The Stratospheric Balloon's Thermal Journey · AI Generator #GEN-4540EA82

Physics · thermodynamics

Problem

A scientific weather balloon is launched from sea level carrying a sealed container of helium gas. The container has rigid walls and contains

0.50\;\text{mol}

of helium at an initial temperature of

T_0 = 288\;\text{K}

and pressure

P_0 = 1.01 \times 10^5\;\text{Pa}

. As the balloon rises through the atmosphere, the container's temperature drops linearly with altitude according to

T(h) = T_0 - \alpha h

, where

\alpha = 6.5 \times 10^{-3}\;\text{K/m}

is the atmospheric lapse rate and

h

is the height above sea level. The container is equipped with a pressure relief valve that opens when the internal pressure exceeds

P_{\text{max}} = 2.5 \times 10^5\;\text{Pa}

. When the valve opens, helium escapes isothermally until the pressure drops back to

P_{\text{max}}

, then the valve closes. (a) At what altitude does the pressure relief valve first open? (b) How many moles of helium remain in the container when the balloon reaches an altitude of

12\;\text{km}

? Take the molar gas constant

R = 8.31\;\text{J/(mol·K)}

and assume helium behaves as an ideal gas.

Related theory

thermodynamics

Browse the problem library