Formulation of Kinetic Theory of Gas

After knowing the experimental gas lows, it is of interest to develop a theoretical model based on the structure and properties of gases, which can be correlated to the experimental facts. Fortunately, such theory has been developed for Formulation of Kinetic Theory of Gases based upon the certain postulates which are supposed to be applicable to an Ideal Gas.

Assumptions of Kinetic Theory of Gases

  1. The gas is composed of very small discrete particles, now called molecules. For gas, the mass and size of the molecules are the same and different for different gases.
  2. The molecules are moving in all directions with the verity of speeds. Some are very fast while others are slow.
  3. Due to random motion, the molecules are executing collision with the walls of the container (wall collision) and with themselves (intermolecular collision). These collisions are perfectly elastic and so there occurs no loss of kinetic energy or momentum of the molecules by this collision.
  4. The gas molecules are assumed to be point masses, that is their size is very small in comparison to the distance they travel.
  5. There exist no intermolecular attraction especially at low pressure, that is one molecule can exert pressure independent of the influence of other molecules.
  6. The pressure exerted by a gas is due to the uniform wall collisions. Higher the frequency of the wall collision greater will be the pressure of the gas.
  7. This explains Boyle's law since when the volume is reduced, wall collision becomes more frequent and pressure is increased.
  8. Through the molecular velocity are constantly changing due to the intermolecular collision, average kinetic energy(є) of the molecules remains fixed at a given temperature. This explains the Charl's law that when T is increased, velocity is increased, wall collision become more frequent and pressure(P) is increased when T kept constant or Volume(V) is increased when P kept constant.

Root Mean Square Speed

    Root mean square(RMS) speed is defined as the square root of the average of the squares of speeds.
CRMS2 = (N1C12 + N2C22 + ...)/N

Formulation of Kinetic Gas Equation

    Let us take a cube of edge length l containing N molecules of gas of molecular mass m and RMS speed is CRMS at temperature T and Pressure P.
Molecular Velocity and its Components in Formulation of Kinetic Theory of Gas
Formulation of Kinetic Theory of Gas
    Let in gas molecules,
    N₁ have velocity C₁,
    N₂ have velocity C₂,
    N₃ have velocity C₃, and so on
    .
    Let us concentrate our discussion to a single molecule among N1 that have resultant velocity C1 and the component velocities are Cx, Cy and Cz.
    So that, C12 = Cx2 + Cy2 + Cz2
    The Molecule will collide walls A and B with the Component Velocity Cx and other opposite faces by Cy and Cz.
    Change of momentum along X-direction for a single collision,
    = mCx - (- mCx)
    = 2 mCx
    Rate of change of momentum of the above type of collision,
    = 2 mCx × (Cx/l)
    = 2 mCx2/l
    Similarly, along Y and Z directions, the rate of change of momentum of the molecule is 2 mCy2/l and 2 mCz2/l respectively.
    Total rate change of momentum for the molecule,
    = 2 mCx²/l + 2 mCx²/l +2 mCz²/l
    = 2 (m/l) (Cx² + Cy² + Cz²)
    = 2 mC₁²/l
    For similar N₁ molecules, it is 2 mN₁C₁²/l
    Taking all the molecules of the gas, the total rate of change of momentum,
    = (2 mN₁C₁²/l)+(2 mN₂C₂²/l)+(2 mN₃C₃²/l)+ ..
    = 2 mN {(N₁C₁² + N₂C₂² + N₃C₃² ..)/N}
    = 2 m N CRMS2
    Where CRMS2 = Root Mean Square Velocity of the Gases.
    According to Newton's 2nd Low of Motion, rate of change of momentum due to wall collision is equal to force developed within the gas molecules.
    That is, P × 6l² = 2 mN CRMS2/l
    or, P × l³ = 1/3 m N CRMS2
∴ PV = 1/3 m N CRMS2
    Here, l³ = Volume of the Cube Contains Gas Molecules.
    The other form of the equation is,
    P = 1/3 × (mN/V) × CRMS2
∴ P = 1/3 d CRMS2
    Where (mN/V) is the density(d) of the gas molecules.
Formulation of Kinetic Theory of Gas and Kinetic gas equation
Kinetic Gas Equation
    This equation is also valid for any shape of the gas container.

Derivation of root mean square velocity

    Let us apply the kinetic equation for 1 mole Ideal Gas.In that case mN = mN₀ = M and Ideal Gas Equation, PV = RT.
    Hence from Kinetic Gas Equation,
    PV = 1/3 m N CRMS2
    or, RT = 1/3 m N CRMS2
or, CRMS2 = 3RT/M
∴ CRMS2 = √3RT/M
    Thus root means square velocity depends on the molar mass(M) and temperature(T) of the gas.

Expression of Average Kinetic Energy:

    The average kinetic energy(Ē) is defined as,
    Ē = 1/2 m CRMS2.
    Again from Kinetic Gas Equation,
    PV = 1/3 m N CRMS2
    or, PV = 2/3 N × 1/3 m CRMS2
    or, PV = 2/3 N Ē
    For 1 mole ideal gas,PV = RT and N = N₀
    Thus RT = 2/3 N₀ Ē
or, Ē = 3/2R/N₀T
∴ Ē = 3/2 kT
    Where k = R/N0 and is known as the Boltzmann Constant. Its value is 1.38 × 10-23 JK-1.
    The total kinetic energy for 1 mole of the gas is,
ETotal = N0 (Ē) = 3/2RT
    Thus Average Kinetic Energy is dependent on T only and it is not dependent on the nature of the gas.
  • Problem 1:
    Calculate the pressure exerted by 1023 gas particles each of mass 10-22 gm in a container of volume 1 dm3. The root means square speed is 105 cm sec-1.
  • Solution:
    We have, N = 1023,
    m = 10-22 gm = 10-25 Kg,
    V = 1 dm3 = 10-3 m3
    and CRMS2 = 105 cm sec-1 = 103 m sec-1
    Therefore, from Kinetic Gas Equation,
    PV = 1/3 mNCRMS2
    or, P = 1/3 × m NV × CRMS2
    Putting the value we have,
    P=(1/3)(10-25 Kg×1023/10-3 m3)×(103 m sec-1)2
    ∴ P = 0.333 × 107 P
  • Problem 2:
    Calculate the root mean square speed of oxygen gas at 270C.
  • Solution:
    We know that CRMS2 = (3RT/M).
    Here, M = 32 gm mol-1, and T = 270 C = (273+27)K = 300 K.
    Thus,CRMS2 = (3 × 8.314 × 107 erg mol-1K-1 × 300 K)/(32 gm mol-1)
    ∴ CRMS = 48356 cm sec-1
  • Problem 3:
    Calculate the RMS speed of NH3 at N.T.P.
  • Answer:
    At N.T.P, V = 22.4 dm3 mol-1 = 22.4 × 10-3 m3 mol-1,
    P = 1 atm = 101325 Pa
    and M = 17 × 10-3 Kg mol-1.
    Thus, CRMS2 = 3RT/M
    = (3 × 101325 × 22.4 × 10-3) /(17 × 10-3)
    ∴ CRMS = 632 m sec-1
  • Problem 4:
    How the root means square velocity for Oxygen compares with that of the Hydrogen?
  • Answer:
    We know that, CRMS2 = 3RT/M
    Hence at a given temperature,
    (CRMS of H2)2/(CRMS of O2)2 = MO2/MH2
    = 32/2
    = 16
    That is CRMS of O2 = 4 × CRMS of H2.
  • Problem 5:
    Calculate the kinetic energy of translation of 8.5 gm NH3 at 270C.
  • Solution:
    We know that total kinetic energy for 1 mole of the gas is,
    ETotal = (3/2)RT
    = (3/2)(2 cal mol-1 K-1 × 300 K)
    = 900 cal mol-1
    Again 8.5 gm NH3 = (8.5/17) mol = 0.5 mol
    Thus Kinetic Energy of 8.5 gm NH3 at 270C is,
    (0.5 × 900) cal
    = 450 cal
  • Problem 6:
    Calculate the RMS velocity of oxygen molecules having a kinetic energy of 2 K.cal mol-1. At what temperature the molecules have this value of KE?
  • Solution:
    T = 673.9K or 400.90C

Assumptions and Formulation of Kinetic Theory of Gas, Derivation of Kinetic Gas Equation, Root Mean Square Velocity and Kinetic Energy with problems

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