5.4The Ideal Gas Law 209We can replace the proportionality sign with an equals sign by incorporating R, a pro-portionality constant called the ideal gas constantV=RnTPRearranging, we get the equation:PV=nRT[5.5]This equation is the ideal gas law, and a hypothetical gas that exactly follows this law is an ideal gas. The value of R, the ideal gas constant, is the same for all gases and has the value:R=0.08206 #atmmol#KThe ideal gas law contains within it the simple gas laws that we have discussed. For example, recall that Boyle’s law states that V∝1>Pwhen the amount of gas (n) and the temperature of the gas (T) are kept constant. We can rearrange the ideal gas law as follows:PV=nRTFirst, divide both sides by PV=nRTPThen put the variables that are constant, along with R, in parentheses:V=(nRT)1PSince nand Tare constant in this case, and since Ris always a constant, we can write:V∝(constant)*1Pwhich means that V∝1>PThe ideal gas law also shows how other pairs of variables are related. For example, from Charles’s law we know that V∝Tat constant pressure and constant number of moles. But what if we heat a sample of gas at constant volumeand constant number of moles? This question applies to the warning labels on aerosol cans such as hair spray or deodorants. These labels warn against excessive heating or incineration of the can, even after the contents are used up. Why? An “empty” aerosol can is not really empty but con-tains a fixed amount of gas trapped in a fixed volume. What would happen if we were to L = litersatm = atmospheresmol = molesK = kelvinsnRTP=Vconstantnand Ideal Gas LawconstantPand constantnand V∝TnRTP=VV∝1PnRTP=VV∝nBoyle’s LawCharles’s LawAvogadro’s LawPV=nRT▲The ideal gas law contains the simple gas laws within it.:L:.:PTT.