= Solution
\b[Entropic <elasticity>.] A rubber contains flexible <polymer> strands tied into a network by <cross-links>. Stretching biases the strands toward relatively extended configurations, reducing their configurational <entropy>. The restoring <entropic force> follows from the increase in <Helmholtz free energy> $F=U-TS$. For a Gaussian strand with $N_s$ statistical steps of length $b$, its deformation-dependent <free energy> is approximately $3k_BTR^2/(2N_sb^2)$. Its stiffness therefore increases with absolute <temperature>, even if the underlying bond stiffness changes little.
In the ideal affine-network model of <rubber elasticity>, the <shear modulus> is
$$
\boxed{G=\nu_s k_BT,}
$$
where $\nu_s$ is the <number density> of elastically active network strands. Thus heating increases the modulus approximately in proportion to $T$, provided the network density and rubbery state remain approximately unchanged. <Thermal expansion>, changing relaxation times or damage can modify that simple trend.
Examples are \b[lightly crosslinked <natural rubber>], based on cis-rich <polyisoprene>, and \b[crosslinked <polydimethylsiloxane>]. The crosslinks prevent the long-time macroscopic flow of an unconnected chain solution.
\b[Density estimate and the unspecified modulus convention.] With room <temperature> $T\simeq298\,\mathrm K$, the <Boltzmann constant> gives $k_BT\simeq4.11\times10^{-21}\,\mathrm J$. If the stated modulus is $G=2\times10^5\,\mathrm{Pa}$, then
$$
\boxed{\nu_s=\frac{G}{k_BT}\simeq4.9\times10^{25}\,\mathrm{m^{-3}}
\simeq81\,\mathrm{mol\,m^{-3}}.}
$$
The molar value counts moles of active strands. If “modulus” instead means <Young's modulus> $E$, a nearly incompressible isotropic rubber has $E\simeq3G$, and the inferred value is
$$
\boxed{\nu_s\simeq\frac{E}{3k_BT}=1.6\times10^{25}\,\mathrm{m^{-3}}
\simeq27\,\mathrm{mol\,m^{-3}}.}
$$
Thus the robust order-of-magnitude estimate is $10^{25}\,\mathrm{m^{-3}}$. The <cross-link density inferred from rubber elasticity> is usually an effective strand density; the actual junction density is $\nu_j=2\nu_s/f$ for junction functionality $f$ in an ideal network without inactive loops or dangling ends. For four-functional junctions this would be half the strand density. The paper supplies neither modulus type nor junction functionality, so a unique chemical junction count cannot be extracted.
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