Hot air balloon - Ideal gas law

[andyfry]

Homework Statement

Estimate the required temperature of the air inside a (rigid) balloon in order for it to achieve lift off.
Volume 2000m³
$$\sum$$mass, m_tot=920kg

Homework Equations

pV=NkT
p=$$\frac{Force}{Area}$$
$$\rho$$=$$\frac{m}{V}$$

The Attempt at a Solution

We require
F_up>(m_air+m_tot)g (assume equal to solve)

and F_up=$$\rho$$Vg

Not entirely sure how to relate this to the ideal gas law, but have tried
F_up=$$\frac{NkTA}{V}$$
(from p=$$\frac{F}{A}$$)

so T=$$\frac{Vg}{NkA}$$(m_air+m_tot)

But I'm left with an N which I don't know, and also an m_air which I also don't know. I assume m_air=N(mass of 1 molecule) but subbing this into the equation would still leave one N and a the mass of 1 molecule of air, which i do not know.

Hopefully my attempt is at least going in the right direction...

 

[Redbelly98]

Homework Equations

pV=NkT
p=$$\frac{Force}{Area}$$
$$\rho$$=$$\frac{m}{V}$$

The Attempt at a Solution

We require
F_up>(m_air+m_tot)g (assume equal to solve)

and F_up=$$\rho$$Vg

This looks good up to this point. Can you find an expression to substitute for m_air, the mass of hot air? It involves the ideal gas law and the molar mass of air.

 

[andyfry]

well m_air=nM
where M=molar mass of air
so subbing into pV=nRT
gives m_air=$$\frac{pVM}{RT}$$
Is this what you mean?
Can't really see how this is helpful though. Still leaves me with unknowns of p, $$\rho$$, and M after subbing into intitial equations.
How does using the molar mass of air, M, instead of m_air help? They are both unknown.

EDIT:
$$\rho$$_inV=$$\rho$$_outV+m_tot

$$\frac{PMV}{RT_{in}}$$=$$\frac{PMV}{RT_{out}}$$+m_tot

(assuming P in and out are equal)

Rearanging for T_in

T_in=$$\frac{PMV}{R}$$($$\frac{RT_{out}}{PMV}$$+$$\frac{1}{m_{tot}}$$)

T_in=T_out+$$\frac{PMV}{Rm_{tot}}$$

assume T_out=283k, P=101kPa, M=28.97, R=8.314

T_in=283+$$\frac{101000*28.97*10^{-3}*2000}{8.314*920}$$=1048k
Which doesn't seem to be quite plausible? Would expect a value of maybe half that...
Can someone check my method please??

 

[Redbelly98]

EDIT:
$$\rho$$_inV=$$\rho$$_outV+m_tot

Actually, it's

ρ_inV + m_tot = ρ_outV​

The mass of the balloon (including the hot air inside it) equals the mass of the displaced air, ρ_outV.

Can someone check my method please??

You're on the right track, be careful with the algebra.

 

[andyfry]

Actually, it's

ρ_inV + m_tot = ρ_outV​

Ah, that might help!

rearranging (properly!) for T_in now gives an answer of 450k, which I believe is the right answer.

Thanks a lot for your help! :)

 

[Redbelly98]

450k

Yup, looks good to me

Thanks a lot for your help! :)

You're welcome.