# Ideal gas temperature definition

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?

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 $$T = 273.16\,\mathrm{K}$$.