Kinetic Theory of Gases

 GAS LAWS

  1. \toThe volume of a given mass of a gas is inversely proportional to its pressure\\at constant temperature, i.e 𝑽∝\frac{𝟏}{P} (if mass of gas and temperature of gasare constant)\\or, PV = constant Thus, P_1 V_1 = P_2 V_2 \\\to Real gas obeys Boyle’s law at low pressure and high temperature.\\\to Boyle’s law is represented graphically as follows:\\

  2. \textbf{Charle’s Law:}\\\toThe volume of given mass of a gas at constant pressure increases or\\decreases by a constant fraction 𝛼(= \frac{1}{273} ) of its volume at 0^oC for rise or\\fall of temperature by 1^oC.\\i.e 𝑽 = 𝑽_𝟎 [𝟏 + 𝜶𝒕]. This law is also known as Charle’s volume law. \toThe volume of given mass of a gas at constant pressure is directly proportional to its absolute temperature i.e 𝑽∝𝑻 or, V/T = constant Thus, 𝑽_𝟏 = 𝑽_𝟐 \to Charle’s law is represented graphically as follows:

  3. \textbf{Dalton’s law of partial pressure:}\\\toThe pressure exerted by a mixture of non-reacting gases at constant temperature\\is equal to sum of partial pressure of each components in the mixture.\\P=P_1 + P_2 + P_3 +..... \toPartial pressure = mole fraction x total pressure\\

  4. \textbf{Gay-Lussac’s law:}\\\toThe volume of given mass of a gas at constant volume increases or decreases\\by a constant fraction𝛼(= \frac{1}{273} ) of its pressure at 0^oC for rise or\\fall of temperature by 1^oC. i.e 𝑷 = 𝑷_𝟎 [𝟏 + 𝜶𝒕]. This law is also known as Charle’s pressure law.\\But, in Gay-Lussac’s law flexible container is used whereas in Charle’s law rigid container is used.\\\toThe pressure of given mass of a gas at constant volume is directly proportional to its\\absolute temperature i.e 𝑷 ∝ 𝑻 or, P/T = constant\\Thus, \frac{𝑷_𝟏}{𝑻_𝟏} = \frac{𝑷_𝟐}{𝑻_𝟐} \\


Combined Gas Equation:
Here, \to \frac{𝑷𝑽}{𝑻}= 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 \to \frac{𝑷_𝟏𝑽_𝟏}{𝑻_𝟏}=\frac{𝑷_𝟐𝑽_𝟐}{𝑻_𝟐}

Ideal Gas Equation

  1. Here,\\\to For n moles of gas, PV = nRT\\(R = 8.314 Jmol^{-1} K^{-1} is universal gas constant and is same for all gases)\\PV = nRT\\=> 𝑷𝑽 = 𝒎 𝑹𝑻 𝑴\\=> 𝑷𝑽 = 𝒎(\frac{𝑹}{𝑴})𝑻\\=> 𝑷𝑽 = 𝒎𝒓𝑻 (m is mass of gas, M is molecular mass, r = R/M is specific gas constant or\\gas constant per unit mass.)\\=>𝑷=𝝆𝒓𝑻 ( 𝜌 =\frac{m}{v} is density of gas)\\Also,\\PV = nRT\\=> 𝑷𝑽 = \frac{𝑵}{𝑵_a}𝑹𝑻 (N = number of molecules, N_a = avogadro’s number)\\=> 𝑷𝑽 = 𝑵𝒌𝑻 (K = R/N_a = 1.3806 x 10-23 J/K is Boltzmann’s constant)


Degree of freedom

Total number of independent quantities (or coordinates)\\required to define the state of a system is degree of freedom (f).\\It can also be defined as total number of independent ways in which gas molecule can have energy.\\f = 3N-R (N= no of atoms, R= no of restrictions or number of independent relation among the atom)\\\underline{{For} {Monoatomic}}: N=1, R=0\\f= 3 i.e. only translation motion.\\\underline{{For} {Diatomic}}: N=2, R=1\\f= 5 i.e. translation and rotational motion\\\underline{{For} {Triatomic}}: Linear (Eg: CO_2): N=3, R=2, f= 7\\Non Linear (Eg: H_2O): N=3, R=3, f= 6

Average Kinetic Energy

  1. \to \underline{{For} {Monoatomic}{Gases}}:(f=3)\\Average KE per mole = \frac{𝑓}{2}𝑓 𝑅𝑇 = \frac{3}{2} 𝑅𝑇\\Average KE per molecule = \frac{𝑓}{2}𝐾𝑇 = \frac{3}{2} 𝐾𝑇\\\to \underline{{For} {Diatomic} {Gases}}: (f=5) :\\Average KE per mole = \frac{𝑓}{2}𝑅𝑇 = \frac{5}{2}𝑅𝑇\\Average KE per molecule =\frac{𝑓}{2}𝐾𝑇 = \frac{5}{2}𝐾𝑇\\

    \to \underline{{For} {Triatomic} {Gases}}: \\

    For linear (f=7): Average KE per mole = \frac{𝑓}{2}𝑅𝑇 = \frac{7}{2}𝑅𝑇\\

    Average KE per molecule = \frac{𝑓}{2}𝐾𝑇 = \frac{7}{2}𝐾𝑇\\

    For linear (f=6): Average KE per mole =\frac{𝑓}{2}𝑅𝑇 = 3𝑅𝑇\\

    Average KE per molecule = \frac{𝑓}{2} 𝐾𝑇 = 3𝐾𝑇\\

  2. \textbf{Note:}\\Average translation KE of gas molecule depends only on its temperature but not on its nature.


Law of equipartition of energy:

Here,\\

According to this law, total energy of gas molecule is equally distributed among all degrees of freedom.\\

Total KE per mole = \frac{𝑓}{2}𝑅𝑇\\

Total KE per molecule = \frac{𝑓}{2}𝐾𝑇\\

\underline{For Monoatomic}:\\

Total KE per mole = \frac{3}{2}𝑅𝑇\\

Total KE per molecule = \frac{3}{2}𝐾𝑇\\

(Translation only)\\

\underline{For Diatomic}:\\

Total KE per mole = \frac{5}{2}𝑅𝑇 = \frac{3}{2}𝑅𝑇 + 𝑅𝑇\\

Total KE per molecule = \frac{5}{2}𝐾𝑇 = \frac{3}{2}𝐾𝑇 + 𝐾𝑇\\

(Three translation and two rotational)


Mixing of two gases

\toThis law is based on conservation of energy.\\

\toTotal KE before mixing two gases is equal to total KE after mixing two\\

gases. (Note: PE is zero so total energy is kinetic)\\

\toIf n perfect gases at absolute temperature T_1, T_2, ....T_n are mixed and\\

number of respective molecules/moles are n_1, n_2, ...n_n then for no energy loss, temperature of mixture is:\\

T_{mix} = \frac{𝒏_𝟏𝑻_𝟏+𝒏_𝟐𝑻_𝟐+⋯+𝒏_𝒏𝑻_𝒏}{𝒏_𝟏+𝒏_𝟐+⋯+𝒏_𝒏}


Kinetic Theory of Gases

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