Question #43564

A sealed cylindrical chamber with a movable piston contains an ideal diatomic gas ($\gamma = 1.4$). Initially, the gas has a volume $V_0 = 32 \text{ L}$ and pressure $P_0 = 100 \text{ kPa}$. The refractive index of the gas at this state is $n_0 = 1.00625$. Inside the chamber, two thin lenses are placed in contact: Lens 1 is equiconvex with a refractive index $n_1 = 1.5$ and a vacuum focal length $f_1 = 20 \text{ cm}$. Lens 2 is equiconcave with a refractive index $n_2 = 1.6$ and a vacuum focal length $f_2 = -40 \text{ cm}$. An external agent does $24 \text{ kJ}$ of work on the gas to compress it adiabatically and reversibly. Assuming the refractive index $n$ of the gas depends on its density $\rho$ according to the Gladstone-Dale relation ($n - 1 \propto \rho$), what is the equivalent focal length of the lens combination in the gas after the compression?