what is the expansion rate of helium versus altitude?

Question

what is the expansion rate of helium versus altitude? Forexample, at what altitude does a 1 m3 helium balloon expand to become 2 m3 helium balloon?

Answer 1

V2 = V1(P1/P2)(T2/T1)
Both pressure and temperature vary with altitude.
Using the formulas for US Standard Atmosphere (1976) up to 11 km:
T = T0 (1 – h / 44329 m)
P = P0 (1 – h / 44329 m)^5.255876
V2/V1 = (1 – h / 44329 m) / (1 – h / 44329 m)^5.255876
V2/V1 = (1 – h / 44329 m)^-4.255876

2 = (1 – h / 44329)^-4.255876
2^(1/(-4.255876) = 1 – h / 44329
44329*2^(1/(-4.255876) = 44329 – h
h = 44329(1 - 2^(1/(-4.255876))
h ≈ 6,662.5 m