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I am trying to solve the problem:

What is the lifting force of in kN for a 10 m diameter spherical balloon with helium inside at 101 kPa and 320 K surrounded by air at 101 kPa and 298.15 K?

The given answer is 5.28 kN.

I start from the Buoyancy Formula:

$$F_{b} = (p_{air} - p_{gas})*g*V$$

And Ideal Gas Law: $$PV = mRT$$ $$P/RT = m/V$$

Density of Air: $101/0.287*298.15=1.18$

Density of He: $101/0.287*320=1.10$

Volume of Helium Balloon: $(4\pi/3)*(10/2)^3=(500/3\pi)$

Substitute:

$$F_b = (1.18 - 1.10)*9.81*(500/3\pi)$$

I get 0.4101 N; am I missing something or is the given answer wrong?

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    $\begingroup$ In the ideal gas law for helium, you have an incorrect value for the specific gas constant $R$. $\endgroup$
    – John1024
    Jul 4 '15 at 6:13
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Firstly, Gas constant of $He$ equals $2077 \ \mathrm{J/kg\cdot K}$ (engineering toolbox), so density of $He$ becomes $101/2.077*320 = 0.152\ \mathrm{kg/m^3}$.

Secondly, $Volume = 523.6 \ \mathrm{m^3}$ not as you calculated.

Thirdly, you need to subtract balloon weight from the buoyancy force $$F_b = \rho_{air}*g*V$$ $$F_b = 1.18 *9.81*523.6 = 6061 \ \mathrm{N}$$ $$F_{weight} = \rho_{He} Vg = 0.152*523.6*9.81 = 780.75 \ \mathrm{N}$$ $$F_{net} = F_b - F_{weight} = 6061 - 780.75 = 5280\ \mathrm{N} = 5.28\ \mathrm{kN}$$

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A useful rule of thumb for plausibility checking of gas balloon lift is that helium and hydrogen provide roughly 1kg of lift per cubic metre. This is close enough for back of the envelope sums and to check that you haven't dropped a decimal point in more detailed calculations.

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