Question #42108

A theoretical model of evaporation proposes that the effective cohesive 'friction' force resisting a molecule's escape from a liquid surface is given by $F_c = \alpha v^4$, where $v$ is the molecule's speed and $\alpha$ is a positive constant. The surface energy density $\gamma$ (in $\text{J/m}^2$) is defined as the product of the surface number density $n_s$ and the true average work done against $F_c$ over a characteristic escape distance $d$. The molecules in the liquid follow a 3D Maxwell-Boltzmann velocity distribution at temperature $T$.

A student incorrectly calculates the surface energy as $\gamma'$ by assuming all molecules escape with exactly the root-mean-square speed $v_{\text{rms}}$.

Find the ratio $\gamma / \gamma'$. Furthermore, if the measured temperature $T$ has a maximum relative error of $1.5\%$ and the molecular mass $m$ has a maximum relative error of $2.0\%$, what is the maximum percentage error in the true surface energy $\gamma$?