Question #81259

A sealed spherical elastomeric shell contains air (assumed to be an ideal gas) initially at atmospheric pressure $P_0 = 1.0 \times 10^5 \text{ Pa}$ and temperature $T_0 = 300 \text{ K}$. The shell is submerged in a lake to a depth $h = 2.0 \text{ m}$, where the water temperature is $T_f = 294 \text{ K}$. The fractional change in the cavity's volume $\delta = \frac{\Delta V}{V_0}$ is governed by the relation $\delta = \frac{P_{int} - P_{ext}}{K_{eff}} - \gamma \Delta T$, where $P_{int}$ and $P_{ext}$ are the final internal and external pressures, $K_{eff} = 9.0 \times 10^5 \text{ Pa}$ is the effective elastic modulus of the shell, $\gamma = 5.0 \times 10^{-4} \text{ K}^{-1}$ is the coefficient of volume expansion, and $\Delta T = T_0 - T_f$. Assuming $|\delta| \ll 1$, calculate the magnitude of the fractional change in volume $|\delta|$ of the cavity. (Take the density of water $\rho = 10^3 \text{ kg/m}^3$, $g = 10 \text{ m/s}^2$, and use the first-order approximation for small changes, ignoring cross terms).