Riemann surface
matlab

P2.6 Using (2.36),
(a) Find the Helmholtz free energy $F$ of the DNA as a function of X. At what value of $X$ is the free energy minimum?
(b) By how much does the entropy change when the DNA is quasi-statically extended from $X=0$ to $X=L / 2$ at a fixed temperature $T$.
(c) If you increase the temperature slightly by $\Delta T$ with the extension force held fixed as $f$, how would the extension $X$ change?

P3.7 What are the Gibbs and Helmholtz free energies for the chain extended with the tension $f$ and the distance $X$ for the case $f \gg k_B T / l$ ?
Solution: Because $f=\frac{\partial F}{\partial X}$, we integrate the $(3.60)$ over $X$ to find the Helmholtz free energy
$$F(X, T, N)=-N k_B T \ln {1-X /(N l)}$$
where the irrelvant constant is omitted. On the other hand the Gibbs free energy is $G(f, T, N)=F(X, T, N)-f X \approx N\left{-f l+k_B T \ln \left(f l /\left(k_B T\right)\right)\right}$ where $F$ and $X$ are expressed as functions of $f$. Alternatively $G$ is directly obtained from the partition function expression
$$G(f, T, N)=-N k_B T \ln \left{\frac{e^{\beta f l}-e^{-\beta f l}}{\beta f l}\right} \approx-N k_B T \ln \left{\frac{e^{\beta f f}}{\beta f l}\right}$$

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