol\;K}\]

If you rearrange the IGL like this you have all the variables on one side and the constant (\(R\)) on the other:

\[{PV\over nT} = R\]

Now assume you are just changing ONE of those variables and want to see how it affects another (single) variable. First rewrite the equation above with subscripts meaning condition 1, condition 2, etc...

\[{P_1V_1\over n_1T_1}={P_2V_2\over n_2T_2}={P_3V_3\over n_3T_3}=\cdots \]

This is affectively the combined gas law where you can predict any final outcome of a variable as long as you know the starting ones and which variables remain constant. Here's an example of keeping both temperature and moles constant for a condition 1 vs condition 2 type problem. This means that \(n_1 = n_2\) and \(T_1 = T_2\) so those variables will cancel out leaving you with Boyle's Law:

\[\require{cancel} \newcommand\ccancel[2][black]{\color{#1}{\bcancel{\color{black}{#2}}}} {P_1V_1\over \ccancel[red]{\color{gray}{n_1T_1}}}={P_2V_2\over \ccancel[red]{\color{gray}{n_2T_2}}}\]

\[P_1V_1 = P_2V_2\]

This is Bolye's Law where the pressure times volume is a constant when the number of moles and temperature are held constant. If you keep playing this game of hold two constant and vary the others, you will get all the "named" gas laws.

Boyle's Law

\[P_1V_1 = P_2V_2\]

constant \(n,T\)

Charles' Law

\[{V_1\over T_1} = {V_2\over T_2}\]

constant \(n,P\)

Avogadro's Law

\[{V_1\over n_1} = {V_2\over n_2}\]

constant \(P,T\)

Gay-Lussac's Law

\[{P_1\over T_1} = {P_2\over T_2}\]

constant \(n,V\)

Inflate a Tire* Law

\[{P_1\over n_1} = {P_2\over n_2}\]

* constant \(V,T\) named by dr mccord

Nobody Does This* Law

\[n_1T_1 = n_2T_2\]

*constant \(P,V\) named by dr mccord

Combined Gas Law

\[{P_1V_1\over T_1} = {P_2V_2\over T_2}\]

constant \(n\)

Ideal Gas Law (rearranged)

\[{PV\over nT} = R\]

*equation of state

Common \(R\) Values

\(R = 0.08206\;{\rm L\;atm/mol\;K}\)
\(R = 0.08314\;{\rm L\;bar/mol\;K}\)
\(R = 62.36\;{\rm L\;torr/mol\;K}\)
\(R = 8.314\;{\rm m^3\;Pa/mol\;K}\)

*all these values will be given on exams

In Dr. McCord's class, you will not have to memorize the names of all the laws. You will need to be able to use any and all the laws to solve gas problems though. Really this just means you are using the ideal gas law in many different ways.



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