Ballooning

The volume of the ballon results to $V = \frac{4}{3} \pi r^3 = 9203 \,\text{m}^3$ With the help of the ideal gas equation $p V = n R T$ we can write for the air density $\rho = \frac{M n}{V} = \frac{M p}{RT}$ Therefore, the buoyancy force reads $F_b = (\rho_\text{ext} - \rho) g V = \frac{M g V}{R} p \left( \frac{1}{T_0} - \frac{1}{T} \right)$ with the densities of the atmosphere, $\rho_\text{ext}$ , and of the air inside the ballon, $\rho$ . At maximal altitude, the buoyancy is compensated by the weight of the balloon $\begin{aligned} & & F_b &\stackrel{!}{=} F_g = m g \\ \Rightarrow & & p &= p_0 e^{-h/h_0} = \frac{m R}{M V} \frac{T T_0}{T - T_0}\\ \Rightarrow & & e^{-h/h_0} &= \frac{m}{\rho_0 V} \frac{T}{T - T_0} \\ \Rightarrow & & h &= - h_0 \log \left(\frac{m}{\rho_0 V} \frac{T}{T - T_0} \right) = h_0 \log \left(\frac{\rho_0 V}{m} \frac{T - T_0}{T} \right) = 6391 \,\text{m} \end{aligned}$ with the air density $\rho_0 = \frac{M p_0}{R T_0} = 1.16 \,\text{kg}/\text{m}^3$ on the ground.