Kinetic Interpretation of Temperature and RMS Speed of Gas Molecules

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Give the Interpretation of Temperature According to Kinetic Theory

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 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.

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 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 = m x N

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

                      P = 1/3 M/V. C2 or PV =1/3MC2

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

So, 

                              1/3MC2 = RT or 1/2MC2 = 3/2RT

                               Since, M = m x N

                               1/2mC2 = 3/2R/NT 

                            1/2mN = 3/2kT        

                                   (∵ R/N = k)

Kinetic Interpretation of Temperature

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

            P = 1/3 M/V. C2 = PV = 1/3MC2 or 1/3MC2 = RT

            =    C2 = 3RT/M or C2α T ( R and M are constants)

            =     Cα√T or √T α 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,

                          1/3M/NC2 = R/N T = kT

Or,                       1/2mC2 = 3/2kT or 1/2mC2 α T

Where, 3/2 k is constant.

But, 1/2mC2 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.

 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 = m x N

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The random velocities of gas molecules is A₁, A₂, A₃,...,Aₙ, = (x₁, y₁, z₁),  (x₂, y₂, z₂),...., (xₙ, yₙ, zₙ) be the rectangular components of the velocities  c₁, c₂, c₃,...,cₙ along three mutually perpendicular directions OX, OY and OZ.

So, x2 +y2 +z2 =c2......  x2+ y2+ z2 =c2

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 = D/S = 2a/x₁

So, momentum transferred by  A₁ = 2mx₁ * x₁/2a = m x2/a

Since,  Pₓ (Pressure along x-axis) = F/a2 = m/a3 (x2 +x2+ x2)

Similarly, Pᵧ = m/a3 (y2+ y2+ y2)

So, P = (Pₓ + Pᵧ + Pz)/3

 P = 1/3m/a3 ( (x2 +x2+...+ x2)+ (y2+ y2+...+ y2) + (z2 + z2 +...+z2)

    = 1/3m/V(c2 + c2+..+c2) =  1/3mn/V(c2 + c2+..+c2)/n

As, C = (c2 + c2+..+c2)/n

We get,

       P = M/3VC2

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, M/V = ρ, we get,      

P =  1/3 ρ C2

Or C = √3P/ρ

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

FAQ (Frequently Asked Questions)

Q1: Why do We Use RMS Speed?

Ans: We use rms speed instead of average speed, because the molecules of a gas have a random motion. Therefore, their net velocity comes out to be zero. The rms speed formula defines both the diffusion and effusion rates.

Q2: Which Gas has the Highest Kinetic Energy?

Ans: At 100°C, the gases named nitrogen and helium have the highest kinetic energy because they have the highest temperature. Also, kinetic energy is directly proportional to the temperature.

Q3: What is Boyle's Law of Gas?

Ans: Boyle’s law is one of the fundamental gas laws, discovered by Robert Boyle in 1962.

It States That: 

At constant temperature, the volume V of given mass of gas is inversely proportional to its pressure, P, given by,

                            V α 1/P = K/P or PV = K

Where, K is a constant, which depends on the nature and temperature of the gas.

Q4: What is the Kinetic Gas Equation?

Ans: The kinetic gas equation is simple and straightforward, given by,

                                PV =nRT

Where P = Pressure

             V = Volume

             n  = Number of moles of gas

             R = Universal gas constant

             T =  Temperature