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I have been doing some statistical mechanics and at the beginning of the course, I had seen this statement:

$$T^\circ (K) = 273.16\frac{\lim_{V\rightarrow \infty} (PV)_{\text{system}}}{\lim_{V\rightarrow \infty} (PV)_{\text{triple point of water}}}$$

Where, $T^{0}$ is the ideal gas temperature in Kelvin, $P$ is the pressure, and $V$ is the molar volume.

I don't understand this definition of the temperature of a gas in the ideal regime of $V\to \infty$, and why this is considered a rigorous definition of temperature of an ideal gas.

My question is, what is the inspiration behind such a definition of ideal temperature? Why does the triple point of water come into the picture here? Is this trying to tell us that the term $$\frac{\lim_{V\to \infty} (PV)_{\text{triple point of water}}}{273.16} = R$$ Where $R$ is the ideal gas constant? If yes, then how?

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The reason that there is a triple point is that temperature is not easy to define as an absolute thing. We were only able to do so because of our knowledge of an absolute zero temperature. The 0th law of thermodynamics implies the existence of an empirical temperature, but more commonly it will be easier to define it relative to temperatures of other systems.

The 0th law states that if two systems are in equilibrium with another, then they are at the same temperature. For an ideal gas, it is useful to define a reference temperature, to which all others are measured. The triple point happens to be a very good one, because it is a singular temperature at any given Pressure and Volume, whereas all other phase transitions in water have a number of different P/V combinations. For instance, the equation of state for an ideal gas shows that at any given temperature, there is a P/V curve which is an isotherm.

We are required to use a point of reference to determine a practically usable temperature scale. Since the triple point of water is a known value in an absolute temperature scale, we use it to fix a practical temperature scale.

EDIT: https://ocw.mit.edu/courses/physics/8-333-statistical-mechanics-i-statistical-mechanics-of-particles-fall-2013/lecture-notes/MIT8_333F13_Lec1.pdf pages 2-3 give a good explanation of the thought process for this definition.

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The limit $V \to +\infty$ comes from the fact that for a given number of moles/gas particles, effects of interactions go down as the volume increases, because molecules are less and less likely to run into each other. This means that $\lim_{V \to \infty} (PV)$ can be deduced from the $PV$ of the equivalent ideal gas. But for an ideal gas, $PV \propto T$. So all things put together, $\lim_{V \to \infty} (PV)_{\mathrm{system}} \propto T_{\mathrm{system}}$.

The other terms just correspond to the definition of temperature in Kelvins. By (historical) convention, we have decided that the so-called triple point of water (which correspond to the unique values of $P$ and $T$ such that all three phases of water exist in equilibrium at the same time) was $T = 273.16\,\mathrm{K}$ (the pressure is not relevant here). So by convention/definition, if the system sits at the same temperature as the triple point of water, its temperature is $T_{\mathrm{system}} = 273.16\,\mathrm{K}$. You can check indeed in your formula that because the numerator and the denominator cancel when the system is at the triple point of water, the formula yields $T = 273.16\,\mathrm{K}$.

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