Nernst Heat Theorem

 

Describe Nernst Heat Theorrem.

How to find a quantitative relationship between ∆G and ∆H in a chemical reaction?

Nernst Heat Theorem:

In the earlier development of chemical thermodynamics, the difficult task was to ascertain a quantitative relationship between ∆G and ∆H in a chemical reaction and to find out ∆G from thermal data. i.e., from ∆H. To deal with this problem a preliminary discussion will be given which will deal with this problem ultimately leading to the Nernst heat theorem and third law of thermodynamics.

1.   Joule and Thomson’s concept: According to this concept, the two terms ∆G and ∆H are identical. To support their statement they gave well known example of Daniel cell where ∆G and ∆H had been found to be the same.

2.   Berthelot’s concept: Berthelot came forward to give his famous concept as:

“When heat is given out in a reaction, the free energy of the system decreases.”

Although this concept described relation between ∆G and ∆H in a qualitative manner and was found to be true in a number of examples such as condensed systems at ordinary temperatures, this system got failed in a number of other cases.

3.   Gibb’s Helmholtz concept: For the first time Gibb’s and Helmholtz deduced a quantitative relation between ∆G and ∆H for a chemical reaction when it is carried out at constant pressure and called it Gibb’s Helmholtz equation.

                                 ∆G = ∆H +T[δ(∆G)/δT]_P               eq……(1)

Case I : When δ(∆G)/δT = 0 eq (1) becomes

                                  ∆G = ∆H

An example of this case is found in Daniel cell where ∆G is found to be the same as ∆H.

Case II : When ∆G ≈ 0  eq (1) becomes as follows

                                  ∆H = -T[δ(∆G)/δT]_V  

From the above equation it follows that ∆H must acquire a large value. An example of this is the fusion of solids at their melting point where ∆G is almost zero and ∆H is very large.

Case III : When T =0K eq (1) yields,

                                   ∆G = ∆H

This condition has little practical importance because it is impossible to carry out a process at the absolute zero.

Except in the above three cases an inequality exists between ∆G and ∆H in all the processes

Equation (1) therefore allows the calculation of ∆H from ∆G. But the reverse seems too to be practicable. Thus the main limitation of Gibb’s Helmholtz equation is that it does not allow the calculation of ∆G from ∆H.

4.   Richard’s concept: In 1902 Richard measure the EMF of cells at low temperature and concluded that δ(∆G)/δT gets decreased gradually with the lowering of temperature. In other words ∆G and ∆H approach each other more closely at extremely low temperatures i.e.

                                Lim_T→0 ∆G =Lim_T→0  ∆H

5.   Nernst Heat Theorem: In 1906 Nernst relied on data given by Richard and made a bold statement for a process in condensed system that the value of δ(∆G)/δT approaches zero asymptotically as the absolute zero is approached. In other words ∆G and ∆H curves meet each other at a short region above absolute zero and run coinciding with each other up to absolute zero. This behavior is shown by full lines not dotted lines. The dotted line curve reveals that the two values ∆G and ∆H not only become equal to each other at absolute zero but their approach to each other becomes rapid and not gradual.

Mathematically it can be written as

                               Lim_T→0[δ(∆G)/δT]_P  = Lim_T→0[δ(∆H)/δT]_P  = 0        eq……(2)

Nernst heat theorem can also be explained as “There is no change in entropy of the system when the process occurs in the vicinity of absolute zero”. This statement can be proved using equation

                                dG = VdP – SdT

At constant pressure we have

                                [δG/δT]_P  = -S

For a process G and S can be replaced by ∆G and ∆S respectively, hence

                                [δ(∆G)/δT]_P  = -∆S

According to eq (2) left hand side of the equation will become zero at 0 K, hence

                               Lim_T→0 ∆S = 0

            Lim_T→0 (S₂ – S₁) =0

This implies that when T =0, then S₁ = S_2.

Nernst heat theorem can be used for the study of variation in molar specific heat at constant pressure for the process occurring in the neighborhood of absolute zero. The molar heat capacity at constant pressure can be defined as

                              C_P = (δH/δT)_P

For the change in H and C_p, the above equation can be written as

                              ∆C_P =[δ(∆H)/δT]_P

In the vicinity of absolute zero, the above expression can be written as

                           Lim_T→0(∆C_P) = Lim_T→0[δ(∆H)/δT]_P    

According to Nernst heat theorem the right hand side of the equation is equal to zero, therefore              

                                          Lim_T→0(∆C_P) = 0

                           Lim_T→0(C_P2-C_P1) = 0

                         Lim_T→0C_P2 = Lim_T→0C_P1

According to above equation the molar heat capacity at constant pressure during any transformation at absolute zero remains unchanged.