[Physics Class Notes] on Kinetic Interpretation of Temperature and RMS Speed of Gas Molecules Pdf for Exam

The kinetic theory states that the average kinetic energy of gas molecules of an ideal gas is directly proportional to the absolute temperature of the molecules.

It is independent of the pressure, volume, and nature of the gas.

Such a kind of interpretation of temperature is called the kinetic interpretation of temperature.

The entire structure of the kinetic theory is based on the following assumptions that were stated by Classius.

What are the Assumptions Made About Molecules in an Ideal Gas?

The major assumptions made about the molecules in an ideal gas are:

• There are a large number of molecules present in gas and all the molecules of gas are identical to each other and have the same mass.

• The molecules of gases obey Newton’s law of motion and are therefore in a state of continuous motion which is not only random but also isotropic, that is, the motion of the molecules is the same in all directions.

• The molecules of the gases are much smaller in size as compared to the average distance between the molecules of the gases. Also, the total volume of the gas is much less than the volume of the container in which they are contained.

• The molecules of gases make elastic collisions with not only each other but also with the walls of the container in which they are contained.

• Van der Waal forces like gravitation and attraction acting on the molecules of the gases are assumed to be negligible.

Here, We Considered Ideal Gas, Which is a Perfect Gas With the Characteristics:

1. The size of the molecule of a gas is zero i.e., it has a point mass with no dimensions.

2. The molecules of gas do not exert any force of attraction or repulsion on each other except during collision. 

 

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Consider one mole of an ideal gas at absolute temperature T, of volume V and molecular weight M. Let N be Avogadro’s number and m be the mass of each molecule of gas.

Then by the relation:  

                                [M=mtimes N]

If C is the r.m.s velocity of the gas molecules, then pressure P exerted by an ideal gas is

                     [P=frac{1}{3}frac{M}{V}C^2] or [PV=frac{1}{3}frac{M}{C^2}]

For a gas equation, PV = RT, and R is a universal gas constant 

So, 

 [frac{1}{3}frac{M}{C^2} = RT] or [frac{3}{2}RT]

  Since, M = m x N

      [frac{1}{2}mC^2] = [frac{3}{2} frac{R}{NT}] 

               [frac{1}{2}mN = frac{3}{2}kT]       

                                   (∵ R/N = k)

Kinetic Interpretation of Temperature

According to the kinetic theory of gases, the pressure P exerted by one mole of an ideal gas is given by,

[P=frac{1}{3}frac{M}{V}C^2]or[PV=frac{1}{3}frac{M}{C^2}]or[frac{1}{3}MC^2=RT]

           [=C^2=frac{3RT}{M} or C^2alpha T] (∵ R and M are constants)

           [=Calpha sqrt{T}alpha C]      

We can say that the square root of absolute temperature T of an ideal gas is directly proportional to the root mean square velocity of its molecules.

Also,

                        [frac{1}{3}frac{M}{NC^2}= frac{R}{NT}=kT]

Or,                [frac{1}{2}mC^2=frac{3}{2} kT or frac{1}{2}mC^2alpha T]

Where, 3/2 k is constant.

But, [frac{1}{2}mC^2] is the average translational kinetic energy per molecule of a gas.

Therefore, the average kinetic energy of a translational per molecule of a gas is directly proportional to the absolute temperature of the gas.

Hence, we can define absolute temperature as the temperature at which the root mean square velocity of the gas molecules reduces to zero, which means molecular motion ceases at absolute zero.

What is the Significance of Kinetic Interpretation of Temperature and Root Mean Square Speed of Gaseous Molecules?

Kinetic interpretation of temperature is very significant in the field of the kinetic theory of gases. The average kinetic energy of a molecule does not depend on the type of molecule under consideration.  The average translational kinetic energy is only dependent upon the absolute temperature of the surroundings. Compared to macroscopic energies, the kinetic energy is very small because of which we do not feel when an air molecule hits our skin. 

On the other hand, the gravitational potential energy acting on a molecule is comparatively negligible when the molecule moves from the top of a room to the bottom of a room. Therefore, with this observation, we can conveniently neglect gravitational energy in typical real-world situations. The RMS speed of the gaseous molecules is surprisingly large. These large molecular velocities do not allow macroscopic movement of air because the molecules move in all directions. The mean free path (that is, the average distance of molecules between collisions) of molecules in the air is very small, therefore, the molecules move rapidly but do not cover a very large distance in a second. 

What are the Properties Defined by the Kinetic Theory of Gases?

The kinetic theory of gases helps us understand and define various properties of gases like temperature, volume, pressure, viscosity, mass diffusivity, and thermal conductivity.

 RMS Speed of Gas

Consider an ideal gas contained in a cubical container such that the volume,

V = a3.

Let n be the number of molecules and mass of each molecule be m. 

So, [M=mtimes N]

 

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The random velocities of gas molecules are A₁, A₂, A₃,…, An, = (x₁, y₁, z₁), (x₂, y₂, z₂),…., (x_n, y_n, z_n) be the rectangular components of the velocities  c₁, c₂, c₃,…,c_n along with three mutually perpendicular directions OX, OY and OZ.

So, x₁2 + y₁2 + z₁2 = c₁2……  x_n2 + y_n2 + z_n2 = c_n2

If x₁ is the component of velocity of the molecule A₁ along OX, and the initial momentum of  A₁ along OX = mx₁,

Momentum after collision = – mx₁

So, total momentum = – mx₁ – mx₁ = – 2mx₁

However, according to the law of conservation of momentum in one dimension, momentum is transferred to the wall by molecule A₁ = 2mx₁.

Time between two successive collisions,

                  [T= frac{D}{S}=frac{2a}{x_1}]

So, momentum transferred by  A₁ = 2mx₁ * x₁/2a = m x₁2/a

Since,  P_x (Pressure along x-axis)[= frac{F}{a^2}=frac{m}{a^3}(x_1^2+x_2^2+x_n^2)]

Similarly, [P_y=frac{m}{a^3}(P_x+P_y+P_z)]

So, [P=frac{(P_x+P_y+P_z)}{3}]

[P=frac{1}{3}frac{m}{a^3}left((x_1^2+x_2^2+..+x_n^2)+(y_1^2+y_2^2+..+y_n^2)+(z_1^2+z_2^2+..+z_n^2) right )]

    [=frac{1}{3}frac{m}{V}(c_1^2+c_2^2+..+c_n^2)=frac{1}{3}frac{mn}{V}frac{(c_1^2+c_2^2+..+c_n^2)}{n}]

As, [C=frac{(c_1^2+c_2^2+..+c_n^2)}{n}]

We get,

       [P=frac{M}{3V}C^2]

 

RMS Velocity of Gas

The RMS or root means square velocity of a gas is defined as the square root of the means of the squares of the random velocities of the individual molecules of a gas.

Putting, [frac{M}{V}=rho ], we get,      

[P=frac{1}{3 rho}C^2]

Or [C=frac{sqrt{3P}}{ rho}]

Hence, knowing the values of P and ρ, the RMS velocity of the gas molecules at a given temperature can be determined.

RMS Speed of Gas Molecules

The RMS speed of gas molecules is the measure of the speed of the particles in a gas. It is the average squared velocity of molecules in a gas.

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