This document discusses key concepts in thermodynamics including: 1. Degree of freedom is defined as the number of possible independent ways a system can have energy. 2. Equipartition theorem states that each degree of freedom contributes 1/2kT to the internal energy of a molecule. 3. The internal energy of an ideal gas can be written as a function of the number of moles, degrees of freedom, and temperature. 4. Real gases are better modeled by the van der Waals equation of state, which accounts for molecular size and attractive forces between molecules.

1. B.Sc. Physics THERMODYNAMICS
Muhammad Azhar (MS Physics, NEDUET) azharphysicist@gmail.com
Why a fog appears just after opening a bottle of a
cold drink?
When the bottle of cold drink (Carbonated drink) is
opened. The pressure inside the bottle drops and
hence the temperature inside the bottle also
decreased, due to which the amount of moisture that
air can hold also drops. Then due to condensation the
droplets of water condense out of the air and the fog
is appears just after opening the bottle.
DEGREE OF FREEDOM
It is defined as the number of possible independent
ways in which a system can have energy.
EQUIPARTITION OF ENERGY
Equipartition of energy is basically a theorem that
derived by James Clark Maxwell according to which;
“When thenumber of molecules is large, the average
energy per molecule is
1
2
𝑘𝑇 for each independent
degree of freedom”
Simply we can say “Each degree of freedom
contributes
1
2
𝑘𝑇 to the internal energy.”
INTERNAL ENERGY OF AN IDEAL GAS
The total internal energy of “n” moles of an ideal gas
can be written as the number of molecules (=n𝑁𝐴)
times the average energy per molecule:
𝑈 = 𝐸𝑖𝑛𝑡 = (n𝑁𝐴)( 𝐾𝑡𝑟𝑎𝑛𝑠) = (n𝑁𝐴)(
𝑓
2
𝑘𝑇)
𝑈 = 𝐸𝑖𝑛𝑡 =
𝑓
2
𝑛𝑅𝑇
Where 𝑓 is the degree of freedom.
For monoatomic gas 𝑓 = 3
𝑈 = 𝐸𝑖𝑛𝑡 =
3
2
𝑛𝑅𝑇
For diatomic gas 𝑓 = 5
𝑈 = 𝐸𝑖𝑛𝑡 =
5
2
𝑛𝑅𝑇
For polyatomic gas 𝑓 = 6
𝑈 = 𝐸𝑖𝑛𝑡 =
6
2
𝑛𝑅𝑇 = 3𝑛𝑅𝑇

2. B.Sc. Physics THERMODYNAMICS
Muhammad Azhar (MS Physics, NEDUET) azharphysicist@gmail.com
KINETIC MOLECULAR THEORY OF GAS
REAL GAS: Vander Waals Equation of State
The ideal gas law works if the gas density is low enough. For many real gases, however, the ideal gas is not a
good approximation. In those cases, we must use the Vander Waals Equation of state, which frees us from two of
the assumptions of the kinetic theory:
1. The molecules are particles with no physical extant
2. The molecules only exert contact forces on one another
The assumption 1 can be disregarded as we model gas molecules as sphere of radius “r”.
The assumption 2 can be also disregarded as the molecules actually exert an attractive on one another due to
electromagnetic force.
Think about the motion of a single gas molecule. It cannot move through the entire volume “V” of the container
because other molecules take up some of that volume. We can write the relation:
𝑉𝑎𝑣𝑎𝑖𝑙𝑏𝑙𝑒 = 𝑉 − 𝑉 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠
For convenient we can write the volume occupied by the molecules in terms of “b” is the volume of molecule per
mole. The volume occupied by the molecules is then the number of moles “n” multiplied by “b”.
𝑉𝑎𝑣𝑎𝑖𝑙𝑏𝑙𝑒 = 𝑉 − 𝑛𝑏
Because molecules takes up some volume, we correct the ideal gas law 𝑃𝑉 = 𝑛𝑅𝑇 as:
𝑃 𝐶𝑙𝑎𝑢𝑠𝑖𝑢𝑠(𝑉 − 𝑛𝑏) = 𝑛𝑅𝑇
Above equation known as Clausius Equation of State.

3. B.Sc. Physics THERMODYNAMICS
Muhammad Azhar (MS Physics, NEDUET) azharphysicist@gmail.com
This equation models the molecules as sphere “r”, but it does not take into account the field force they exert on
one another.
We can modify the Clausius Equation of State to include that attractive filed force. Imagine the motion of one
molecule as it approaches one of the container’s walls, where it pulled back by attractive force of the other
molecules. The force molecule exerts on the wall is smaller than it would be if there were no field forces. Because
all the molecules experience such a force, the pressure exerted on the walls is reduced:
𝑃 = 𝑃 𝐶𝑙𝑎𝑢𝑠𝑖𝑢𝑠 − 𝑃𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛