Thermodynamic Temperature Scale
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[edit] Definition
The best known thermodynamic temperature scale is the kelvin scale.
[edit] Discussion
By the Second Carnot Principle, any two reversible heat engines running between thermal reservoirs of temperatures TH and TC must have the same efficiency, where efficiency is defined as η th = 1 - (QC / QH). So of course any two such reversible heat engines must likewise have the same ratio of QC / QH. So there is a function f relating the temperatures of the two thermal reservoirs to the ratio QC / QH, that is, some function f such that
f(TH , TC) = QC / QH
and this function holds good for any reversible heat engine.
Suppose this function takes is its input temperatures in degrees celsius. We shall use this function to derive the kelvin scale from the celcius scale. Let us define the temperature in kelvin of a body at T ºC to be given by 273.16 f(TwºC, T ºC), where TwºC is the triple point temperature of water in centigrade. We shall explain later where the number 273.16 comes from.
Now QC / QH = QCQw / QHQw = (QC / Qw) / (QH / Qw)
It follows from the defintion of f that
QC / Qw = f(TwºC , TCºC)
and that
QH / Qw = f(TwºC , THºC)
Substituting these functions into the equation for QC / QH above, we then have
QC / QH = f(TwºC , TCºC) / f(TwºC , THºC)
Multiplying the top and bottom of the right hand side of the equation by 273.16, we get
QC / QH = 273.16 f(TwºC , TCºC) / 273.16 f(TwºC , THºC)
Or, by definition of the kelvin scale
QC / QH = TC in kelvin / TH in kelvin.
You will notice that the constant factor 273.16 cancels out: we might have chosen any number at all and we would still have produced a thermodynamic temperature scale. The kelvin scale is defined by this choice: it allows us to convert temperatures from Celsius to kelvin simply by adding a constant, so it is a convenient choice in practice. The Rankine scale, produced by choosing the constant to be 491.67, stands in a similar relationship to the Fahrenheit scale.
