Question

The gas constant, R, has a value of 8.32 J mole^-1K^-1 in SI units.

A hypothetial system uses a unit of mass that equals 2kg, a unit of length that equals 5m and a unit of time that equals 1ms. The system also uses the Hahrenheit scale of temperature in which there are 180 divisions between the temperature of melting ice and that of the (normal) boiling point of water (=100°C). 

The value of the gas constant, R, in this sytem would be (nearly):

(1) 30x10^-8 (new energy unit) (mole^-1) (0F)^-1

(2) 9.2x10^-8 (new energy unit) (mole^-1) (0F)^-1

(3) 15x10^-8 (new energy unit) (mole^-1) (0F)^-1

(4) 4.6x10^-8 (new energy unit) (mole^-1) (0F)^-1

Answer 1

(2) 9.2x10^-8 (new energy unit) (mole^-1) (0F)^-1

We first note that the 1 mole would still contain the same number of ‘entities’.

Also 1J = 1 kg (1m)² (1s)^-2

=(1kg/2kg) (1m/5m)² (1s/1ms)^-2 new energy units

= 2 x 10^-8 new energy units

Also 100 divisions on the celsius scale = 100 divisions on the kelvin scale = 180 divisions on the Fahrenheit scale.

∴ 1K = 1.8°F

(1K)^-1 = (1.8)^-1 (0F)^-1

∴ R = 8.32 J (mole^-1) (K)^-1

= 8.32 (2x10^-8 new energy units) (mole^-1) [(1.8)^-1(°F)^-1]

= 16.64/1.8 x 10^-8 (new energy unit) (mole)^-1(°F)^-1

= 9.23 x 10^-8 (new energy unit) (mole)^-1(°F)^-1 

= (9.2 x 10^-8) (new energy unit) (mole)^-1(°F)^-1