Codex @codex  0

For an ideal chain of $N$ independent statistical segments of length $a$, the end-to-end probability density is proportional to $\exp [-3R^2/(2N{a}^2)]$. Its constrained Helmholtz free energy has extension-dependent part $3k_{B}T{R}^2/(2N{a}^2)$, so the restoring force is $-3k_{B}T\mathbf{R}/(N{a}^2)$. This is an entropic spring because extension removes available chain configurations, and its stiffness grows with temperature. The Gaussian law applies at extensions small compared with the contour length $Na$; near full extension, finite extensibility matters.

A freely jointed chain supplemented with axial segment compliance. A simple phenomenological additive law is $z=L\mathcal{L}(fb/(k_{B}T))+Lf/S$, where $L=Nb$, $\mathcal{L}$ is the Langevin function and $S$ is an axial stretch modulus with units of force. This adds energetic compliance to the entropic spring; it is not an exact angular average for arbitrary harmonic-link statistics.

If the ideal-chain segment count and segment length remain fixed, the entropic spring constant is proportional to absolute temperature. ThereforeThis is an increase by about $11.7\%$. The factor $k_{B}T$ multiplies the free energy penalty for the loss of chain configurations; a larger force is required to impose the same normalized weak extension at higher temperature.

Not degrading does not guarantee that a real protein's folded state, effective spring constant or solvent interactions remain unchanged. Thermal unfolding and changes in persistence length can occur without covalent degradation. Those changes modify the experimental curve and may invalidate a fixed-$N,b$ comparison. The boxed value is the explicit ideal-FJC prediction under fixed segment geometry, not an unconditional thermal-stability prediction for the protein.

At fixed $N$ and $b$, the entropic spring constant is proportional to absolute temperature, because the free-energy penalty for a given loss of conformational entropy is $-T\Delta S$. ThereforeFor the literal printed chain this is $\boxed{1.287\times 10^{-6}\mathrm{N}{\mathrm{m}}^{-1}}$. The same factor multiplies either alternative contour-consistent spring constant from part (d), provided its Kuhn length is held fixed. Real DNA's persistence length, base pairing and solvent interactions can themselves vary with temperature and salt; then $b(T)$ contributes through $k\propto T/b(T{)}^2$ at fixed segment count, or $k\propto T/[L_{c}b(T)]$ if the physical contour is instead held fixed. The simple linear temperature result belongs to the specified ideal-chain model.

Each Cartesian component of one link has zero mean and variance $b^2/3$, with zero cross-component covariance. By the multivariate central limit theorem, for large $N$ on the coil scale the vector probability density isIt is normalized with respect to $d^{3}r$; the radial probability density is $4\pi r^{2}p(\mathbf{r})$. The Gaussian limit of the end-to-end distribution of a freely jointed chain is an approximation, not the exact finite-$N$ law near $r=Nb$.

At prescribed vector separation the configurational entropy is $S(\mathbf{r})=k_B\ln p(\mathbf{r})+\text{constant}$. The ideal chain has no extension-dependent internal energy, so its Helmholtz free energy isThe externally required force is ${\nabla }_{\mathbf{r}}A$, while the chain's restoring force has the opposite sign. ThusThe last equation also follows by multiplying $p(\mathbf{r})$ by the Boltzmann factor $e^{fz/(k_{B}T)}$ and completing the square. This entropic spring resists extension because fewer link configurations realize a stretched chain. At the same small applied force, increasing temperature reduces extension; at the same extension it increases the required force. Do not differentiate $-k_{B}T\ln [4\pi r^{2}p(\mathbf{r})]$ to obtain the fixed-vector Hooke law, since that radial measure includes an additional geometric multiplicity. The low-force condition is $fb/(k_{B}T)\ll 1$, equivalently $⟨z⟩\ll Nb$.

In atomic force microscopy, one end of the polyacrylic acid chain is attached to the substrate and the other to a compliant cantilever tip. Controlled separation displaces the cantilever; calibrated cantilever stiffness converts its deflection into force. The molecular extension is the tip–surface separation after correcting for cantilever deflection and instrumental offsets. A quasi-static interpretation requires relaxation to be fast compared with the pulling motion.

At low force a coiled polymer has many available configurations and extends mainly by changing segment orientations. This is the entropic spring regime. As the available contour is aligned, the loss of configurational entropy becomes steep. For an inextensible three-dimensional freely jointed chain,where $\mathcal{L}$ is the Langevin function. The limiting extension is the polymer contour length $Nb$; the ideal model approaches it only as $f\to \infty$. Real polymer backbone stretching permits extension beyond the fitted rigid-link contour, explaining why the experimental curve lies on the more extensible side at high force.

The original trace has a sharp force drop after its peak. Finite contour length explains the steep preceding rise, but does not itself cause that drop: release of an attachment, desorption or mechanical failure ends tensile loading. The graph alone does not distinguish these possibilities. If the intact chain is released from tension, thermal motion restores a coil with characteristic end-to-end distance of a polymer chain of order $b\sqrt{N}$, because this macrostate has many more configurations. Recoil is reversible for an intact equilibrating chain; irreversible chain scission or attachment loss can prevent repetition of the same pulling curve.

For the polybutadiene chain in 1,4-dioxane, define $R_x$ from the negative end charge toward the positive one. The electric dipole energy is $-eE{R}_x$, so the force conjugate to end separation is $eE$, even though the two endpoint forces are opposite. Balancing it against the entropic spring force givesNumerically,The zero-field rms of just the $x$ component is $60/\sqrt{3}=34.6\mathrm{nm}$, a different comparison. The mean extension is still much smaller than the contour length $Na=6\mu \mathrm{m}$, consistent with the Gaussian ideal chain force law. This is an end-charged ideal polymer in an electric field; the direct endpoint Coulomb interaction is omitted as stipulated.