Ideal gas law

The ideal gas law is useful for calculating temperatures, volumes, pressures or number of moles for many gases over a wide range of temperatures and pressures. However, the law fails at low temperatures or high pressures. The ideal gas law is the combination of Boyle's law, Charles's law and Avogadro's law and is expresses as:

Ideal gas law: PV = nRT,

where P = pressure, V = volume, n = number of moles, R = 0.082057 (L x atm)/(K x mol), the constant of proportionality relating the molar volume of a gas to T/P (the "molar gas constant"), and T = the absolute temperature, in degrees Kelvin.

Boyle's law: PV = constant (at fixed temperature and amount of gas)

Charles's law: V/T = constant (at fixed pressure and amount of gas)

Boyle's + Charles's PV/T = constant (at fixed amount of gas)

Avogadro's law: V = nV_m (at fixed temperature and pressure, where V_m is the same for all gases)

The gas laws started, in the 1660's, with Robert Boyle's law, stating "the volume of a sample of gas at a given temperature varies inversely with the applied pressure, or V = constant/P (at fixed temperature and amount of gas). Then Jacques Alexandre Charles' experiments with hot-air balloons, and additional contributions by John Dalton (1801) and Joseph Louis Gay-Lussac (1802) showed that a sample of gas, at a fixed pressure, increases in volume linearly with the temperature, or V/T = a constant. This is known as Charles's law. Extrapolations of volume/temperature data for many gases, to a volume of zero, all cross at about -273 degrees C, which is absolute zero. Of course, the gases would liquify before reaching this temperature and so the law does not really apply in this temperature region. In 1811 Amedeo Avogadro re-interpreted Gay-Lussac's law of combining volumes (1808) to state Avogadro's law , Equal volumes of any two gases at the same temperature and pressure contain the same number of molecules. The molar volume of gas, at standard temperature ( 0 Celcius) and pressure (1 atm) is 22.4 L.

Values of the Molar Gas Constant (R) in Different Units
0.082057 liter atm / (K mol)
8.31441 J / (K mol)
8.31441 kg m2/(s2 K mol)
8.31441 dm3 kPa / (K mol)
1.98719 cal / (K mol)

Example problems

PROBLEM 1) Two liters of gas at 1 atm and 25C is placed under 5 atm of pressure at 25C. What is the final volume of gas?

Using Boyle's law:

Eq. 1.1) P_iV_i = constant = P_fV_f or

Eq. 1.2) V_f= P_iV_i/P_f

Eq. 1.3) V_f= (1 atm)(2 L) / (5 atm) = 0.4 L

Using Ideal gas law:

Eq. 1.4) n = P_iV_i / RT_i = P_fV_f / RT_f

But since T_i = T_f and R is fixed, this reduces to Eq. 1.1 shown above.

PROBLEM 2) How many moles of nitrogen are present in a 50L tank at 25C when the pressure is 10 atm? (Note: Kelvin = Celcius + 273.15). Numbers include only 3 significant figures.

Eq 2.1) n = PV/RT = (10.0 atm)(50 L) / [(0.0821 L atm / (K mol)](298K) = 20.4 mol