March 2011 - Career Point

Perfect gas equation :
From the kinetic theory of gases the equation of an
ideal gas is
pV = RT for a mole
and
pV = M
m RT for any mass m
Avogadro number (N) and Boltzmann constant (k) :
The number of entities in a mole of a substance is
called the Avogadro number. Its value is
6.023 × 10 23 mol –1 .
The value of the universal gas constant per molecular
is called Boltzmann constant (k). Its value is
1.38 × 10 –23 J K –1 .
Degrees of Freedom : Principle of equipartition of
energy :
The number of ways in which energy may be stored
by a system is called its degrees of freedom.
Principle of Equipartition of Energy : This
principle states that the total energy of a gas in
thermal equilibrium is divided equally among its
degrees of freedom and that the energy per degree of
freedom is kT/2 where T is the temperature of the
gas. For a monoatomic atom the number of degrees
of freedom is 3, for a diatomic atom it is 5, for a
polyatomic atom it is 6.
Hence the energy of a mole of a monoatomic gas is
⎛ 1 ⎞ 3
µ = N ⎜3 × kT ⎟ = RT
⎝ 2 ⎠ 2
Which is the same as that given by the kinetic theory.
For a mole of diatomic gas µ
⎛ 1 ⎞ 5
= N⎜5 × kT ⎟ = RT
⎝ 2 ⎠ 2
For a mole of polyatomic gas µ
⎛ 1 ⎞
= N⎜6 × kT ⎟ = 3RT
⎝ 2 ⎠
When the irrational degrees of freedom are also taken
into account, the number of degrees of freedom
= 6n – 6 for non-linear molecules
= 6n – 5 for linear molecules
where n = number of atoms in a molecule.
Kinetic Temperature :
The kinetic temperature of a moving particle is the
temperature of an ideal gas in thermal equilibrium
whose rms velocity equals the velocity of the given
particle.
Maxwellian distribution of velocities :
In a perfect gas all the molecules do not have the
same velocity, rather velocities are distributed among
them. Maxwell enunciated a law of distribution of
velocities among the molecules of a perfect gas.
According to this law, the number of molecules with
velocities between c and c + dc per unit volume is
dn = 4πna 3 2
bc
e − c 2 dc where
m m
b = and a =
2kT 2πkT
and the number of molecules with the velocity c per
unit volume is
n c = 4πna 3 2
bc
e − c 2
The plot of n c and c is shown in the figure. The
velocity possessed by the maximum number of
molecules is called the most probable velocity
α = 2 kT / m
The mean velocity
c =
α c C rms
8 kT / mπ
and v rms = 3 kT / mπ
Conduction :
The transfer of heat through solids occurs mainly by
conduction, in which each particle passes on thermal
energy to the neighboring particle but does not move
from its position. Very little conduction occurs in
liquids and gases.
θ 1 θ 2
Q
d A
Consider a slab of area A and thickness d, whose
opposite faces are at temperature θ 1 and θ 2 (θ 1 > θ 2 ).
Let Q heat be conducted through the slab in time t.
⎛ θ1 − θ2
⎞
Then Q = λA ⎜ ⎟ t
⎝ d ⎠
where λ = thermal conductivity of the material.
This has a fixed value for a particular material, being
large for good conductors (e.g., Cu, Ag) and low for
insulators (e.g., glass, wood).
Heat Current : The quantity Q/t gives the heat flow
per unit time, and is called the heat current.
In the steady state, the heat current must be the same
across every cross-section. This is a very useful
principle, and can be applied also to layers or slabs in
contact.
Q dθ dθ θ
= – λA where the quantity =
1 − θ 2
t dx
dx d
called the temperature gradient.
Q
is
XtraEdge for IIT-JEE 25 MARCH 2011