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NEET Physics Kinetic Theory of Gases

Kinetic Theory of Gases PDF Notes, Important Questions and Synopsis

SYNOPSIS

  • The kinetic theory of gases relates the macroscopic properties of gases such as pressure and temperature to the microscopic properties of gas molecules such as speed and kinetic energy.
  • Assumptions of the kinetic theory of gases:
    • Collisions between gas molecules or between a molecule and the container are completely elastic.
    • Mass of gas molecules is negligible. So, the effect of gravity can be neglected.
    • Volume of molecules is considered negligible.
    • A collision between molecules is governed by Newton’s Law of Motion (i.e. Net force = Change in momentum per unit time).
    • Interaction force between particles is negligible. (That is, they exert no forces on one another.)
    • Molecules are considered to be in constant, random and rapid motion. So, average velocity of particles in all the 3-directions will be zero, i.e. <vx> = <vy> = <vz> = 0.
  •  An ideal gas is one in which the pressure p, volume V and temperature T are related by pV = μRT, where R is called the gas constant.
  • Real gases satisfy the ideal gas equations only approximately, more so at low pressures and high temperatures.
  • The kinetic theory of an ideal gas gives the relation

    begin mathsize 12px style straight P equals 1 third straight n text    end text straight m stack straight v squared with bar on top end style
    Where n is the number density of molecules, m is the mass of the molecule and   is the mean of squared speed.
  • Root mean square (RMS) speed Vrms: It is defined as the square root of the mean of squares of the speed of different molecules.

    begin mathsize 12px style straight V subscript rms equals square root of fraction numerator 3 kT over denominator straight m end fraction end root equals square root of fraction numerator 3 RT over denominator straight M end fraction end root equals square root of fraction numerator 3 straight P over denominator straight rho end fraction end root end style
  • The temperature of a gas is a measure of the average kinetic energy of molecules independent of the nature of the gas or molecule. In a mixture of gases at a fixed temperature, the heavier molecules have lower average speedThe pressure exerted by n moles of an ideal gas in terms of the speed of its molecules is P = 1/3 nm vrms2.

  • The average kinetic energy of a molecule is proportional to the absolute temperature of the gas. Molecules of different gases such as begin mathsize 12px style He ,  straight H subscript straight 2 ,  straight O subscript straight 2 end style at same temperature will have same translational kinetic energy, though their rms speed is different.

  • The degrees of freedom of a gas molecule are independent ways in which the molecule can store energy.

  • Law of equipartition of energy: Every degree of freedom of a molecule has associated with it, on average, an internal energy of (½) kT per molecule.
    For f degree of freedom, the equipartition of energy can be given as
    Total energy possessed by N molecule = N(f/2)kT
    Total energy possessed with each mole = (f/2)RT
    Total energy possessed by each molecule = (f/2)kT

  • Specific heat of a gas does not have a single or unique value.
    In an adiabatic process, the gas is compressed suddenly and no heat is supplied.
    Therefore,   

    begin mathsize 12px style table attributes columnalign left end attributes row cell text c= end text fraction numerator text ΔQ end text over denominator text mΔT end text end fraction text =0 end text end cell row cell text ΔQ end text equals 0 end cell end table end style
    In an isothermal process,

    begin mathsize 12px style text ΔT=0;c= end text fraction numerator text ΔQ end text over denominator text mΔT end text end fraction text = end text straight infinity end style
    Molar specific heat at constant volume
    Error converting from MathML to accessible text.
    Molar specific heat at constant pressure
    Error converting from MathML to accessible text.

  • Monatomic gases only have three translational degrees of freedom.

  • Diatomic gases in general have three translational, two rotational and two vibrational degrees of freedom.

  • Value of begin mathsize 12px style straight gamma end style is different for monatomic, diatomic and triatomic molecules.

  • The mean free path is the average distance covered by a molecule between two successive collisions:

    begin mathsize 12px style straight l equals fraction numerator straight 1 over denominator square root of straight 2 nπd to the power of straight 2 end fraction end style where n is the number density and d is the diameter of the molecule.

  • If two non-reactive gases are enclosed in a vessel of volume V, where the number of moles of one gas is  and another gas is n1 and n2 , then

    Total moles = begin mathsize 12px style straight n subscript 1 plus straight n subscript 2 end style

    Molecular weight of mixture
    begin mathsize 12px style straight M equals fraction numerator straight n subscript straight 1 straight M subscript straight 1 plus straight n subscript straight 2 straight M subscript straight 2 over denominator straight n subscript straight 1 plus straight n subscript straight 2 end fraction end style

    Specific heat of a mixture at constant volume
    begin mathsize 12px style straight C subscript Vmix straight equals fraction numerator straight n subscript straight 1 straight C subscript straight v 1 end subscript plus straight n subscript straight 2 straight C subscript straight v 2 end subscript over denominator straight n subscript straight 1 plus straight n subscript straight 2 end fraction end style
    Specific heat of a mixture at constant volume
    begin mathsize 12px style straight C subscript Pmix straight equals fraction numerator straight n subscript straight 1 straight C subscript straight P 1 end subscript plus straight n subscript straight 2 straight C subscript straight P 2 end subscript over denominator straight n subscript straight 1 plus straight n subscript straight 2 end fraction end style

    begin mathsize 12px style straight gamma subscript mix straight equals straight C subscript Pmix over straight C subscript Vmix end style