Consider a cubical vessel with perfectly elastic walls containing large number of molecules say N let l be the dimension of each side of the cubical vessel.
v1x , v1y, v1z be the x, y, and z component of a molecule with velocity v.
Consider the motion of molecule in the direction perpendicular to the face of cubical vessel.
Molecule strikes the face A with a velocity v1x and rebounds with the same velocity in the backward direction as the collisions are perfectly elastic.
If m is the mass of molecule, the change in Momentum during collision is
mv1x - (-mv1x) = 2 mv1x (1)
The distance travelled parallel to x-axis an is between A to A´ and when molecule rebounds from A´ and travel towards A is 2L
Time taken by molecule to go to face A´ and then comeback to A is
Δt = 2l/v1x
Number of impacts of this molecule with A in unit time is
n = I/Δt = v1x/ 2l (2)
Rate of change of momentum is
ΔF = ΔP/Δt
from (1) and (2)
ΔF = mv1x2 / l
this is the force exerted on wall A due to this molecule.
Force on wall A due to all other molecules
F = Σmv1x2/L (3)
As all directions are equivalent
Σv1x2= 1/3Σ((v1x)2 + (v1y)2 +( v1z)2 )
= 1/3 Σv12
Thus F = (m/3L) Σv12
N is total no. of molecules in the container so
F = (mN/3L) (Σ(v1)2/N)
Pressure is force per unit area so
P = F/L2
where ,M is the total mass of the gas and if ρ is the density of gas then
since Σ(v1)2/N is the average of squared speeds and is written as vmq2 known as mean square speed
Thus, vrms=√(Σ(v1)2/N) is known as root mean squared speed rms-speed and vmq2 = (vrms)2
Pressure thus becomes
P = (1/3)ρvmq2 (4)
or PV = (1/3) Nmvmq2 (5)
from equation (4) rms speed is given as
vrms = √(3P/ρ)
= √(3PV/M) (6)
link to this page by copying the following text Also Read
Thanks for visiting our website. DISCLOSURE: THIS PAGE MAY CONTAIN AFFILIATE LINKS, MEANING I GET A COMMISSION IF YOU DECIDE TO MAKE A PURCHASE THROUGH MY LINKS, AT NO COST TO YOU. PLEASE READ MY DISCLOSURE FOR MORE INFO.
Our aim is to help students learn subjects like
physics, maths and science for students in school , college and those preparing for competitive exams.