On plotting a fraction (ΔN / N) of molecules having different speeds against the speeds of the molecules (along x-axis) a curve known as Maxwell's distribution curve is obtained. The important features of which are:
The fraction of molecules with very low or very high speeds is very small.
The fraction of molecules possessing higher and higher speeds goes on increasing till it reaches a peak and then starts decreasing.
The maximum fraction of molecules possess a speed, corresponding to the peak in the curve which is referred to as most probable speed.
Fig: - Maxwell-Boltzmann's distribution of speeds
The increase in temperature of the gas results in increase in the molecular motion. Consequently, the value of the most probable speed increases with increase in temperature. It may be noted that as long as the temperature of a gas is constant, the fraction having the speed equal to most probable speed remains the same but the molecules having this speed may not be the same. In fact, the molecules keep on changing their speed as a result of collisions.
Molecular motions may be described in terms of different types of molecular speeds. These are defined and described below.
Most probable speed
The most probable speed of a gas is the speed possessed by the maximum fraction of gas molecules at a given temperature denoted by a.
where, R is the gas constant
T is the absolute temperature and
M is the molecular mass.
Average speed
This is the average of speeds possessed by the molecules in a sample of any gas. This is defined as,
It can be shown that
Root mean square speed
The root mean square speed is expressed by the relationship,
Root mean square speed in any gas is given by the relationship,
The root mean square speed is commonly used and can be calculated from the following relations:
Relation between most probable, average and root mean square speeds
The three molecular speeds are expressed as
Root mean square speed , arms =
Average speed aav= 
Most probable speed a = 
This, mrms: aav: a = 1.00:0.92:0.82
or mrms > mav > a
Thus for gases, the root mean square speed is directly proportional to ÖT. and inversely proportional to ÖM. Therefore heavier molecules move slower.
Average kinetic energy of a gas
From the kinetic model of gases,
PV = (1/3) mNu2
For 1 mole of the gas, this equation becomes,
PV = (1/3) mNA m2 = (1/3)mm2
For an ideal gas,
PV = (2/3)x(1/2) mNA m2 = (1/3)Mm2
(Average kinetic energy per mole of the gas, Ek = (1/2)mNAm2)
or
PV = (2/3) x Ek
Since PV = RT for 1 mole of gas
Therefore
RT = 2/3 x Ek or Ek = (3/2) RT
For 'n' moles of gas, the kinetic energy is
Ek = (3/2)nRT
The units of Ek depend upon the units of R. Hence the following point holds true:
The assumption of kinetic theory that the average kinetic energy or molecular velocity of any gas is directly proportional to its absolute temperature.
K.E a u2
or
u2 a T or u a ÖT
Problem
8. Calculate the kinetic energy of 2g of oxygen at -23°C.
Solution
Kinetic energy is given as
Ek = (3/2) nRT
n = (2/32) = (1/16) mol
R = 8.314 JK-1 mol-1 , T = 273 – 23 = 250k
Ek = (2/32) x (1/16) x 8.314x250 = 194.86 Jmol-1
9. Calculate the root mean square speed of methane molecules at 27°C.
Solution
Root mean square speed,
T = 27+ 273 = 300 K, M= 16. R= 8.314 x 107 erg k-1 mol-1
= 683.9 ms-1
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