= Solution
\b[Structure through pair correlations.] For a homogeneous isotropic monatomic liquid of <number density> $n=N/V$, the <radial distribution function> $g(r)$ is defined by
$$
\boxed{dN_{\rm neighbours}=4\pi n r^2g(r)\,dr.}
$$
It compares the conditional density around a tagged particle with the bulk density. A repulsive core makes close neighbours unlikely; a first peak identifies a preferred coordination shell, and damped peaks at larger radii describe short-range order. Far away, $g(r)\to1$ in the bulk limit. Unlike a crystal, an ordinary liquid has no indefinitely repeated positional order. An ideal uncorrelated gas has $g(r)=1$ away from the tagged particle. The shell factor $4\pi r^2$ must not be confused with $g(r)$ itself.
\Image[/past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2017-q2-liquid-correlations.png]
{title=Original schematic radial distribution function showing a core exclusion region and damped liquid-like coordination shells; the curve is illustrative rather than measured data}
The same pair structure enters the <static structure factor> measured through scattering. In the bulk isotropic convention,
$$
S(k)=1+4\pi n\int_0^\infty[g(r)-1]r^2\frac{\sin kr}{kr}\,dr.
$$
Thus real-space coordination and reciprocal-space scattering peaks describe the same correlations.
\b[Energy from the structure.] If interactions are sums of a central <pair potential> $v(r)$, the <equipartition theorem> gives the classical <kinetic energy> $3Nk_BT/2$. Count the interaction of a tagged particle with its neighbours and divide by two to avoid counting each pair twice. The <pair-correlation expression for internal energy> is
$$
\boxed{U=\frac32Nk_BT+2\pi Nn\int_0^\infty r^2v(r)g(r)\,dr.}
$$
This makes structure thermodynamically useful: short-distance repulsion and the attractive region contribute with different weights. Additional internal molecular modes or many-body interactions require extra terms. The formula is a classical bulk pair-potential result, not a universal <energy> formula for every liquid.
\b[Pressure and the dilute expansion.] The mechanical <virial theorem> relates the pair forces to <pressure>. The <virial pressure from pair correlations> is
$$
p=nk_BT-\frac{2\pi n^2}{3}\int_0^\infty r^3v'(r)g(r)\,dr.
$$
At low density, $g(r)\simeq e^{-\beta v(r)}$, giving the <virial expansion>
$$
\boxed{\frac p{nk_BT}=1+B_2(T)n+B_3(T)n^2+\cdots,\qquad
B_2(T)=-2\pi\int_0^\infty[e^{-\beta v(r)}-1]r^2\,dr.}
$$
The <second virial coefficient> has units of <volume> per particle; higher coefficients encode additional cluster correlations. A hard-sphere exclusion diameter $d$ gives $B_2=2\pi d^3/3>0$. Attractions contribute negative terms and can make $B_2$ negative. Although the pair-correlation formulas can describe a dense liquid when its actual $g(r)$ is known, truncating the virial series after a few terms is normally a dilute-fluid approximation. It should not be silently extrapolated through condensation or into a dense liquid.
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