Thermo dynamical Consideration of Peltier Effect

Consider a copper-iron thermocouple with eth cold junction at the absolute temperature T₁ and the hot junction at the absolute temperature T₂. The thermoelectric current I flows in the direction. The flow of this current provides the necessary condition for the Peltier effect to take place and consequently the heat is absorbed at the hot junction and generated at the cold junction. Let π₁ and π₂ be the Peltier coefficients of the cold and the hot junctions respectively.

Heat energy absorbed in t seconds at the hot junction = π2/t.
Heat energy evolved in t seconds at the cold junction = π2/t.

. :    Net energy absorbed = (π2 – π1)it                                … (1)

This energy is used in setting up a thermo emf E_p in the circuit.
The electrical energy developed in the thermocouple = Ep it.
. :    (π₂ – π₁) it = Ep it
Or    Ep = (π₂ – π₁)                                        … (2)

The thermocouple acts like a heat engine in which an amount of heat energy π₂  is absorbed from the source at a higher temperature T₂ and an amount π₁ is rejected top the sink at the lower temperature T1. Applying second law of thermodynamics, we have
    π₁/T₁ = π₂/T₂, or π₂/π₁ = T₂/ T₁
or    π₂ – π₁/π₁ = T₂ – T₁ /T₁
or     (π₂ – π₁) = π₁/T₁ (T₂ – T₁)
or    (π2 – π₁) ∞ (T₂ – T₁)                (π1/T1 is constant)                … (3)
From Eqs. (2) and (3), we have
        Ep ∞ (T₂ – T₁)

Thus the emf developed in the thermocouple is directly proportional to (T₂ – T₁), the temperature difference between the two junctions. The graph between Ep and T should be a straight line.
But it has been found experimentally that the graph between Ep and T is a parabolic curve. This shows that Peltier effect alone cannot explain the thermo-electric phenomenon in a thermocouple. This led to the discovery of Thomson effect. Thomson said that there must be some additional source of EMF which has not been taken into account in Peltier effect. This additional EMF is due to the different parts of the same metal being at different temperatures. The combined EMF due to the Pletier effect and Thomson effect showed the agreement with the experimental results.