A hypothetical gas sample has its molecular speed distribution graph as shown in the figure. The speed (u) and `(dN)/(du)` have appropriate units. Find the root mean square speed of the molecules. Do not worry about units.

Options

A. `sqrt((5)/(3))`
B. `sqrt((8)/(3))`
C. `sqrt((7)/(3))`
D. None of these

Answer

B

Explanation

The equation of the straight line is
`(dN)/(du) = -u + 4`
`u_(rms)^(2)=(intu^(2)dN)/(intdN) = (overset(4)underset(0)(int)u^(2)(dN)/(du)du)/(overset(4)underset(0)(int)(dN)/(du)du)`
`= (overset(4)underset(0)(int)u^(2) (-u + 4)du)/("area under the graph")`
`u_(rms)^(2) = (intu^(2)dN)/(intdN) = (overset(4)underset(0)(int)u^(2)(dN)/(du)du)/(overset(4)underset(0)(int)(dN)/(du)du)`
`(overset(4)underset(0)(int)(-u^(3))du+4overset(4)underset(0)intu^(2)du)/((1)/(2)xx4xx4) = [-u^(4)/(4)+4.(u^(3))/(3)]_(0)^(4)/(8)`
`u_(rms)^(2) = (8)/(3) rArr u_(rms) = sqrt((8)/(3))`