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What is the trick to derive the ideal gas law and gas constant from the "combination" of Charles', Boyle's, and Avogado's laws? The general chemistry books I have seen tell me this is how they derived the ideal gas law and constant but they do not show how that is achieved. I tried to go about the process and failed. Let me show you what I did and maybe you can help me out.
The book says:
Boyle's Law => V = k/P (the constant is k, and the units seem to be L*atm)
Charle's Law => V = bT (b is the constant, and the units seem to be L/Kelvin)
Avogadro's Law => V = an (a is the constant, and the units seem to be L/mol)
But if you try to combine these three laws in a straightforward fashion I think you will fail as I did. What I tried was to multiply Boyle's and Charle's equations and then divide that result by Avogadro's equation to get V = (k/P)(bT)/(an) . . . but that is not right! (Note: n needs to be in the numerator on the right side of the equation). (Additionally, the units won't work-out for the ideal gas constant R, where kb/a = R has units of (L*atm*mol)/Kelvin which is wrong, i.e. should be (L*atm)/(mol*Kelvin).)
So, what is the trick I am missing here? Is there a simple and straightforward way to derive the ideal gas law and gas constant from the "combination" of Boyle's, Charles', and Avogadro's laws? Or is the process actually quite involved, and for that reason most general chemistry texts (i.e. all that I have seen, including physical chemistry texts) avoid providing the "derivation"? Alternatively, does someone know of a book or a webpage or someplace I can look to find this derivation of the ideal gas law and constant from Charles', Boyle's, and Avogadro's laws?
Thanks!
The book says:
Boyle's Law => V = k/P (the constant is k, and the units seem to be L*atm)
Charle's Law => V = bT (b is the constant, and the units seem to be L/Kelvin)
Avogadro's Law => V = an (a is the constant, and the units seem to be L/mol)
But if you try to combine these three laws in a straightforward fashion I think you will fail as I did. What I tried was to multiply Boyle's and Charle's equations and then divide that result by Avogadro's equation to get V = (k/P)(bT)/(an) . . . but that is not right! (Note: n needs to be in the numerator on the right side of the equation). (Additionally, the units won't work-out for the ideal gas constant R, where kb/a = R has units of (L*atm*mol)/Kelvin which is wrong, i.e. should be (L*atm)/(mol*Kelvin).)
So, what is the trick I am missing here? Is there a simple and straightforward way to derive the ideal gas law and gas constant from the "combination" of Boyle's, Charles', and Avogadro's laws? Or is the process actually quite involved, and for that reason most general chemistry texts (i.e. all that I have seen, including physical chemistry texts) avoid providing the "derivation"? Alternatively, does someone know of a book or a webpage or someplace I can look to find this derivation of the ideal gas law and constant from Charles', Boyle's, and Avogadro's laws?
Thanks!