Thermodynamic Temperature Scale

A perfect thermometer would show an equal amount of temperature difference for equal amount of heat transfer for all the substances, but this was impossible because the specific heat of all the substances is different.

       Before the invention of absolute temperature scale, the temperature was measured mostly by air or gas thermometers because the difference in readings of temperature in gas thermometers at different pressures was in-appreciably small (which was not possible for mercury and alcohol thermometers). This temperature scale was relative as it was dependent on the property (specific heat) of the substances.

             William Thomson thought of developing an absolute temperature scale which would be independent of the specific heat of the substance so that any substance used in the thermometer should show the same temperature difference for the same amount of heat transfer. This was the purpose of developing an absolute thermodynamic temperature scale.

Thermodynamic Temperature Scale

"A Thermodynamic temperature scale is defined as a temperature scale that is independent of the properties of the substance which is used to measure the temperature in the measuring devices such as thermometer, etc". It is also called as "Absolute Temperature Scale". 

More specifically, we can define a thermodynamic temperature scale that is independent of the working medium. Such a temperature scale offers great convenience for thermodynamic calculations. 

DERIVATION

To define a Thermodynamic temperature scale that does not depend on the thermometric property of a substance, Carnot principle can be used since the Carnot engine efficiency does not depend on the working fluid. It depends on the temperatures of the reservoirs between which it operates.

For two given heat/thermal reservoirs, all reversible heat engines (such as Carnot Engine) have the same thermal efficiency, regardless of the working substance. This fact provides the basis of the thermodynamic temperature scale.

Take the example of Carnot Engine, 

"Carnot's principle implies that the efficiency of a reversible heat engine is a function of temperature of hot and cold reservoirs". 

OR

➩ ηᵣₑᵥ ＝ f (Tᴄ,Tʜ)

➩ 1– Qᴄ/Qʜ ＝ f (Tᴄ,Tʜ)

         ➩ Qᴄ/Qʜ ＝ Φ (Tᴄ,Tʜ)    

                 ----------------------(1) 

( Q₂= Qᴄ; Q₁= Qʜ; t₁= Tʜ; t₂= Tᴄ )

There are many functional relations that would satisfy this equation but for the thermodynamics temperature scale, Kelvin selected one relation;

-------------------(2)

This is the "Thermodynamic Temperature Scale". This observation led Lord Kelvin to propose a Thermodynamic temperature scale. He proposed that "the thermodynamic temperature of a hot and cold reservoir be defined in such a way that their ratio was equal to the ratio of heat transfer from the hot reservoir to the heat transfer to the cold reservoir". 

According to equation (2), the ratio of energy absorbed to the energy rejected as heat by a reversible engine is equal to the ratio of the absolute temperature of the source and the sink. Thus, the Carnot efficiency;

☛ Earlier we derived the Carnot efficiency with an ideal gas as a medium. Carnot engine are thus not limited to an ideal gas and hold for Carnot cycles with any medium.

☛ The absolute zero or zero of the thermodynamic temperature scale is the value of T₂ at which the Carnot efficiency becomes unity.

   ( T₂＝ 0 ; η ＝ 1 )

Concepts Of Absolute Scale

Concept of absolute scale is applicable to anything which is scalar, any scalar can not have negative value in absolute terms. Thus, an absolute temperature scale having only positive values so the minimum possible value must be zero whether we can achieving or not, such scale is absolute scale. Temperature is a scaler so it's theoretical minimum value has to be zero.

Concepts Of Absolute Zero

An absolute temperature scale is one in which the lowest possible temperature is zero. An example of one used in the sciences is the Kelvin scale, where zero represents.

Absolute zero — the coldest temperature that is thermodynamically possible, when all molecules would theoretically have 0 energy.

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