Kinetic Theory: Laws and Equations

Kinetic theory of gases is a fundamental, historical model of the thermodynamic behavior of gases that provides several fundamental principles of thermodynamics.

• The kinetic theory of ideal gases is based on the molecular picture of matter.
• The kinetic theory of gases describes macroscopic characteristics such as volume, pressure, and temperature.
• An ideal gas is a gas that follows Boyle's law, Charles' law, Gay Lussac's law, and Avogadro’s law.
• Gases have no shape or size and can be contained in vessels of any shape and size.
• Since the molecules of the gases are apart from each other, they have negligible force of molecular interaction.
• Therefore, gases expand indefinitely and uniformly to fill the available space.
• Many scientists like Boyle and Newton tried to explain the behavior of gases. 
• However, the real theory was developed in the 19^th century by Maxwell and Boltzmann.
• This theory is known as the Kinetic theory, which explains the behavior of gases.

Key Terms: Boyle's law, Charles' law, Gay Lussac's law, Avogadro’s law, Ideal Gas, Degree of freedom, Thermal conductivity, rms speed, Dalton’s law of partial pressure, Kinetic energy, Specific heat.

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Molecular Theory of Matter

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About 200 years ago, John Dalton proposed the atomic theory.

According to this theory

• The smallest constituents of an element are atoms.
• Atoms of one element are identical but differ from those of other elements.
• A small number of atoms of each element combine to form a molecule of a compound.

He suggested this theory to explain the “Law of definite proportions” and “Law of multiple proportions”, which are obeyed by the elements when they combine to form compounds.

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Kinetic Theory

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The kinetic theory explains the behavior of an ideal gas on the basis of the fact that the gas consists of rapidly moving molecules and atoms.

• In general, the molecules of gases are in constant motion and they attack each other and the surfaces of vessels.
• This theory also shows that the particles of gas collide with each other and cause gaseous pressure due to the surface of the vessel.
• The kinetic theory of gases also describes properties such as temperature, coherence, and thermal conductivity. 
• In the 19th century Maxwell, Boltzmann, and others proposed this theory.

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Ideal Gas Equation

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The ideal gas strictly adheres to the basic principles of Boyle's law, Charles' law, Gay Lussac's law, and Avogadro’s law.

• An ideal gas is a gas that contains a group of randomly charged particles that meet only in an elastic collision.
• The relationship between pressure P, volume V, and absolute temperature T of gas can express the ideal gas equation.

The ideal gas equation is given below:

PV = nRT

Where

• P = Pressure
• V = Volume
• T = Temperature
• n = Number of moles (gas)
• R = Gas constant = 8.314JK⁻¹mol⁻¹

Now, any gas that follows this equation is called an Ideal gas.

Boyle’s Law

According to Boyle's law, the pressure exerted by a gas (mass kept at a constant temperature) is inversely proportional to the volume it occupies.

In other words, the pressure and volume of a gas are inversely proportional to each other as long as the temperature and volume of a gas are constant.

This law can be represented mathematically as follows:

P ∝ (1/V) ⇒ PV = k

Where

• P = Pressure applied by the gas
• V = Volume occupied 
• k = constant

The graph between pressure and a given volume of gas kept at a constant temperature is shown below:

According to Charles' law, the volume of the ideal gas is directly proportional to the absolute temperature at constant pressure.

The law states that the Kelvin temperature and volume are in direct proportion when the pressure on the dry gas sample is kept constant.

This law can be represented mathematically as follows:

V∝T ⇒ V/T = Constant ⇒ V_I /T_I = V_F /T_F

Where

• V_I = Initial volume
• V_F = Final volume
• T_I = Initial absolute temperature
• T_F = Final absolute temperature

Here we must remember that temperature is measured in Kelvin‌ and is the absolute temperature rather than in F or °C.

Gay Lussac’s Law (Pressure Law)

According to Gay-Lussac's law, the pressure exerted by a gas (kept at a given mass and constant volume) is directly proportional to the absolute temperature of the gas.

In other words, the pressure of a gas is proportional to the temperature of the gas when the volume is constant.

Mathematically this law can be represented as

P ∝ T ⇒ P/T = k

Where

• P = pressure applied by the gas
• T = temperature of the gas
• k = constant

The relationship between pressure and absolute temperature for a given mass (in a constant volume) can be graphically depicted as follows:

The Avogadro Law, also known as the Avogadro principle or the Avogadro hypothesis, states that at a constant temperature and pressure, the volume of a gas is directly proportional to the total number of atoms/molecules of gas occupied (that is the volume of the gaseous substance).

At constant pressure and temperature, Avogadro's law can be represented as:

V ∝ n ⇒ V/n = k

Where

• V = Volume of the gas
• n = amount of gaseous substance in moles
• k = constant

As the volume of the gaseous substance increases, the corresponding increase in the volume occupied by the gas can be calculated with the help of the following formula:

V₁/n₁ = V₂/n₂ = k (According to Avagadro’s rule)

The graphical representation of Avogadro's law (with the amount of the substance on the X-axis and the volume on the Y-axis) is shown below.

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Dalton’s Law of Partial Pressure

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According to Dalton's law of partial pressure, the entire pressure applied by a mixture of non-interacting gases is equal to the sum of the partial pressures applied by each gas within the mixture.

Let us consider a mixture of non-interacting ideal gases and μ_1, μ_2, μ₃….…… be the number of moles of the gases respectively, in a vessel of volume V at temperature T and pressure P. Then we can write

     PV = (μ₁ + μ₂ + μ₃ +…..)RT

⇒ P = μ₁RT/V + μ₂RT/V + μ₃RT/V +……

⇒ P = P₁ + P₂ + P₃ +……

Where, P₁ = μ₁RT/V is the pressure exerted by gas 1 at temperature T, volume V if no other gases were present.

Thus, the total pressure of a mixture of ideal gases is the sum of the partial pressures. This is known as “Dalton’s law of partial pressure”.

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Kinetic Interpretation of Temperature

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According to the Kinetic Interpretation of Temperature, the total average kinetic energy of all the molecules of a gas is directly proportional to its absolute temperature (T).

The pressure exerted by an ideal gas is given by

P = 1/3 nmv²

Where n is the number of molecules (N) of gas per unit volume (V) and v is the velocity of the gas molecules.

\(\Rightarrow PV={1\over 3}Nmv^2\)……. (i)

The internal energy E of an ideal gas is purely kinetic, therefore

E = N x 1/2 mv² ⇒ Nmv² = 2E ……… (ii)

From equation (i) and (ii), we get

PV = 2/3 E

From the ideal gas equation, PV = μRT = NK_BT

Where

• μ is the number of moles of the gas
• N is the number of molecules of the gas
• K_B is Boltzmann's constant

⇒ 2/3 E = NK_BT

Therefore, the average kinetic energy of a gas molecule

E/N = 3/2 K_BT

The above equation shows that:

• The average kinetic energy of a molecule is proportional to the absolute temperature of the gas.
• It is independent of the pressure, volume, or nature of the ideal gas.
• This result relates the internal energy and temperature of a molecule.
• This is known as the Kinetic interpretation of temperature.
• According to this kinetic interpretation of temperature, the average kinetic energy U is zero at T = 0, i.e., the motion of molecules ceases altogether at absolute zero.

RMS Speed

The root mean square speed is defined as the square root of the mean of the squares of the random speeds of the individual molecules of gas. It is given by

\(v_{rms}=\sqrt{ v^2}\)

Also, it is given by

\(v_{rms}=\sqrt{3K_BT\over m}=\sqrt{3RT\over M}\)

Where

• m = mass of one molecule of gas
• M = mass of one mole of gas

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Degrees of Freedom

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The degree of freedom can be defined as the total number of independent coordinates required to specify the position of a molecule or the number of independent motions possible with any molecule.

• If a molecule moves in a line, it has one degree of freedom.
• Molecules of monatomic gases like neon, argon, helium, etc. have three degrees of freedom.
• Diatomic molecules like O_2, N_2, H_2, etc. have six degrees of freedom.

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Law of Equipartition of Energy

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The law of Equipartition of Energy states that If a dynamic system is in thermal equilibrium, the energy of the system is equally distributed amongst the various degrees of freedom and the energy associated with each degree of freedom per molecule is 1/2 k_bT, (where k_b is Boltzman constant).

The kinetic energy of a single molecule along the x-axis, the y-axis, and the z-axis is given as

1/2 mv_x² (along the x-axis)

1/2 mv_y² ,(along y-axis)

1/2 mv_z²(along z-axis)

When the gas is at thermal equilibrium, then according to the kinetic theory of gases, the average kinetic energy of a molecule is given by,

1/2 mv_rms² = 3/2K_bT

1/2 mv_x² + 1/2 mv_y² + 1/2 mv_z² = 3/2 K_bT

where v is the root-mean-square velocity of the molecules, Kb is the Boltzmann constant and T is the temperature of the gas.

A mono-atomic gas has three degrees of freedom, so the average kinetic energy per degree of freedom is given by 

KE_x = 1/2k_bT

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Mean Free Path

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The mean free path of a molecule in a gas is the average distance traveled by the molecule between two successive collisions.

• Inside the gas, there are several molecules that are randomly moving and colliding with each other.
• The distance that a particular gas molecule travels without colliding is known as the mean free path.
• The mean free path is represented by λ and its SI unit is meter.

Expression of Mean Free Path

Consider each molecule of gas as a sphere of diameter (d).

• The average speed of each molecule is v.
• Suppose the molecule suffers a collision with another molecule within the distance (d).
• The molecule which comes within the distance range of its diameter will have a collision with that molecule.

Let the number of molecules per unit volume = n

Therefore, the total number of collisions in time Δt = vΔtπd² * n

Rate of collision =vΔtπd² * n/Δt = vπd²n

Let the time between collision τ =1/vπd²n

Average distance between collision = τv = 1/πd²

1/πd²n this value was modified and a factor was introduced.

Mean free path(l) = 1/√2 π d²n

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Things to Remember

• The kinetic theory explains the behavior of an ideal gas on the basis of the fact that the gas consists of rapidly moving molecules and atoms
• According to the Kinetic theory of gases, a given amount of gas is a mixture of a very large number of identical molecules of the order of Avagadro’s number.
• According to Boyle’s law, the pressure and volume of a gas are inversely proportional to each other.
• Charle’s law states that the temperature and volume are directly proportional to each other.
• Dalton’s law of partial pressure states that the entire pressure is equal to the sum of the partial pressures.
• The mean free path of a molecule in a gas is the average distance traveled by the molecule between two successive collisions.

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Previous Years Questions

1. For a monatomic gas, the molar specific heat at constant pressure divided by the molar gas constant R is equal to
2. If m represents the mass of each molecule of a gas and T, its absolute temperature, then the root mean square velocity of the gaseous molecule is proportional to
3. The total number of degrees of freedom of a rigid diatomic molecule is​
4. If the pressure and the volume of a certain quantity of ideal gas are halved, then its temperature​
5. To decrease the volume of a gas by 5% at a constant temperature, the pressure should be​
6. The respective speeds of five molecules are 2, 1.5, 1.6, 1.6, and 1.2km/s, the most probable speed in km/s will be​
7. The volume thermal expansion coefficient of an ideal gas at constant pressure is
8. The heat absorbed by the gas is
9. The ratio of kinetic energy and rotational energy in the motion of a disc is:
10. A real gas behaves as an ideal gas,​
11. AT NTP the rms velocity of hydrogen molecules is 1.8 km/s. Under the same conditions, the rms velocity of oxygen molecules in km/s will be​
12. At constant pressure, the ratio of the increase in the volume of an ideal gas per degree rise in kelvin temperature to its original volume is​
13. If the temperature is doubled and oxygen molecules dissociate into oxygen atoms, the rms speed becomes​
14. The figure shows graphs of pressure versus density for an ideal gas at two temperatures T1 and T2, then​
15. What will be the molar-specific heat at the constant volume of an ideal gas consisting of rigid diatomic molecules?​