A futuristic spacecraft uses the solar wind as fuel for a novel thermodynamic engine. The engine works by collecting charged particles from the solar wind in a magnetic funnel, then using the thermal energy of these particles to run a heat engine cycle.
At Earth's orbital distance, the solar wind has a particle density of n=5.0×106 particles/m3 and an average speed of v=4.0×105 m/s. The particles are primarily protons with kinetic energy that corresponds to an effective temperature.
The spacecraft's magnetic funnel has a collection cross-sectional area of A=100 m2 and collects particles into a cylindrical chamber of volume V=2.0 m3. The engine operates in cycles: it collects particles for time tc=300 s, then uses the collected gas as the hot reservoir for a Carnot engine operating between the particle temperature and the cold temperature of space (Tc=2.7 K).
Assuming the collected particles behave as an ideal gas and that the kinetic energy of the solar wind particles represents their thermal energy, calculate the maximum mechanical work output per cycle of this engine.
*Take the proton mass as mp=1.67×10−27 kg and Boltzmann constant kB=1.38×10−23 J/K.*