Value of R, The Gas Constant

The equation of ideal gas is 

PV = nRT

From the ideal gas equation we can write

R = PV/nT

    = (Pressure x Volume)/ (Mole x Temperature)

Since P is defined as force per unit area, hence

R = (Force/Area) (Volume)/ (Mole) (Temperature)

Area = (Length)²

Volume = (Length)³

R =[Force/(Length)²] (Length)³/(Mole)(Temperature)

R = (Force) (Length)/(Mole)(Temperature)

R = Work/(Mole)(Temperature)   [Force x Length]

The physical significance of R is work per degree per mol. It may be expressed in any set of units repressing work or energy.

Numerical value of R:

Numerical value of gas constant can be obtained readily at 1 atm and 273.16 K. Since 1 mole of an ideal gas occupies a volume of 22.414 liters at STP, therefore

R =(1 atm) (22.414 L) /(1 mol) (273.16 K)

R = 0.0821 L atm K^-1 mol^-1

If the volume is expressed in cubic centimetres instead of litres, the value is

R = 82. 1 cc atm K^-1 mol^-1

When the pressure is expressed in dyne/cm² and volume in the cubic centimetres, then R can be expressed in units of ergK^-1mol^-1. A pressure of 1 atm is the pressure of a column of mercury 76 cm high and 1 square centimetre of cross section at 273.16 K. Since the density of mercury is 13.6 g/cm³, the mass of the column is 76 x 13.6 g. The pressure in dyne/cm² will be its mass multiplied by the acceleration due to gravity, 981 cm/S². Inserting the values, we get pressure as

1 atm = hdg

            = (76 cm)(13.6 g/cm³)(981 cm/S²)

            = 1.0133 x 10⁶ dyne/cm² (Where 1 dyne = 1 g.cm/S²)

Volume of the gas can be expressed in cubic centimetre, e.g., 1 L = 1000 cm³. Substituting these values of P and V, we get

R = (1.0133 x 106 dynes/cm²) (22414 cm³)/ (1 mol)(273.16 K)

   = 8.314 x 10⁷ erg K^-1mol^-1     (dyne x cm = erg)

  = 8.314 Joule K^-1 mol^-1          (10⁷ ergs = 1 J)

  = 1.987 cal K^-1 mol^-1               (1  cal = 4.184 J)

In the SI units, pressure is expressed in pascal Pa, which is defined as the pressure produced by a force of 1 N on an area of 1 m2. Hence, the pressure corresponding to 1 atm in Pa is given by

P = (0.76 m)(13.6 x 10³ kgm^-3)(9.81 ms^-2)

  = 101325 Pa =101325 Nm^-2

Thus the gas constant R is expressed as

R = PV/nT = (101325 Nm^-2) (22.414 x 10^-3 m³)/ (1 mol)(273.16 K)

   = 8.314 N m K^-1 mol^-1

   = 8.314 J K^-1 mol^-1

Hence the value of R in different units is

R = 0.0821 L atm K^-1 mol^-1

    = 8.314 x 10⁷ erg K^-1 mol^-1

    = 8.314 J K^-1 mol^-1

   = 1.987 cal K^-1 mol^-1

It should be clearly understood that although R may be expressed in different units, for pressure-volume calculations involving gases R must be taken in the same units as those used for pressure and volume.