In an "ideal gas thermometer", the relation $PV=nRT,$ or equivalently $PV=NkT,$ is used to deterime $T.$ Typically, the volume is held fixed, and the pressure is measured with a pressure gauge. The thermometer has to be calibrated using known fixed points.

Call the fixed volume $V_0,$ and take as the fixed point the pressure at the freezing and boiling points of water, assumed to be at $0\mathrm{.}0C$ and $100\mathrm{.}0C.$ Call these pressures $P_b$ and $P_f.$

Then the temperature in kelvin, given by the thermometer, would be

$T[K]=\frac{100}{P_b-P_f}P.$

In this question, assume that $0C=273\mathrm{.}15K.$

$($a$)$ Show that upon substitution of $PV=NkT,$ this formula correctly gives $T[K]=T.$

$($b$)$ Show that such a thermometer, using an imperfect gas that obeys the equation

$(P+a\frac{N^2}{V_{\text{@}}^2})(V_{\text{@}}-Nb)=NkT,$

will still give a correct reading for the temperature, provided that it is calibrated at $0C$ and $100C$ assuming a linear equation of the form $T[K]=T_{\text{@}}+mP.($i$.$e$.$ find $T_{\text{@}}$ and $m,$ and show that the thermometer works.$)$

$($c$)$ Explain why the imperfect gas thermometer fails in the low temperature limit$.$