In a gaseous helium-filled cooling cell of an ion trap mass spectrometer, injected molecular ions are slowed down by a series of n n\,n random elastic collisions with helium atoms.
The kinetic energy En E_n\,En of a molecular ion after n n\,n collisions is modelled by the formula:
En=E0e−αn E_n = E_0 e^{-\alpha n} En=E0e−αnwhere E0 E_0\,E0 is the initial kinetic energy of the molecular ion and α=0.18\alpha = 0.18α=0.18.
A molecular ion is considered thermalised when its kinetic energy is reduced to the mean kinetic energy of a helium gas atom. The helium buffer gas is maintained at a temperature of 135 K.
A particular molecular ion enters the cooling cell with an initial kinetic energy of 3.5 keV3.5\text{ keV}3.5 keV.
Calculate the minimum number of collisions n n\,n required for this molecular ion to become thermalised.
(Boltzmann constant kB=1.38×10−23 J K−1k_B = 1.38 \times 10^{-23}\text{ J K}^{-1}kB=1.38×10−23 J K−1, elementary charge e=1.60×10−19 Ce = 1.60 \times 10^{-19}\text{ C}e=1.60×10−19 C)
