(a) Calculate (i) root-mean-square speed and (ii) the mean energy of 1 mol of hyderogen at STP given that density of hydrogen is `0.09 kg//m^(3)`. (b) Given that the mass of a molecule of hydergen is `3.34 xx 10^(-27)` kg, calculate Avogadro's number. (c ) Calculate Boltmann's constant.

Explanation

(i) We know that the pressure of a gas is given as
`P = (1)/(2) rho_(rms)^(2)`
`v_(rms) = sqrt(((3 P)/(rho)))`
`= sqrt(((3 xx 0.76 xx 13.6 xx 10^(3) xx 9.8)/(0.09)))`
`= 1837 m//s = 1.837 m//s`
(ii) Kinetic energy `= (1)/(2) mv_(rms)^(2)`
Here `M = 2 g = 2 xx 10^(-3) kg`
`KE = (1)/(2) xx 2 xx 10^(-3) xx (1837)^(2)`
`= 3374.56 J`
(b) Mass of one mulecule of `H_(2) = 3.34 xx 10^(-2) kg`
Molecule mass of hydrogen `= 2 xx 10^(-3) kg`
Avogadro's number `N_(A)`, which is the number of molecules in one gram molecule of hyderogen, givne by
`N_(A) = (2 xx 10^(-3))/(3.34 xx 10^(-27)) = 5.988 xx 10^(23)` molecules
c. We know that `K = R//N_(A) = 8.3//(5.988 xx 10^(23))`
`= 1.37 xx 10^(-23) J//mol K`