# Which one are correct statements, if \[{u_{r.m.s}}\], \[{u_{MP}}\] and \[{u_{AV}}\] are roots mean square, most probable and average speed of an ideal monatomic gas at absolute temperature T, having a mass of one molecule m?

This question has multiple correct options.

A.No, molecule can have speed greater than \[\sqrt 2 {\text{ }}{u_{r.m.s}}\]

B. No, molecules can have speed less than \[\dfrac{{{u_{MP}}}}{{\sqrt 2 }}\].

C. \[{u_{MP}} \prec {u_{AV}} \prec {u_{r.m.s}}\]

D. The average kinetic energy of a molecule is \[\dfrac{3}{4}m{\left( {{u_{MP}}} \right)^2}\].

Answer

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**Hint:**Root mean square speed is the square root of the mean speed of each individual particle that moves in a random direction. The most probable speed is the minimum speed of the particles. Thus we will find the relation between them. Also, we will find the kinetic energy of particles due to their random motion.

**Complete answer:**

When gas is completely filled in a container then the particles move in random motion due to collision and thermal conditions. The velocity with which particles move is not equal for all particles. Thus we calculate the root mean square speed of all the particles which move in a random direction. The root means square speed, \[{u_{r.m.s}}\] is the square root of the mean speed of the speed of all particles. The most probable speed is the minimum speed of the particle which is moving inside the container. The average speed of a particle is the average of all the speeds of all particles in the container.

All these speeds are interrelated as:

\[{u_{rms}}:{u_{AV}}:{u_{MP}} = \sqrt {\dfrac{{3RT}}{M}} :\sqrt {\dfrac{{8RT}}{{\pi M}}} :\sqrt {\dfrac{{2RT}}{M}} = \sqrt 3 :\sqrt {\dfrac{8}{\pi }} :\sqrt 2 \]

Thus we can say that the root means the square speed of the particle is greatest and the most probable speed is the lowest among all of the speed of particles. Also the kinetic energy of a particle is given by \[\dfrac{3}{4}m{\left( {{u_{MP}}} \right)^2}\]. Therefore we can say that the correct option among the above-given options is option C and option D.

**Note:**It must be noted that molecule can have speed greater than \[\sqrt 2 {\text{ }}{u_{r.m.s}}\] and also molecule can have speed less than \[\dfrac{{{u_{MP}}}}{{\sqrt 2 }}\]. The speed of a particle is dependent on the molar mass of the gas. When the molar mass of gas increases then the speed of the particle of gas decreases as the mass of gas is inversely proportional to the speed of the particle.

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