Question

Helium atoms have a mass of 4u and oxygen molecules have a mass of 32u, where u is defined as an atomic mass unit (u=1.660540 x 10^−27kg). Compare a gas of helium atoms to a gas of oxygen molecules.
1) At what gas temperature would the average translational kinetic energy of a helium atom be equal to that of an oxygen molecule in a gas of temperature 300 K?
2) At what gas temperature would the root-mean-square (rms) speed of a helium atom be equal to that of an oxygen molecule in a gas at 300 K?

Answer 1

Answer:

300 K

37.5 K

Explanation:

$k_b$ = Boltzmann constant = $1.38\times 10^{-23}\ J/K$

T = Absolute temperature

$M_h$ = Mass of Helium atom = 4 u

$M_o$ = Mass of Oxygen atom = 32 u

$u=1.660540\times 10^{−27}\ kg$

The kinetic energy of a gas is given by

$K=\frac{3}{2}k_bT$

So, the temperature would be 300 K

RMS velocity is given by

$v=\sqrt{\frac{3k_bT}{M}}$

$\frac{T_h}{M_h}=\frac{T_o}{M_o}\\\Rightarrow T_h=\frac{T_o}{M_o}\times M_h\\\Rightarrow T_h=\frac{300\times 4}{32}\\\Rightarrow T_h=37.5\ K$

The gas temperature is 37.5 K