Ask a Teacher

17.Gas molecules exert pressure on the surface of the container when they collide with the walls of the container. a. What is the expression for the pressure exerted by an ideal gas (1) b. Deduce the relationship between kinetic energy and absolute temperature of an ideal gas from the above expression (2) c. State law of equipartition of energy. What is the average kinetic energy of a monatomic gas molecule at temperature T. (1½) d. 30kg of water at 200C is mixed with 20kg of water at 300C. find the neutral temperature (1½)

.Pressure Exerted by a Gas: When a gas is filled into a vessel, its molecules move in all possible directions. They then collide with the walls of the container and hence exert pressure on the walls of the vessel.

Let v be the velocity of a gas molecule at any istant, and v_x, v_y and v_z be the components of its velocity along x,y and z-axes respectively,

Let n be the total number of molecules in a gas and m be the mass of each molecule. The vessel having the gas is a cube of side l.

Consider the motion of the above molecule along x-axis.
Then its momentum before collision with the wall

                    = mv_x

After collision its momentum

                    = -mv_x

?     Change in its momentum per collision

                   = -2mv_x

Rate of change of momentum along x-axis

Then, rate of change of momentum of all molecules along x-axis

Also, force exerted by molecules on each wall perpendicular to x-axis is (ignoring -ve sign).

Pressure exerted by molecules on each wall perpendicular to x-axis is,

Similarly, pressures along y-axis and z-axis are,

Particles are free to move in all possible directions, therefore,

            P_x = P_y = P_z = P

                  (i)

[where , mean square velocity of gas molecules]

    l³ = Volume (V)

or                                                        (ii)

where M = mass of the gas

and = density of gas (?)

                                           (iii)

Kinetic energy and temperature

Consider one mole of a gas occupying a volume V. If ? is the density and  is the mean square velocity of the molecules of the gas, the pressure exerted by the gas,
         
But P V = RT and = Average kinetic energy of one mole of the gas

of one mole of the gas = ;

where R is the universal gas constant.

of one molecule of the gas = ;
where N is the Avogadro number. Since R/N = k, the Boltzman's constant,
Average K.E of one molecule of the gas =
                 ?   ? T
Note: (i) Average KE of 1 kg of gas =   ; where R is the ordinary gas constant.
Note: (ii) Consider any amount of a gas. Let P, V, T and n be the pressure, volume, temperature and number of molecules of th gas.

  K.E. of a molecule of the gas =

              ? K.E of the gas =

Law of Equipartition of Energy states that Total energy of a dynamical system in thermal equilibrium is equally divided among its degrees of freedom.

                    Energy per degree of freedom = (1/2)kT;
where k = R/N, the Boltzman's constant and T is the absolute temperature of the gas.