  Low-dimensional quantum magnets, due to the existence of abundant exotic
quantum phases therein and experimental feasibilities in laboratories,
continues intriguing people in condensed matter physics. In this work, a
comprehensive study of Cu(NO$_3$)$_2$ $\cdot$ 2.5H$_2$O (copper nitrate
hemipentahydrate, CN), a spin chain material, is performed with multi-technique
approach including thermal tensor network (TTN) simulations, first-principles
calculations, as well as magnetization measurements in experiments. Employing a
cutting-edge TTN method developed in the present work, we determine the
couplings $J=5.13$ K, $\alpha=0.23(1)$ and Land\'e factors
$g_{\parallel}=2.31$, $g_{\perp}=2.14$ in an alternating Heisenberg
antiferromagnetic chain model, with which one can fit strikingly well the
magnetothermodynamic properties. Part of the fitted experimental data are
measured on the single-crystal CN specimens synthesized by us. Based on
first-principles calculations, we reveal explicitly the spin chain scenario in
CN by displaying the calculated electron density distributions, from which the
distinct superexchange paths are visualized. On top of that, we investigated
the magnetocaloric effect (MCE) in CN by calculating its isentropes and
magnetic Gr\"ueisen parameter (GP). Prominent quantum-criticality-enhanced MCE
was uncovered, the TTN simulations are in good agreements with measured
isentropic lines in the sub-Kelvin region. We propose that CN is potentially a
very promising quantum critical coolant, due to the remarkably enhanced MCE
near both critical fields of moderate strengths as 2.87 and 4.08 T,
respectively.
