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Gases can be studied by considering two theories, either by considering the small scale activity of the individual molecules present in the gas or by considering the gas as a whole in a large scale activity. To study the small-scale activity of the molecules present in the gas, we must have to use a theoretical model. This theoretical model is known as the Kinetic theory of gases. It was founded by the British scientist James Clerk Maxwell and the Austrian physicist Ludwig Boltzmann in the 19th century.

This theory assumes that molecules are tiny in size in relation to the distance between the molecules and the molecules are in constant random motion and regularly collide with each other as well as the walls of the container in which they are stored. This is what you learn in the kinetic theory of gases for JEE.

The individual molecules present in gases possess the standard physical properties of an element, such as mass, momentum, energy. The sum of masses of all the molecules divided by the volume that the gas occupies gives the density of the gas. The measure of the linear momentum of the gas is considered as the pressure of the gas. When the molecules of the gas collide against the walls of the container they are in, the molecules impart some momentum on the walls that produce a force that can be measured. The temperature of a gas is a measure that shows the mean kinetic energy of the gas. The molecules of the gas are in constant motion and produce energy from that motion. The temperature is directly proportional to the motion. These are some important points covered under the topic of the kinetic theory of gases IIT JEE syllabus.

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The kinetic theory of gases is based upon some assumptions. Let us discuss some of these assumptions.

The molecules or atoms of the gases are in constant random motion; they do not remain still at any time.

The molecules or atoms of the gas are repeatedly colliding against each other as well as the walls of the container the gas is stored in.

The particles of gases are very small, and the total volume occupied by these particles is considered negligible if compared with the total volume of the container.

There is a negligible force of attraction between the particles of the gases.

The average kinetic energy of gas particles is directly proportional to the actual temperature of that particular gas, and all gases at the same temperature have equal kinetic energy.

Boltzmannâ€™s Constant

k_{B} = nR/N

k_{B} is the Boltzmannâ€™s constant

R is the gas constant

n is the number of moles

N is the number of particles in one mole (the Avogadro number)

Total Translational K.E of Gas

K.E = (3/2)nRT

n is the number of moles

R is the universal gas constant

T is the absolute temperature

Maxwell Distribution Law

V_{rms} ï¼ž Vï¼ž Vp

V_{rms} is the RMS speed.

V is the Average speed.

V_{p} is the most probable speed.

RMS Speed (Vrms)

V_{rms} = \[\sqrt{8kt/m} = \sqrt{3RT/M}\]

R is the universal gas constant.

T is the absolute temperature.

M is the molar mass.

Average Speed

\[\overrightarrow{v} = \sqrt{8kt/\pi m} = \sqrt{8RT/\pi M}\]

Most Probable Speed (Vp)

\[V _\rho = \sqrt{2kt/m} = \sqrt{2RT/M}\]

The Pressure of Ideal Gas

\[P = \frac{1}{3} V^{2} rms\]

P is the density of molecules.

Equipartition of Energy

\[K = \frac{1}{2} K_{B}T\] for each degree of freedom.

K = (f/2) KÐ²T for molecules having f degrees of freedom.

K_{B} is the Boltzmannâ€™s constant.

T is the temperature of the gas.

Internal Energy

U = (f/2) nRT

For n moles of an ideal gas.

Example 1Â

What is the average velocity or root mean square velocity of a molecule in a sample of oxygen at 0-degree celsius?

Solution

R = 8.3145 (kg.m^{2}/sec^{2})/ K.mol

T = Absolute temperature in kelvin

M = mass of the moles present in the gas considered in kilograms.

T = â„ƒ + 273

T = 0 + 273

T = 273 K

Molar mass of oxygen = 2 ï½˜ 16

Molar mass of oxygen = 32 g/mol

Convert this to kg/mol

Molar mass of oxygen = 32g/mol ï½˜ 1 kg/1000g

Molar mass of oxygen = 3.2 ï½˜10^{-2}kg/mol

Vrms = \[\sqrt{3RT/M}\]

= [ 3( 8.3145 (kg.m^{2}/sec^{2})/ K.mol)(273 K) / 3.2 ï½˜10^{-2}kg/mol ]^{1/2}

=( 2.128 ï½˜ 10^{5}m^{2}/sec^{2})^{1/2}

= 461 m/sec

RMS = 461 m/sec.

FAQ (Frequently Asked Questions)

1. What is Boltzmann Constant?

Answer: Max Plank introduced the Boltzman Constant. It was named after Ludwig Boltzmann. It is considered as a physical constant, which is obtained by taking the ratio of two constants that are designated as gas constant and Avogadro number. The Boltzmann constant is used in various diverse disciplines of physics like it is used for expressing the equipartition of the energy of an atom, and it is also used to express the Boltzmann factor. This constant plays a significant role in the statistical definition of entropy. You can learn all this in the kinetic theory of gases IIT JEE syllabus.

2. What is RMS?

Answer: RMS stands for Root Mean Square. The root mean square voltage or current can also be considered as time-averaged voltage or current in an AC system. The root mean square that can also be considered as a quadratic mean is defined as a statistical measure of the magnitude of a varying quantity. It basically is useful when the function deviates between negative and positive values. The RMS value of a set of values is the square root of the arithmetic mean of the squares of the original values. This is also taught in the kinetic theory of gases JEE syllabus.