Kelvin

 ( 1848 Introductions ) 

The scale was designed on the principle that "a unit of heat descending from a body at the temperature ° of this scale, to a body at the temperature , would give out the same mechanical effect, whatever be the number ." Specifically, Thomson expressed the amount of work necessary to produce a unit of heat (the thermal efficiency) as $\backslash mu(t)\; (1\; +\; E\; t)/E$, where $t$ is the temperature in Celsius, $E$ is the coefficient of thermal expansion, and $\backslash mu(t)$ was "Carnot's function", a substance-independent quantity depending on temperature, motivated by an obsolete version of Carnot's theorem. The scale is derived by finding a change of variables $T\_\{1848\}\; =\; f(T)$ of temperature $T$ such that $dT\_\{1848\}\; /\; dT$ is proportional to $\backslash mu$. When Thomson published his paper in 1848, he only considered Regnault's experimental measurements of $\backslash mu(t)$. That same year, James Prescott Joule suggested to Thomson that the true formula for Carnot's function was $$\backslash mu(t)\; =\; J\; \backslash frac\{E\}\{1+E\; t\},$$ where $J$ is "the mechanical equivalent of a unit of heat", now referred to as the specific heat capacity of water, approximately . Thomson was initially skeptical of the deviations of Joule's formula from experiment, stating "I think it will be generally admitted that there can be no such inaccuracy in Regnault's part of the data, and there remains only the uncertainty regarding the density of saturated steam". Thomson referred to the correctness of Joule's formula as "Mayer's hypothesis", on account of it having been first assumed by Mayer. Thomson arranged numerous experiments in coordination with Joule, eventually concluding by 1854 that Joule's formula was correct and the effect of temperature on the density of saturated steam accounted for all discrepancies with Regnault's data. Therefore, in terms of the modern Kelvin scale $T$, the first scale could be expressed as follows: $$T\_\{1848\}\; =\; 100\; \backslash frac\{\backslash log(T\; /\; \backslash text\{273\; K\})\}\{\backslash log(\backslash text\{373\; K\}\; /\; \backslash text\{273\; K\})\}$$ The parameters of the scale were arbitrarily chosen to coincide with the Celsius scale at 0° and 100 °C or 273 and 373 K (the melting and boiling points of water).: "The arbitrary points which coincide on the two scales are 0° and 100°" On this scale, an increase of approximately 222 degrees corresponds to a doubling of Kelvin temperature, regardless of the starting temperature, and "infinite cold" (absolute zero) has a numerical value of negative infinity.

To explain this definition, consider a reversible Carnot cycle engine, where $Q\_\backslash mathrm\{H\}$ is the amount of heat energy transferred into the system, $Q\_\backslash mathrm\{C\}$ is the heat leaving the system, $W$ is the work done by the system ($Q\_\backslash mathrm\{H\}\; -\; Q\_\backslash mathrm\{C\}$), $t\_\backslash mathrm\{H\}$ is the temperature of the hot reservoir in degrees Celsius, and $t\_\backslash mathrm\{C\}$ is the temperature of the cold reservoir in Celsius. The Carnot function is defined as $\backslash mu\; =W\; /\; Q\_\backslash mathrm\{H\}\; /\; (t\_\backslash mathrm\{H\}\; -\; t\_\backslash mathrm\{C\})$, and the absolute temperature as $T\_\backslash mathrm\{H\}\; =\; J/\backslash mu$. One finds the relationship $T\_\backslash mathrm\{H\}=\; J\; Q\_\backslash mathrm\{H\}\; (t\_\backslash mathrm\{H\}-t\_\backslash mathrm\{C\})/W$. By supposing $T\_\backslash mathrm\{H\}-T\_\backslash mathrm\{C\}=J\; (t\_\backslash mathrm\{H\}-t\_\backslash mathrm\{C\})$, one obtains the general principle of an absolute thermodynamic temperature scale for the Carnot engine, $Q\_\backslash mathrm\{H\}\; /\; T\_\backslash mathrm\{H\}\; =\; Q\_\backslash mathrm\{C\}\; /\; T\_\backslash mathrm\{C\}$. The definition can be shown to correspond to the thermometric temperature of the ideal gas laws.