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).

Question

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).

Infinity Learn · Accepted Answer

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 $$=$$  ∫$$dNdudu=1244=8=N0Hence$$,  $$vRMS=$$∫$$04u24$$−$$udu8=182563$$−$$64=83$$.

Kinetic theory of ideal gases

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A hypothetical gas sample has its molecular speed distribution graph as shown in the figure. Speed u and $\frac{d N}{d u}$ have appropriate limits. The root mean square speed of the molecules is $\sqrt{\frac{N}{3}}$ . Find N. (Do not worry about units).

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Kinetic theory of ideal gases

Remember concepts with our Masterclasses.

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).

Ready to Test Your Skills?

free study material

A hypothetical gas sample has its molecular speed distribution graph as shown in the figure. Speed u and $\frac{d N}{d u}$ have appropriate limits. The root mean square speed of the molecules is $\sqrt{\frac{N}{3}}$ . Find N. (Do not worry about units).