Confusion on meaning of fugacity in scientific publication

in my research project in statistical mechanics, in the context of phase transition and condensation, I was reading the seminal paper of Yang and Lee titled: "Statistical theory of equations of state and phase transitions I. Theory of condensation” as freely available at this link and there are two things here confusing me:

1. y is defined as the following familiar expression ${(\frac{2\pi m k t}{h^2})}^{\frac{3}{2}} \exp{(\frac{\mu}{kT})}$, so for given mass of particles we clearly have that the fugacity is a function of temperature, but then when the authors analyze the grand partition function in the thermodynamic limit of infinite volume they take it as a function of y indeterminate and as a function of volume, but when they proceed to look at its analytical behavior they say for a given temperature and plot it as a function of y, but for a given temperature y should not be constant? (page 3 for example). On page 3 also, for the second case they mention they say as T varies the accumulation points $t_1,t_2$ also move along the y axis and they mention a critical temperature seemingly ignoring y determining T and vice versa.

2. Finally, in figures 1 and 2 they plot the density $\rho$ as a function of volume v which I though was taken to be $V \to \infty$ as I previously thought as the density is defined in this limit. This is in the first case they mention on page 3 where a region R is free of roots as $V \to \infty$ which is confusing me.

These two confusions are preventing me from understanding the authors' intentions and I cannot pinpoint their source, therefore I am asking here in the hopes someone can clarify these things for me. I thank all helpers.

• $T$ should be thought as being fixed in the analysis; the curves that are plotted are isotherms. Then, $y$ is used instead of $\mu$ because it is more convenient, but these two quantities are related by the very simple definition you gave, so one can use one or the other as one prefers (when they plot various quantities as functions of $\log y$, they are basically plotting them as functions of $\mu$). They do not plot $\rho$ (the density) as a function of $v$ (the specific volume, $v=1/\rho$), they plot $p$ (the pressure) as a function of $v$. This gives the standard $p-v$ isotherms. – Yvan Velenik Jun 12 '16 at 16:48
• @YvanVelenik this seems like it could be an answer – Rococo Jun 12 '16 at 17:46
• @Rococo Well, I prefer to leave answers as comments when I don't have time to polish them to a satisfactory state. But thanks! – Yvan Velenik Jun 12 '16 at 17:54