(a) Kinetic theory of gases is based on some simplifying assumptions. Molecules of the gas are assumed to behave as hard elastic identical spheres.

Q#388: [Gases > Kinetic theory of gases] (Past Exam Paper – June 2012 Paper 41 Q2)

(a) Kinetic theory of gases is based on some simplifying assumptions.

Molecules of the gas are assumed to behave as hard elastic identical spheres.

State assumption about ideal gas molecules based on

(i) nature of their movement

(ii) their volume

(b) Cube of volume V contains N molecules of an ideal gas. Each molecule has a component c_X of velocity normal to one side S of the cube, as shown.

Pressure p of the gas due to component c_X of velocity is given by expression

pV = Nmc_X²

where m is mass of a molecule.

Explain how expression leads to the relation

pV = (1/3) Nm<c²>

where <c²> is mean square speed of the molecules.

(c) Molecules of an ideal gas have a root-mean-square (r.m.s.) speed of 520 m s^–1 at a temperature of 27 °C.

Calculate r.m.s. speed of the molecules at a temperature of 100 °C.

Solution 388:

(a)

(i)

EITHER The gas molecules are in random motion

OR The gas molecules have constant velocity until they hit the wall / other molecules

(ii)

EITHER The (total) volume of the molecules is negligible compared to the volume of the containing vessel

OR The radius / diameter of a molecule is negligible compared to the average intermolecular distance.

(b)

{Consider a single molecule of the gas in motion. This molecule will have 3 components of velocity (since it is able to move in 3 dimensions).}

EITHER A molecule has components of velocity in 3 directions OR c² = c_X² + c_Y² + x_Z²

{Now, consider all the gas molecules which are in motion. Since the motion of the gas molecules are random in nature and the number of molecules is large, the components of velocities of all the gas molecules can be averaged as <c_X²> = <c_Y²> = <c_Z²>}

Random motion and averaging, so <c_X²> = <c_Y²> = <c_Z²>

{Since <c_X²> = <c_Y²> = <c_Z²>, each of them can be written as the component of velocities in one specific direction. E.g. <c_X²>. Now, replace this in the average of the equation c² = c_X² + c_Y² + x_Z² . <c²> = <c_X²> + <c_X²> + <c_X²> = 3<c_X²>}

<c²> = 3<c_X²>

{But, since the pressure p of the gas due to component c_X of velocity is given by expression

 pV = Nmc_X². <c_X²> = <c²> / 3}

So, pV = (1/3) Nm<c²>

(c)

{crms = √<c²>}

<c²> α T          or c_rms α √T

(In kelvin,) the temperatures are (27+273 =) 300K and (100+273 =) 373K

{When T = 300K, c_rms = 520ms^-1. When T = 373K, c_rms = √(373/300) x 520 = 580 ms^-1. }

c_rms = 580 ms^-1

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