A rigid-walled storage vessel of volume $V$ holds $n$ moles of an ideal gas at pressure $p$ and absolute temperature $T$. The molar mass of the gas is $M$.

Use the ideal gas equation to show that the density $\rho$ of the gas is given by the expression

where $R$ is the molar gas constant.

The main aerostat envelope is filled with heated nitrogen gas.

• The inflated volume of the envelope is $1600m^3$.
• The temperature of the heated gas inside the envelope is maintained at $-40^{\circ }C$.
• The temperature of the surrounding ambient atmosphere is $-180^{\circ }C$.
• The ambient atmospheric pressure at this altitude is $1.45\times 10^{5}Pa$.
• The average molar mass of the atmospheric gas is $0.028{kg\; mol}^{-1}$.
• Take the acceleration of gravity on Titan $g=1.35{m\; s}^{-2}$ and the molar gas constant $R=8.31{J\; mol}^{-1}{K}^{-1}$.

Calculate the density of the surrounding cool atmospheric gas.

Calculate the density of the heated gas inside the envelope.

Calculate the net lifting force (net upthrust) acting on the aerostat due to the difference in density between the heated gas inside and the cooler ambient gas outside. (Assume the volume of the envelope fabric and instrument payload is negligible compared to the volume of the heated gas).

An auxiliary propulsion thruster on the aerostat intakes ambient atmospheric gas and expels it horizontally as a high-speed jet to assist in maneuvering. The thruster expels $1.8kg$ of gas per second at a speed of $35{m\; s}^{-1}$ relative to the aerostat.

Calculate the horizontal thrust force produced by this expelled gas flow.

When the aerostat climbs to high altitudes in Titan's atmosphere, explain the effect on the maximum horizontal thrust produced by the thruster if the volume flow rate of the intake gas remains constant.