Qu.: THE THERMODYNAMIC SCALE AND ITS REALIZATION

The thermodynamic scale thus defined depends upon the properties of the thermodynamic substance; the perfect gas scale which depends on the properties of perfect gas.

The efficiency of a Carnot’s, however, is a thermodynamic property which is a function of alone. This fact allows us to define a scale of temperature which is absolute in the sense of being independent of the properties of any substance whatsoever and which is based only on the fundamental principles of thermodynamics. It is called the absolute thermodynamic temperature scale.

The symbol T represents temperature on the thermodynamic scale as yet undefined, but we agreed to use T for temperature on the ideal gas scale on the understanding that the scale are numerically identical.

θ Let us consider the C1 absorbs heat Q1 from a reservoir at temperature 1 and θ rejects heat Q2 to a reservoir at the temperature 2 . Its efficiency is a function only of the θ θ 1 and 2 that is

Q 2 θ θ η = 1 - = φ(1 , 2 ) (1) Q1

Q 1 θ θ = f(1 , 2 ) (2) Q2

θ θ where both φ and f are universal functions of 1 and 2 .

θ θ θ> θ > θ Let us consider another Carnot engine C2 operate between temperatures 2 and 3 ( 1 2 3 ), θ and let the amount of working substance be so adjusted that C2 absorbs heat Q2 at temperature 2 θ and rejects Q3 at 3 . Hence

Q 2 θ θ = f(2 , 3 ) (3) Q3 θ When engines C1 and C2 are coupled together, the net heat exchange suffered by 2 reservoir is θ θ zero, and in effect, we have a single Carnot engine C operating between the temperatures 1 and 3 .

Fig. 1 Derivation of thermodynamic temperature using Carnot engine.

Therefore,

Q 1 θ θ = f (1 , 3 ) (4) Q3

Since

Q Q Q 1 = 1 2 (5) Q3 Q 2 Q 3

We have

θθ θθ θθ f(13 , ) = f ( 12 , ) . f ( 23 , ) (6)

θ θ The left side of qu.(6) is independent of 2 , therefore 2 must cancel out from the product on the right. This can happen only if the functions f have the form

T (θ ) f (θ , θ ) = 1 (7) 1 2 θ T (2 )

where T (θ ) is a universal function of the empirical temperature θ . According to a suggestion by Lord , the thermodynamic temperature is defined by taking the value of the function T itself as the temperature, that is, T ( θ1) = T 1 and T ( θ2) = T 2 , which gives us

Q T 1 = 1 (8) Q2 T 2 Thus the ratio of two temperatures on the thermodynamic scale is equal to the ratio of the quantities of heat absorbed and rejected by a Carnot engine operating between these two temperatures.

If one reservoir is at the of water temperature Ttr and the other reservoir has an arbitrary temperature T , and if the heat exchange by a Carnot engine at these temperatures are Qtr and Q respectively, then

Q T = (9) Qtr T tr

When the numerical value 273.16 is arbitrarily assigned to Ttr , we obtain the Kelvin scale on which the temperature T is defined by the equation,

Q T = (273.16 K) (10) Qtr

Now, Q denotes the absolute value of the heat exchange by a Carnot engine with a reservoir at thermodynamic temperature T . Since the Q' s are all positive quantities , it follows from Eq. (8) that that the thermodynamic temperature is necessarily positive. Further we see that = Q2 decreases as T 2 is lowered. The zero of the scale is then that temperature at which Q2 0 . A

Carnot engine operating between an arbitrary temperature T1 and the , T2 = 0 would reject no heat at this temperature Q2 = 0 , and all the heat absorbed would be converted into work. The Carnot engine would violate the second law. This could be interpreted to mean that the absolute zero is not attainable. However it should be noted that since Q2 = 0 , the isothermal process at the = temperature of the sink Q2 0 would also be adiabatic , a contradiction in terms rendering the Carnot cycle ambiguous to say the least. The unattainability of the absolute zero is one of the statements of the third law of thermodynamics. At this juncture, basing our discussion on the 2 nd law, we can say this = = much that Q2 = 0 as Q2 0 and that Q2 0 is the lowest conceivable temperature.

The existence of negative temperatures has been proved to experiments which utilized the magnetic and thermal properties of a nuclear magnetic sub-system. Paradoxically negative temperature do not mean temperatures lower than absolute zero, negative temperatures are hotter than any positive temperature. Moreover negative temperatures never occur in complete systems in equilibrium so for normal purposes we take the thermodynamic temperature as positive. We have seen that the Carnot cycle of an ideal gas gives us the relation

Q T 2 = 2 Q1 T 1 where T2 and T1 were the temperatures of the sink and source, respectively, on the ideal gas scale. We have proved that the ratio of two ideal gas temperatures is equal to the ratio of the corresponding thermodynamic temperatures. Assigning the same numerical value 273.16 to the triple point of water, the two scales are equivalent.

Operation of Carnot cycles for the purpose of determining thermodynamic temperatures is hardly feasible in practice. Quasi-static process can be approximated to in the laboratory but dissipative effects cannot be eliminated altogether to fulfill the stringent conditions reversibility. The thermodynamic temperature as defined in Eq. (20) cannot be determined directly. The thermodynamic temperatures are measured by constant volume gas corrected to give the ideal gas scale.

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