Derive ideal gas equation. Give the ‘R’ value in S.I. & C.G.S units.
Answers
Value of Universal Gas Constant in CGS-unit
Therefore, putting the values of P, V, T, and n in ideal gas law, we have universal gas constant (R) = (7.6 × 13.6 × 981 × 22.4 × 103)/(1 × 273) = 8.314 × 107 dyne cm2 mol-1 K-1 = 8.314 × 107 erg mol-1 K-1, where dyne cm2 = erg.
Definition of Ideal Gas Constant
The ideal gas constant is the proportionality constant in the ideal gas equation. It is the ratio of the product of pressure and volume to the product of mole and temperature.
Formula of Gas Constant
The formula of the gas constant from the ideal gas law equation is
The ideal gas law equation: R = PV/RT
where P is the pressure of an ideal gas,V is the volume the gas occupies,n is the number of moles of the gas,and the T is the temperature in the kelvin.
Value of Universal Gas Constant in CGS-unit
In CGS units, pressure = 1 atm = 76 × 13.6 × 981 dyne cm-2 and volume =22.4 liter = 22.4 × 103 cm3. Therefore, putting the values of P, V, T, and n in ideal gas law, we have universal gas constant (R) = (7.6 × 13.6 × 981 × 22.4 × 103)/(1 × 273) = 8.314 × 107 dyne cm2 mol-1 K-1 = 8.314 × 107 erg mol-1 K-1, where dyne cm2 = erg.
Value of Gas Constant in SI-unit
The Values of universal constant (R) in CGS-system = 8.314 × 107 erg mol-1 K-1. But 1 J = 107 erg. Therefore, the universal constant in SI units = 8.314 J mol-1 K-1. Agin from specific heat relation, 4.18 J = 1 calorie. Therefore universal gas constant from ideal gas law equation = (8.314/ 4.18) cal mol-1 K-1 = 1.987 calories mol-1 K-1 ≃ 2 calories mol-1 K-1.
Number of Molecules Per Unit Volume
PV = nRT = (N/N0) × RT
where N = number gas molecules present
N0 = Avogadro number = 6.023 × 1023
∴ P = (N/V) × (R/N0) × T = N′KT
where N′ = number molecules per unit volume.
k = Boltzmann constant = R/N₀
= 1.38 × 10-16 erg molecules-1 K-1