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Therefore, $\langle V \rangle=\frac{\partial G}{\partial P}$ grows like $N$ and the relative fluctuation of the volume vanishes at the TL. …

of state in the form: $$ \frac{n}{V}= \frac{P}{RT}, $$ where $P,T$, and $R$ are the pressure, temperature and gas constant respectively, and $n$ and $V$ are the quantity of molecules (in mol) and the volume …

In the right hand side of the equation fo $dU$ you have an extensive quantity $dV$ which can be divided by either $N$ or $V$ without problems (the same for $dU$), and a couple af derivatives, $C_V$ a …

If the particles are confined in a finite volume, each cartesian component of the position ${\bf r}_i$ is an element of a subset of the real numbers $R$, each position vector ${\bf r}_i$ is inside a finite … volume $V$, and the $3N$ position $Q$ is an element of a finite volume subset of $R^{3N}$, the $3N$-fold cartesian product of $R$, of volume $V^{N}$. …

expansion coefficient, $\chi_T = - \frac{1}{V} \left( \frac{\partial{V}}{\partial{P}} \right)_T$ is the isothermal compressibility, and $C_V=\left( \frac{\partial{E}}{\partial{T}} \right)_V$ is the constant volume …

The formula $$ U_{thermal}=nf\cdot\frac{1}{2}kT $$ can be derived from the equipartition theorem and holds for systems of molecules such that their Hamiltonian is only made by $f$ quadratic terms. In …

The equation of state does not tell everything about a thermodynamic system. Moreover, the specific heat is not related to the value of the internal energy but to the variation of internal energy when …

Physically, it would correspond to an interval of increasing pressures where the volume would remain constant. That would be a highly unphysical behavior. …

The consequence for thermodynamics is that we are not obliged to integrate $p$ as a function of the volume. … Let's consider an isotherm $p(V)=\alpha V^2$ from a volume $V_1$ to a volume $V_2$. …

The region to the left of the coexistence line (in blue) corresponds to a homogeneous liquid (small volume = high density). … The region to the right of the red curve corresponds to the homogeneous vapor (high volume = low density). …

However, an equation of state connects pressure, volume, and temperature. … For example, if we use the equation of state to get the temperature as a function of volume and pressure, ($T=\tau(p,V)$) we can consider the compound function $$ \tilde U(p,V) = U(\tau(p,V),V). …

The most common method used in modern textbooks is confining the positions inside a finite volume $V$ with a one-body potential equal to zero inside the volume and equal to $ +\infty$ outside. … to the volume $V^N$. …