Question:

If average velocity becomes $4$ times then t-hat will be the effect on rms velocity at that Temperature?

Updated On: May 19, 2022
  • 1.4 times
  • 4 times
  • 2 times
  • $ \frac{1}{4} $ times
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The Correct Option is B

Solution and Explanation

Ratio of $v_{a v} / v_{r m s}$ remains constant.
Average speed is the arithmetic mean of the speeds of molecules in a gas at a given temperature,
i.e., $v_{a v}=\left(v_{1}+v_{2}+v_{3}+\ldots\right) / N$
and according to kinetic theory of gases,
$v_{a v}=\sqrt{\frac{8 R T}{M \pi}} $... (i)
Also, rms speed (root mean square speed)
is defined as the square root of mean of squares of the speed of different molecules, i.e., $v_{r m s}=\sqrt{\left(v_{1}^{2}+v_{2}^{2}+v_{3}^{2}+\ldots\right) / N}$
$=\sqrt{(\bar{v})^{2}}$
and according to kinetic theory of gases,
$v_{r m s}=\sqrt{\frac{3 R T}{M}} $... (ii)
From Eqs. (i) and (ii), we get
$v_{a v}=\sqrt{\left(\frac{8}{3 \pi}\right)} v_{r m s}$
$=0.92 v_{r m s} \ldots$ (iii)
Therefore, $\frac{v_{a v}}{v_{r m s}}=$ constant
Hence, root mean square velocity is also become $4 $ times.
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Concepts Used:

Kinetic Molecular Theory of Gases

Postulates of Kinetic Theory of Gases:

  • Gases consist of particles in constant, random motion. They continue in a straight line until they collide with each other or the walls of their container. 
  • Particles are point masses with no volume. The particles are so small compared to the space between them, that we do not consider their size in ideal gases.
  • Gas pressure is due to the molecules colliding with the walls of the container. All of these collisions are perfectly elastic, meaning that there is no change in energy of either the particles or the wall upon collision.  No energy is lost or gained from collisions. The time it takes to collide is negligible compared with the time between collisions.
  • The kinetic energy of a gas is a measure of its Kelvin temperature. Individual gas molecules have different speeds, but the temperature and
    kinetic energy of the gas refer to the average of these speeds.
  • The average kinetic energy of a gas particle is directly proportional to the temperature. An increase in temperature increases the speed in which the gas molecules move.
  • All gases at a given temperature have the same average kinetic energy.
  • Lighter gas molecules move faster than heavier molecules.