The Gas Equation: This equation combines the Boyle's and Charle's laws.
According to Boyle's law, \(V \propto \frac 1P\) (at constant T)
According to Charle's law, \(V \propto T\) (at constant P)
Combining the two laws, \(V \propto \frac 1 P \times T\)
\(V \propto \frac TP\)
or \(\frac{PV}{T}\) = constant (R)
Nature of R: From the gas equation,
\(R =\frac{P \times V}{T}=\frac{\text { Pressure } \times \text { Volume }}{\text { Moles } \times \text { Temperature }}\)
\(=\frac{\text { Force } / \text { area } \times \text { Volume }}{\text { Moles } \times \text { Temperature }}\)
\(=\frac{\text { Force } \times \text { length }}{\text { Moles } \times \text { Temperature }}\)
\(=\frac{\text { Work }}{\text { Moles } \times \text { Temperature }}\)
\(=\text { Work done per degree per mol }\)
Thus R represents work done per degree per mole.
Value of R: The value of R (gas constant) in the various units are given below.
\(R =0.0821 \text { litre-atm } / \mathrm{K} / \mathrm{mole} \)
\(=8.314 \times 10^7 \mathrm{ergs} / \mathrm{K} / \mathrm{mole} \)
\(=8.314 \text { joules } / \mathrm{K} / \mathrm{mole}\)
\(=1.99 \mathrm{\ calories} / \mathrm{K} / \mathrm{mole}\)
\(=0.002 \mathrm{\ kcal} / \mathrm{K} / \mathrm{mole} \)
\(=5.189 \times 10^{19} \mathrm{eV} / \mathrm{K} / \mathrm{mole}\)
\(=8.314 \mathrm{\ Nm} / \mathrm{K} / \mathrm{mole}\)
\( =8.314 \mathrm{\ kPa\ } \mathrm{dm^3} / \mathrm{K} / \mathrm{mole}\)
\( =8.314 \mathrm{\ MPa\ } \mathrm{cm^3} / \mathrm{K} / \mathrm{mole}\)
Gas constant for a single molecule is called Boltzmann constant (k)
\(\frac{R}{N} = k\)
Values of k = 1.38 x 10-16 erg/deg-abs/molecule
= 1.38 x 10-23 Joule/deg-abs/molecule
For 'n' moles of a gas, the equation becomes
PV = nRT
Initial pressure P1 volume v1 and T1 of a gas may be related with the final pressure P2 volume V2 and temperature T2 as below.
\(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\)
Standard temperature and standard pressure: While using the above relation, standard (or normal) conditions are usually given; standard (normal) conditions mean that temperature is 0°C or 273K or 273°A and pressure is 1 atmosphere or 760 mm. It is because of the fact that the volume of any gas at 0°C and 1 atmospheric pressure is considered as volume at standard (or normal) temperature and pressure, i.e., at STP or NTP
Applications of gas equation:
(i) Calculation of mass and molecular weight of the gas
\(PV = nRT\)
\(PV = \frac mM RT \quad \left(\because n = \frac m{M}\right)\)
\(m = \frac {PVM}{RT}\)
where m = Mass of the gas
M = Mol. wt. of the gas
(ii) Calculation of density (d) of the gas
\(PV = \frac mM RT\)
\(P = \frac mV \times \frac {RT}{M}\)
\(= \frac{dRT}M \quad(\because d = \frac mV)\)
\(\frac{dT}{P} = \frac MR\)
Since M and R are constant for a particular gas,
Thus \(\frac{dT}P\) = constant
Thus, at two different temperature and pressure
\(\frac{d_1T_1}{P_1} = \frac {d_2T_2}{P_2}\)
