Q.1.15 - An Introduction to Thermal Physics

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?

Step-by-Step Solution

Answer

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

Step 1: Given information

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

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 ${\rho }_a$

Step 2: 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,

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