Question

acities at constant pressure for O(g), Odg) and Olg) are 21.90, 29.36 and 39.20 I K 3mol, respectively. Discuss the difference using the equipartition theorem.

Answer 1

According to the equipartition of energy (theorem), every degree of freedom has an average energy of $rac{1}{2}k_{B}T$ in thermal equilibrium and contributes   $rac{1}{2}k_{B}$ to the system's heat capacity. Here $k_{B}$ is Boltzmann constant and T is absolute temperature.

i.e. if degree of freedom of an atom is 'f' then we can write total energy of the atom as

And if there is one moles of the atom then

   ..................................(1)

Where is universal gas constant and N is Avogadro constant.

The molar heat capacity at constant volume ( $C_{v}$ ), is defined as

..............................(2)

or,   

or, ................................(3)

And the relation between molar heat capacity at constant volume  ( $C_{v}$ ) and the molar heat capacity at constant pressure is

  

or, ..............................(4)

Now for oxygen atom (i.e. O) :

As oxygen atom is monoatomic, so it will have only 3 translational degrees of freedom. i.e. f = 3.

So.

Therefore

For oxygen gas (i.e. O₂) :

Total degree of freedom of O ₂ (diatomic molecule) is 5 (3 translation + 2 rotation) i.e f = 5 .

So,

Therefore

For ozone gas (i.e. O₃) :

Total degree of freedom of O₃ (triatomic molecule) is 7 (3 translation + 3 rotation + 1 vibration) i.e f = 7 .

So,   

Therefore   

So, according to equipartition theorem the heat capacity at constant pressure for O (g) , O₂ (g) and O₃ (g) is 20.785 , 29.099 and 37.413 J/K-mol respectively.

Hence, we can see that these calculated values differ from originally given values (which are 21.90 , 29.36 and 39.20 J/K-mol) by 1 to 2 units.

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