# 21. Physics | KTG | Translational Kinetic Energy of a Gas Molecule | by Ashish Arora (GA)

now we’ll study the , translational ki-netic
energy of a gas molecule. as we know, for a gas molecule. its ki-netic energy is written
as. is half , m dash. v r-m-s square , here this m dash is the mass of one molecule. and
v r-m-s is the root mean square velocity, if we substitute the value of this r-m-s velocity
will see. it is half m dash. is three r t by, m. in this situation we know well that,
the molar mass of. gas can be written as mass of one mole of gas molecule that is mass of
avogadro number of gas molecules. so it can be written as m dash multiplied by avogadro
number . if we substitute it here will get half, m dash, three r t by, m dash avogadro
number. here this m dash gets canceled out. r by avogadro number we already studied it
is given as boltzmann constant. so we replace it , the result will be three by two, kt.
this is the total , translation ki-netic energy of a gas molecule. and no matter in this situation
whatever be the type of gas it wont make any difference, so we can say for. all gases.
at a given temperature. translational ki-netic energy. of each molecule. is same. and this,
translational ki-netic energy is directly proportional to the . absolute temperature
of , a gas. again , i repeat this wont make any difference what are be the type of gas,
for all gases at a given temperature translational ki-netic energy of each gas molecule is given
by. three by two k t where k is the boltzmann constant , and t is the absolute temperature
of gas. similarly if we calculate the translational ki-netic energy for n moles of a gas we can
write. for n moles of a gas. total translational ki-netic energy. of gas can be given as. for
n moles it can be written as three by two k–t which is the, translational ki-netic
energy of one gas molecule it is multiplied by, the total number of molecules so for n
moles number of molecules are n , multiplied by avogadro number. here, k and avogadro number
the product . can be written as r, the gas constant. as we know the gas constant is boltzmann
constant , multiplied by avogadro number, so. this can be written as three by two n
r t. this also you need to keep in mind, for n moles of any gas the total translational
ki-netic energy is always, written as three by two n r t.