Lecture Note on VISCOSITY OF A GAS -Dr S P Singh Department of Chemistry, A N College, Patna

A sketchy summary of the main points

Viscosity of gases, relation between mean free path and coefficient of viscosity, temperature and pressure dependence of viscosity, calculation of collision diameter from the coefficient of viscosity

Viscosity is the property of a fluid which implies resistance to flow. Viscosity arises from jump of molecules from one layer to another in case of a gas. There is a transfer of momentum of molecules from faster layer to slower layer or vice-versa. Let us consider a gas having laminar flow over a horizontal surface OX with a velocity smaller than the thermal velocity of the molecule. The velocity of the gaseous layer in contact with the surface is zero which goes on increasing upon increasing the distance from OX towards OY (the direction perpendicular to OX) at a uniform rate . Suppose a layer ‘B’ of the gas is at a certain distance from the fixed surface OX having velocity ‘v’. Two layers ‘A’ and ‘C’ above and below are taken into consideration at a distance ‘l’ (mean free path of the gaseous molecules) so that the molecules moving vertically up and down can’t collide while moving between the two layers.

Thus, the velocity of a gas in the layer ‘A’ ------(i) = + Likely, the velocity of the gas in the layer ‘C’ ------(ii) The gaseous molecules are moving in all directions due= to −thermal velocity; therefore, it may be supposed that of the gaseous molecules are moving along the three Cartesian coordinates each. Consequently, of the total molecule will be moving upward and of the molecules will be moving downward the layer ‘B’ at any instant. There is thus, a continuous interchange of molecules between the ℎ layers A and C. ℎ Let, n = no. of molecules per cc m = mass of each gaseous molecule = average velocity of the gas

The number of molecules crossing the layer ‘B’ downward or upward per unit area per second

The momentum carried by all the molecules moving downward and crossing the layer ‘B’ per unit area ------= ------(iii) Similarly,= the momentum + carried by all the molecules moving upward and crossing the layer ‘B’ per unit area ------(iv) Hence the net transfer of momentum per unit area of the layer per second = −

= + − − ------ ------(v) In the steady state= ∙of flow of the gas, the transfer of momentum per unit area per sec = The tangential viscous force per unit area of the layer due to viscosity

where, (eta) = coefficient of viscosity ⇒ ∙ =

∴ = ∙ ( ---- (vi) The CGS unit of viscosity is Poise or dyne sec/cm2 and the SI unit is Ns.m-2. = ∙ , density = per unit volume) Effect of Pressure and Temperature on Viscosity 1. Effect of Pressure The density of the gas increases with pressure but the mean free path l decreases in the same ratio, so that remains unchanged. Thus the coefficient of viscosity is independent of pressure at a given temperature. It can also be shown as follows;

= ∙ But = ∙ = √

∴ = ∙ √ ------(vii) The equation= (vii)√ clearly shows that the viscosity is independent of pressure. 2. Effect of Temperature

On substituting, the value = ∙ of ‘ =’ and ∙ ‘ ’ in the equation (v),

where, & / / = × √ = = √

/ / √ () / = √

/ = √ where, & / = × = ( = ) ------(vii) / = √ Thus, the viscosity of a gas the square root of the absolute temperature while it is inverse in the case of liquid (The coefficient∝ of viscosity of liquid will be decreased as the temperature increases).

Experimentally, however it is observed that the increase in viscosity is somewhat higher than the predicted by the relation established. The deviation is due to the existence of intermolecular attraction which was not taken into account in the Kinetic Theory. A satisfactory empirical relation between viscosity and temperature has been put forward by in the form

√ = where K and C are the constant. C is called Sutherland’s constant. 1 + Calculation of collision diameter from the coefficient of viscosity

( , = ∙ = ∙ , density = per unit volume) where, & / / = × √ = = √

/ / √ () / = √ ℎ, =

/ = √ where, & / = × = ( = )

(where, N= Avogadro No) ……… (i) (This expression of viscosity/ is in terms of molar mass, M absolute temperature, T and = √ quantity , ) / = √ ……… (ii) / / ∴ = [ √ ] Thus, the collision diameter, can be calculated with the measured value of coefficient of viscosity. Alternatively, the values of Avogadro No, N and collision diameter, can be calculated from the expression given below for a gas obeying vander Waals equation of state; ……… (iii) 3/2 Using equation = (i), N ……… (iv) Using equation (i) & (ii), = ( ( ))( ) ½ ……… (v) = ( ) () ******