A hypothetical gas sample has its molecular speed distribution graph as shown in  the figure. Speed u and dNdu  have appropriate limits. The root mean square speed  of the molecules is N3. Find N. (Do not worry about units).

By definition,  vRMS=∫0N0u2dNN0, where  N0 is total number of molecules.Equation of line is:  dNdu=4−u⇒dN=4−udu ⇒vRMS=∫04u24−uduN0  (Here integration only runs from 0 to 4 because max value  of u is 4.Also, area under the graph =  ∫dNdudu=1244=8=N0Hence,  vRMS=∫04u24−udu8=182563−64=83.