NEET Physics Thermodynamics And Kinetic Theory Of Gases Notes

NEET Physics Thermodynamics And Kinetic Theory Of Gases Notes

Thermodynamics and Kinetic Theory of Gases

Ideal Gas or Perfect Gas

1. Molecules of the ideal gas are point masses, with zero volume.
2. There is no intermolecular force between the molecules of ideal gas.
3. There is no intermolecular potential energy for the molecules of ideal gas.
4. The molecules of ideal gas possess only the kinetic energy.
5. The ideal gas can not be converted into liquids or solids. (This is the consequence of the absence of intermolecular force).
6. The internal energy of an ideal gas depends only on temperature.

Ideal Gas Equation: The equation which relates all the macroscopic variables [P, V, T] of an ideal gas is called ideal gas equation. It is given by, PV = nRT

∴ n → number of moles R = 8.31 J mol^-1 K^-1

For one mole of gas, PV = RT

Read And Learn More: NEET Physics Notes

Avagadro’s Hypothesis

Equal volumes of all gases at the same temperature and pressure contain the same number of molecules.

• One mole of every gas at NTP has same volume equal to 22.4 litres.
• One mole of every gas contains same number of molecules called Avagadro’s number. N_A = 6.023 x 10²⁵
• Avagadro’s number is also equal to the number of atoms in 12g of Carbon -12.

Kinetic Theory Of Gases Equation

Real Gases

• The gases actually found in nature are called real gases.
• The molecules of the real gas have a finite volume.
• There is intermolecular attraction or repulsion between the molecules of the real gas.
• The intermolecular force is attractive at larger intermolecular separation and repulsive when the molecules are too close to each other.
• Molecules of real gas have intermolecular potential energy as well as kinetic energy.
• Real gases can be liquified or solidified.
• The internal energy of real gases depends on volume, pressure as well as temperature.
• Real gases do not obey the equation.

PV = nRT

Real gases obey ideal gas equation at very low pressure and at very high temperature.

NTP or STP

• NTP stands for normal temperature and pressure.
• STP stands for standard temperature and pressure.
• NTP and STP both mean the same.
• They refer to a temperature of 273K or 0⁰C and 1 atm pressure.

Absolute Zero Temperature: The absolute zero refers to zero of the kelvin scale. i.e., absolute zero = 0K = -273.15⁰C

• At the absolute zero all molecular motion ceases.
• The volume of ideal gas becomes zero at the absolute zero.
• The pressure of the ideal gas becomes zero at absolute zero.
• The molecular energy or internal energy of the ideal gas becomes zero at absolute zero.
• All real gases get liquified before reaching the absolute zero.

Degree of Freedom

The number of ways in which a gas molecule can absorb energy is called degrees of freedom. Total degree of freedom f = Translational degree of freedom (f_t) rotational degree of freedom (f_r) + vibrational degree of freedom (f_v)

f_t is present at all temperatures, f_r is present at ordinary temperatures and f_v are present only at high temperatures.

The degree of freedom can be calculated by using the relation, f = 3N – k

Where N = number of atoms in the molecule (atomicity) k is the number of relations or constraints.

For a monoatomic molecule, f= 3 x 1 – 0 =3

For a diatomic molecule, f = 3 x 2 – 1 = 5

• At very low temperatures (<70K), the degrees of freedom corresponding to the rotatory motion are absent.
• Hence, the diatomic molecule possesses only 3 degrees of freedom.
• At very high temperatures diatomic molecules have 7 degrees of freedom.

In triatomic molecules degrees of freedom depend on the structure of the molecules.

For linear triatomic molecules, (k = 2)  f = 3 x 3 – 2 = 7

For a non-linear triatomic molecule, (k = 3) f = 3 x 3 – 3 = 6

Degrees Of Freedom Example:

• CO₂ is a linear molecule with 7 degrees of freedom
• O₃ and H₂O are non-linear molecules with 6 degrees of freedom.

Maxwell’s Law of Equipartition of Energy

This law states that the kinetic energy is equally distributed among all the degrees of freedom and energy associated with each degree of freedom is = \(\frac{1}{2} K_b \mathrm{~T}\)

Where T is the absolute temperature and Kb is the Boltzmann constant.

⇒ \(\mathrm{K}_{\mathrm{B}}=\frac{\mathrm{R}}{\mathrm{N}_{\mathrm{A}}}=1.38 \times 10^{-23} \mathrm{~K}^{-1}\)

The kinetic energy of a molecule having f degrees of freedom is given by

⇒ \(\mathrm{U}_{\mathrm{k}}=\frac{\mathrm{f}}{2} \mathrm{~K}_{\mathrm{b}} \mathrm{T}\)

Total kinetic energy of 1 mole of gas with f degree of freedom is given by

⇒ \(\mathrm{U}_{\mathrm{k}}=\mathrm{N}_{\mathrm{A}}\left[\frac{\mathrm{f}}{2} \mathrm{~K}_{\mathrm{b}} \mathrm{T}\right]=\frac{\mathrm{f}}{2} \mathrm{RT}\)

Where N_A is Avagadro’s number and R is universal gas constant.

Specific Heat Capacity of Gases

In an ideal gas, the total energy of the gas or internal energy U of the gas is equal to the total kinetic energy of all the molecules in the gas.

For one mole of a monoatomic gas, the total energy is U = \(\frac{3}{2} \mathrm{~N}_A \mathrm{k}_{\mathrm{B}} \mathrm{T}=\frac{3}{2} \mathrm{RT}\)

The specific heat at constant volume for a monatomic gas is,

⇒ \(\mathrm{C}_{\mathrm{V}}=\frac{\mathrm{dU}}{\mathrm{dT}}=\frac{\mathrm{d}}{\mathrm{dT}}\left(\frac{3}{2} \mathrm{RT}\right)=\frac{3}{2} \mathrm{R}\)

w.k.t., C_P – C_V = R

∴ \(C_P=C_V+R=\frac{3}{2} R+R=\frac{5}{2} R\)

γ = \(\frac{C_P}{C_V}=\frac{\frac{5}{2} R}{\frac{3}{2} R}=\frac{5}{3}=1.67\)

For a diatomic gas \(\mathrm{U}=\frac{5}{2} \mathrm{RT}\)

⇒ \(\mathrm{C}_{\mathrm{V}}=\frac{5}{2} \mathrm{R} \quad \mathrm{C}_{\mathrm{p}}=\frac{7}{2} \mathrm{R}\)

⇒ \(\gamma=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{C}_{\mathrm{V}}}=\frac{7}{5}=1.4\)

For non-linear triatomic, U = 3RT

⇒ C_V = 3R

⇒ C_P = 4R

In general, for poly atomic gas molecules, \(=\left(1+\frac{2}{f}\right)\)

Note:

• Specific heat capacity of solids = 3R
• Specific heat capacity of water = 9R

Mean Free Path: The average distance travelled by a molecule between two successive collisions is called the mean free path.

It can be shown that, \(\bar{l}=\frac{1}{\sqrt{2} \pi d^2 \mathrm{n}}\)

Where, d is the diameter of each molecule and n is the number of molecules per unit volume If m is the mass of each molecule and ρ is the density of the gas,

⇒ \(\bar{l}=\frac{\mathrm{m}}{\sqrt{2 \pi \mathrm{d}^2 \rho}} \quad\left(because \mathrm{n}=\frac{\rho}{\mathrm{m}}\right)\)

Boyle’s Law

It states that, at constant temperature, the volume of the given mass of the gas is inversely proportional to its pressure P.

i.e., PV = constant.

Charle’s Law

It states that, at constant pressure, the volume of the given mass of the gas is proportional to its absolute temperature.

i.e., \(\frac{\mathrm{V}}{\mathrm{T}}\) = constant

Expression for Pressure Exerted By a Gas

⇒ \(P=\frac{1}{3} \rho v_{\mathrm{vos}}^2\)

P = \(\frac{1}{3} \frac{M}{V} v_{\mathrm{ras}}^2\)

Calorie: The quantity of heat required to raise the temperature of 1g of water from 14.5⁰ C to 15.5⁰ C

Kinetic Molecular Theory Of Gases

First Law of Thermodynamics

When a certain amount of heat is given to a system, a part of it is used to increase the internal energy, and the remaining part is used in doing external work.

∴ dQ = dU + dW

The first law of thermodynamics is in accordance with law of conservation of energy.

Sign Convention

• Work done by the system is taken positive.
• Work done on the system is taken negative.
• Increase in U is taken positive.
• Decrease in U is taken negative.
• Heat added to the system is taken positive.
• Heat given out from the system is taken negative.

Thermodynamic Processes

1. Isothermal process:

PV = constant

dQ = dU + dW

dQ = dW

⇒ \(C_r=\frac{d U}{d T}=\infty\)

Work done by isothermal process is,

W = \(2.303 R T \log _{10} \frac{V_2}{V_1}\)

or,

W = \(2,303 R T \log _{10} \frac{P_1}{P_2}\)

2. Adiabatic process:

dQ = dU + dW

dU + dW = 0

PV^γ= constant

Work done in adiabatic process is \(W=\frac{R}{\gamma-1}\left[T_1-T_2]\right.\)

3. Isochoric process:

ΔV = 0

dQ = dU

4. Isobaric process: ΔP = 0

Heat Engine: A device used to convert heat energy into useful mechanical work is called heat engine.

The efficiency of an engine is the ratio between work done by the engine and the amount of heat absorbed by the system.

⇒ \(\eta=\frac{W}{Q_1}=\frac{Q_1-Q_2}{Q_1}=1-\frac{Q_2}{Q_1}\)

The efficiency of Carnot’s heat engine is given by

⇒ \(\eta=1-\frac{Q_2}{Q_1} \quad \text { Or } \quad \eta=1-\frac{T_2}{T_1}\)

Refrigerator: Coefficient of performance, \(\beta=\frac{Q_2}{W}=\frac{Q_2}{Q_1-Q_2}\)

or, \(\beta=\frac{T_2}{T_1-T_2}\)

Cyclic Process: “A process in which the system after passing through various stages returns to its initial state” is called as cyclic process.

For a cyclic process, PV graph is a closed curve. The area under P-V graph gives work done by the substance. In a cyclic process there will be no change in the internal energy.

i.e., ΔU = 0

Therefore, ΔQ = ΔW

The total heat absorbed by the system equals the work done by the system.