Question #42510

A macromolecule of mass $m$ with $f$ internal degrees of freedom is attached to a spring of force constant $k$ and undergoes weakly damped oscillations along the $x$-axis. It experiences a drag force $F_d = -b v$, where the damping coefficient $b$ is directly proportional to the molecule's instantaneous internal energy $U$, such that $b = \beta U$ ($\beta$ is a positive constant). Initially ($t=0$), the molecule is in thermal equilibrium with the surrounding fluid at temperature $T_0$, and is given an initial macroscopic oscillation energy $E_0$. Assuming the energy dissipated by the drag force is entirely absorbed by the molecule's internal degrees of freedom (with negligible heat transfer to the fluid), which of the following expressions gives the time $t_{1/2}$ required for the oscillation energy to decrease to $E_0/2$? (Assume $k_B$ is the Boltzmann constant).