Kinetic Theory of Gases - Practice Questions & MCQ

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For gaseous state ,if most probable speed is denoted by C*,average speed by $\bar{C}$ and mean square speed by C ,then for a large number of molecules the ratios of these speeds are:

$C^{*}:\bar{C}:C= 1:1.225:1.128$

$C^{*}:\bar{C}:C= 1.225:1.128:1$

$C^{*}:\bar{C}:C=1.128: 1.225:1$

$C^{*}:\bar{C}:C= 1:1.128:1.225$

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Kinetic Energy and Molecular Speeds

Kinetic Energy and Molecular Speeds
As you have studied in the previous section the molecules of a gas are always in motion and are colliding with each other and with the walls of the container. Due to these collisions, the speeds and the kinetic energies of the individual molecules keep on changing. However, at a given temperature, the average kinetic energy Of the gas molecules remains constant.

If at a given temperature, n₁, molecules have speed u₁, n₂, molecules have speed u₂, n₃ molecules have speed u₃, and so on. Then, the total kinetic energy (EK) of the gas at this temperature is given by:
$\mathrm{E_{K}=\frac{1}{2} m\left(n_{1} v_{1}^{2}+n_{2} v_{2}^{2}+n_{3} v_{3}^{2}+\ldots \ldots \ldots\right)}$
where m is the mass of the molecule. The corresponding average kinetic energy $\overline{E_{k}}$ of the gas will be:

$\\\overline{\mathrm{E}_{\mathrm{K}}}=\frac{1}{2} \frac{\mathrm{m}\left(\mathrm{n}_{1} \mathrm{v}_{1}^{2}+\mathrm{n}_{2} \mathrm{v}_{2}^{2}+\mathrm{n}_{3} \mathrm{v}_{3}^{2}+\ldots \ldots \ldots\right)}{\left(\mathrm{n}_{1}+\mathrm{n}_{2}+\mathrm{n}_{3}+\ldots \ldots \ldots\right)}\\\\\\\mathrm{{ If \: the \: term \: } \frac{\left(n_{1} v_{1}^{2}+n_{2} v_{2}^{2}+n_{3} v_{3}^{2}+\ldots \ldots . .\right)}{\left(n_{1}+n_{2}+n_{3}+\ldots . . . .\right)}=\overline{v}^{2}}\\\\\mathrm{then \: the\: average\: kinetic\: energy\: is\: given\: by:}\\\\\mathrm{\overline{E_{\kappa}}=\frac{1}{2} m \overline{v^{2}}}\\\\\mathrm{where\: v \: is\: given\: by }\\\\\mathrm{v=\sqrt{\frac{\left(n_{1} v_{1}^{2}+n_{2} v_{2}^{2}+n_{3} v_{3}^{2}+\ldots \ldots \ldots\right)}{\left(n_{1}+n_{2}+n_{3}+\ldots \dots \dots\right)}}}$
This 'v' is known as root-mean-square speed u_rms

Maxwell-Boltzmann Distribution of speeds

According to it

Molecules have different speeds due to frequent molecular collisions with the walls and among themselves.
Rare molecules have either very high or very low speed.
Maximum number of molecules of the gas have maximum velocity which is called most probable velocity and after Vmp velocity decreases.
Zero velocity is not possible.
All these velocities increase with the increase in temperature but fraction of molecules having these velocities decreases.

Average Speed, u_av
It is the average of different velocities possessed by the molecules

$\begin{aligned} u_{a v} &=\frac{u_{1}+u_{2}+u_{3}}{n}\\ u_{a v} &=\frac{n_{1} u_{1}+n_{2} u_{2}+n_{3} u_{3}}{n_{1}+n_{2}+n_{3}} \end{aligned}$

Here n₁, n₂, n₃ are the number of molecules having u₁, u₂, u₃ velocities respectively.
Relation between U_av, temperature and molar mass is given as

$\mathrm{u_{a v}=\sqrt{8 R T / \pi M}=\sqrt{8 P V / \pi M}}$

Most Probable Speed, u_mp
The most probable speed(u_mp) of a gas at a given temperature is the speed possessed by the maximum number of molecules at that temperature. Unlike average speed and root mean square speed, the most probable speed cannot be expressed in terms of the individual molecular speeds.

The most probable speed(u_mp) is related to absolute temperature (T) by the expression:

$u_{m p}=\sqrt{\frac{2 R T}{M}}=\sqrt{\frac{2 P V}{M}}$

Root Mean Square Speed u_rms
It is the square root of the mean of the square of the velocities of different molecules.
$\begin{array}{l}{u_{r m s}=\frac{\sqrt{u_{1}^{2}+u_{2}^{2}+\ldots \ldots . .}}{n}} \\ {=\frac{\sqrt{n_{1} u_{1}^{2}+n_{2} u_{2}^{2}+n_{3} u_{3}^{2}}}{n_{1}+n_{2}+n_{3}}} \\ {u_{r m s}=\sqrt{3 R T / M}} \\ {u_{r m s}=\sqrt{3 P V / M}=\sqrt{3 P / d}}\end{array}$

Relation between u_av, u_mp and u_rms
The three types of molecular speeds, namely, most probable speed(v_mp), average speed (v_av) and root mean square speed(v_rms) of a gas at a given temperature are related to each other as follows:
$\begin{array}{l}{v_{m p} : v_{a v} : v_{m s}=\sqrt{\frac{2 R T}{M}} : \sqrt{\frac{8 R T}{\pi M}} : \sqrt{\frac{3 R T}{M}}} \\ {v_{m p} : v_{a v} : v_{r m s}=1.414 : 1.596 : 1.732} \\ {v_{m p} : v_{a v} : v_{r m s}=1 : 1.128 : 1.224}\end{array}$

For a particular gas, at a particular temperature:
v_mp < v_av < v_rms

It follows from the above relationship that:
Average speed(v_av) =0.921 x Root mean square speed(v_rms)
Most probable speed(v_mp) = 0.817 x Root mean square speed(v_rms)

Kinetic Molecular Theory of Gases

All the gas laws that we have discussed like Boyle's law, Charles' Law, Avogadro's Law are merely based on the experimental evidence. There was no theoretical background to justify them. So, the scientists were curious to know why the gases behave in a peculiar manner under certain set of conditions. From Charles' law we got to know that the gases expand on heating. But there was no theory to give the reason for such fact. So, there was a need for some theory which could tell about the happenings at the molecular level and so could answer all the questions arising regarding the behaviour of gases.

Later a theory was given called kinetic molecular theory of gases to provide a sound theoretical basis for various gas laws. The kinetic theory of gases is based on the following assumptions or postulates:

Actual volume of gas molecules is negligible in comparison to the total volume of the gas: Postulate says that all the gases are made up of extremely small particles called molecules dispersed throughout the container. These particles are so small that they are regarded as point masses. As they are point masses, so the actual volume occupied by the gas molecules is negligible in comparison to the total volume of the gas.

Support for the assumption. This assumption explains the great compressibility of gases because there is a lot of empty space between the gas molecule.
No force of attraction between the gas molecules: As the distance between the gas molecules is very large, so it is assumed that there is no force of attraction between the gas molecules at ordinary temperature and pressure.
Support for the assumption: Due to no force of attraction between the gas molecules, therefore the gases easily expand and occupy all the space available to them on heating.
Particles of gas are in constant random motion: Particles of gas are in a state of constant random motion.
Support for the assumption: This assumption is supported by the fact that gases do not have a fixed shape because of their random motion.
Particles of gas collide with each other and with the walls of the container: ParticLes of gas move in a straight line with high velocities in all the possible directions. During their this motion, they collide with each other and with the walls of the container in which gas is enclosed and even change direction upon collisions.

Support for the assumption. Gas exerts some pressure. Pressure of the gas exerted is just because of the collisions of particles with the walls of the container.
Collisions are perfectly elastic: When the gas molecules collide with each other they pass on their energies. There is a transfer of energy from one colliding molecule to the other but the total energy of molecules before and after the collision remains the same therefore, the collisions are called perfectly elastic. So, there is no net loss of energy.
Support for the assumption: As there is no loss of kinetic energy, therefore the motion of molecules do not cease so, the gases never settle down.
Different particles of the gas, have different speeds: Different particles of gas possess different kinetic energies, therefore they have different speeds at a particular time.
Support for the assumption: This postulate is reasonable as when the molecules collide, they change their speed. Even though the initial speeds are same, but after collisions, there is transfer of energy from one molecule to the other. So, as the energy changes after the collisions, so do the speeds. But the distribution of speeds remains constant at a particular temperature.
The average kinetic energy of the gas molecules is directly proportional to the absolute temperature: As discussed in the above assumption the speed of a molecule changes with time, i.e. the speed of a molecule is variable. Therefore we talk about the average kinetic energy of the molecules Kinetic molecular theory of gases establishes a link between the molecular motion and temperature. As the temperature increases, so the kinetic energy also increases