Radical polymerization, a type of chain-growth polymerization. Peroxide bond cleavage initiates reactive chains; propagation adds monomer to a radical end while regenerating a radical end on the longer chain. Termination and chain transfer can then stop or redirect growth. In the propagation step there is no obligatory small-molecule condensate, so this is not a step-growth condensation.

Addition consumes the carbon–carbon double bond of vinyl chloride and retains chlorine as a side substituent. The interior repeat of A is thereforenamely polyvinyl chloride. Benzoyl peroxide supplies initiating free radicals; head-to-tail addition gives the usual backbone connectivity. End groups and tacticity are not uniquely specified by this repeat formula.

The crystalline expression diverges at $\varphi ={\varphi }_{\mathrm{cp}}\simeq 0.907$, where the cage free area vanishes. This is physically different from the spurious $\varphi =0.5$ pole produced by extrapolating the low-density fluid approximation. A real compression path cannot be inferred by following that fluid pole and then abruptly switching curves. At low density it follows a fluid equation of state, while at sufficiently high density an ordered arrangement can leave more accessible rattling configurations than a disordered crowded arrangement. This is entropy-driven hard-sphere crystallization, here in two dimensions: allowed hard-core configurations have the same interaction energy, so configurational entropy drives the phase competition. An equilibrium first-order coexistence interval would have common pressure and chemical potential, giving a constant-pressure segment as mean density changes. The two asymptotic expressions alone neither locate that interval nor determine the detailed two-dimensional melting sequence. They require a consistent free-energy comparison and trustworthy approximations in the transition region.

Insert the stipulated near-close-packing relation for $R$ into the hard-disc cage approximation:Writing $y=\sqrt{{\varphi }_{\mathrm{cp}}a/(N{a}_d)}$, we have ${\partial }_{a}y=y/(2a)$ and therefore ${\partial }_a\ln \mathcal{A}=y/[a(y-1)]$. From the supplied crystalline canonical partition function,Its close-packing scale is set by the triangular lattice packing of discs, with ${\varphi }_{\mathrm{cp}}=\pi /(2\sqrt{3})$ in the ideal geometry.

In the circular hard-disc cage approximation, moving the central disc towards a neighbour brings contact after a centre displacement $R-2r$. ThusAt close packing, $R=2r$ and $\mathcal{A}=0$. The radius concerns the accessible positions of the moving centre; subtracting only one disc radius would incorrectly leave a finite rattling area at contact. The real cage is bounded by curved exclusion arcs, so the circle is an inscribed approximation.

The surface pressure follows by differentiating the logarithm of the canonical partition function with respect to container area:Its formal divergence is therefore $\varphi =1/2$. This lies outside the low-density regime used to derive the expression. It is a model pole, not an actual geometrical close-packing fraction.

At sufficiently low packing fraction, neglect overlaps of the excluded regions around different disc centres. Sequential insertion gives the configurational integralTaking logarithms and expanding to first order in density,For large $N$, replace $N-1$ by $N$. The expression $N\ln (a-Nb)$ has the same leading-density expansion when $b=2a_d$. Thus the hard-disc excluded-area approximation yieldsThe factor $N!$ accounts for indistinguishable particles and ${\Lambda }^{2N}$ for two-dimensional momentum integration. The factor one-half between pair-excluded area and $b$ avoids counting every pair twice. Equality of leading expansions does not make this free-volume resummation exact at high density.

With wall effects neglected, $a_1=a$. A second hard disc cannot have its centre within distance $2r$ of the first centre. Its excluded volume in this two-dimensional setting is an excluded area $\pi (2r{)}^2=4a_d$, givingThis is exclusion of the second centre, rather than simply subtraction of the first disc's own area.

The volume integral is a gamma function integral,For the prescribed bridge relation, $\beta P{Q}_P=(\beta P{\Lambda }^3{)}^{-N}$, so $G=N{k}_{B}T\ln (\beta P{\Lambda }^3)$. Since $\Lambda \propto T^{-1/2}$, this gives the macroscopic ideal-gas resultHeating at fixed pressure requires both an increase in internal energy and expansion work; at fixed volume the latter contribution is absent.

The finite-size normalization of the isothermal-isobaric ensemble matters here. Applying part (a) literally to the unprefactored $Q_P$ instead gives $⟨H{⟩}_P=(5N/2+1)k_{B}T$ and $C_P=(5N/2+1)k_B$. The two answers differ because $-{\partial }_{\beta }\ln (\beta P{Q}_P)=-{\partial }_{\beta }\ln Q_P-k_{B}T$. The normalized continuous-volume distribution has $⟨PV⟩=(N+1)k_{B}T$; its variance agrees with the unprefactored result in part (b). The bridge convention subtracts an order-one contribution and recovers the usual extensive thermodynamic answer. Both agree to leading order in the thermodynamic limit, but their exact finite-$N$ derivatives should not be mixed.

Away from singular regimes, assume enthalpy and heat capacity at constant pressure are extensive: $⟨H⟩=Nh$ and $C_P=N{c}_P$, with nonzero intensive $h$. The enthalpy fluctuation formula then givesAbsolute fluctuations grow as $\sqrt{N}$, while the mean grows as $N$. In the thermodynamic limit, enthalpy per particle therefore concentrates around its mean; it is relative, rather than absolute, fluctuations that vanish. This is the usual relative fluctuations of an extensive observable scaling. It assumes ordinary extensive heat capacity and a nonzero mean with a stated energy reference; a critical divergence or an artificially zero mean requires separate analysis.

Differentiate the normalized average, including the derivative of its denominator. The enthalpy fluctuation formula follows fromSince $d\beta /dT=-1/(k_B{T}^2)$,Thus a larger equilibrium spread of enthalpy corresponds to a larger response to heating at fixed pressure. The variance is nonnegative, so this finite-system equilibrium description has $C_P\ge 0$. The relation connects a response coefficient to fluctuations of the appropriate exchanged quantity; at fixed volume the corresponding quantity is internal energy.