Question

Estimate the average temperature of the air inside a hot-air balloon (see Figure 1.1). Assume that the total mass of the unfilled balloon and payload is 500 kg. What is the mass of the air inside the balloon?

Short answer

The mass of air inside the balloon is$m_a=\left(500\right)\frac{T_o}{T_i}$

Step-by-step solution

Given information

mass of unfilled balloons and payload is $m_b=500kg$

The temperature of air inside the balloon is $T_i$

The temperature outside the air balloon is $T_o$

molar mass of dry air is $M_a$

mass of air inside the balloon is $m_a$

density of air is${\rho }_a$

Explanation

The buoyant force of the air and the gravitational pull on the balloon are in equilibrium:

$F_{B,a}=F_{g,a}$

From which follows:

${\rho }_{a}gV=m_{b}g$

Rewriting as an expression for volume:

$V=\frac{m_b}{{\rho }_a}$

Using the ideal gas law in terms of density:

$V=\frac{m_{b}R{T}_o}{M_{a}P}$

Rewriting the ideal gas law, in this case in terms of the amount of moles of air inside the balloon:

$n=\frac{PV}{R{T}_i}=\frac{m_b}{M_a}\frac{T_o}{T_i}$

Finally, the mass of the air inside is given by the relation:

$m_a=m_b\frac{T_o}{T_i}$

Plugging the value of mass of balloon in the above equation, we get,

$m_a=\left(500\right)\frac{T_o}{T_i}$