Question

For each of the following pairs of quantities, tell which one is larger and explain your answer. If they are equal, say so and explain how you can tell. (“KE” means kinetic energy)
a) The rms velocity of nitrogen at 25 C or the rms velocity of oxygen at 75 C.
b) The most probable KE for argon at 25 C or the most probable KE for neon at 25 C.

c) The average velocity of hydrogen or the average velocity of helium at the same temperature.

d) The fraction of CH4 molecules with velocities greater than 500 m/sec, or the fraction of NH3 molecules with velocities greater than 500 m/sec at the same temperature.
e) The fraction of He atoms with kinetic energies between 100 and 200 J/mol, or the fraction of Ne atoms with kinetic energies between 100 and 200 J/mol, if both gases are at
100 C.

Answer 1

a) rms ( root mean square ) velocity formuls is

_rms = √3RT/M_m

_{For Oxygen}

     _rms = √3×8.314(J/K mol) × 348.15K/0.015999kg/mol

= √5.43×10^5 J/kg

=√5.43×10^5(m^2/s^2)

= 736.9m/s

For Nitrogen

_rms= √ 3× 8.314(J/mol K ) × 298.15K/0.014007kg

= 728.7m/s

Therefore,

rms velocity of Oxygen at 75℃ is larger

b) most probable kinetic energy is calculated as follows

KE_mp = (1/2)× K_B × T

K_B is constant ( Boltzeman constant)

T is temperture which same

Therefore,

Most probable KE for Argon and neon will be equal

c) Avarage velocity is calculated as follows

   _avarage = √8kT/πm

Molecular Mass of Hydrogen is lower than atomic mass of Helium

From equation we know that atomic or molecular mass of gas is inversly proportional to average speed

Therefore,

Hydrogen will have the large average speed

d) Molar mass of CH4 is 16g

Molar mass of NH3 is 17g

CH4 is lighter molecule comparing to NH3

Most proble velocity is larger for CH4 than NH3

Most probable velocity means velocity which most of the molecules having

Therefore,

At same temperture ,fraction of CH4 molecule having velocities greater than 500m/s is larger than fraction of NH3 molecule having velocities greater than 500m/s

e) Most probable Kinetic energy is calculated as follows

KE_mp = (1/2)K_BT

where,

K_B is constant

so, K.E depands on temperature and not on molecules

Therefore,

Both Argon and Neon should have the same fraction