Question

at a festival, spherical balloons with a radius of 140. cm are to be inflated with hot air and released. the air at the festival will have a temperature of 25°c and must be heated to 100°c to make the balloons float. 2.00 kg of butane (c₄h₁₀) fuel are available to be burned to heat the air. calculate the maximum number of balloons that can be inflated with hot air.
here are some data you may find useful:
specific heat capacity of air: 1.009 j/(g°c)
density of air at 100°c: 0.946 kg/m³
density of butane at 100°c: 1.898 kg/m³
formation enthalpy of butane at 25°c: -125.7 kj/mol
any other data you need should be taken from the aleks data resource.
balloons

Explanation:

Step1: Calculate the volume of one balloon

The volume \(V\) of a sphere is given by the formula \(V=\frac{4}{3}\pi r^{3}\). Given \(r = 140\space cm=1.4\space m\), then \(V=\frac{4}{3}\pi(1.4)^{3}\space m^{3}\approx11.49\space m^{3}\)

Step2: Calculate the mass of air in one balloon

Using the density formula \(
ho=\frac{m}{V}\), where \(
ho = 0.946\space kg/m^{3}\) (density of air at \(100^{\circ}C\)) and \(V = 11.49\space m^{3}\). Then \(m=
ho V=0.946\times11.49\space kg\approx10.87\space kg = 10870\space g\)

Step3: Calculate the heat required to heat the air in one balloon

The heat formula is \(Q = mc\Delta T\). Here, \(c = 1.009\space J/(g\cdot^{\circ}C)\), \(\Delta T=(100 - 25)^{\circ}C = 75^{\circ}C\), and \(m = 10870\space g\). So \(Q=10870\times1.009\times75\space J\approx8.25\times10^{5}\space J\)

Step4: Calculate the number of moles of butane

The molar mass of \(C_{4}H_{10}\) is \(M=(4\times12 + 10\times1)\space g/mol=58\space g/mol\). Given \(m_{butane}=2000\space g\), the number of moles \(n=\frac{2000}{58}\space mol\approx34.48\space mol\)

Step5: Calculate the total heat released by butane

Assume the heat of combustion of butane (using standard data, for example, the heat of combustion of butane is approximately \(2658\space kJ/mol\)). The total heat \(Q_{total}=n\times\Delta H_{combustion}\). If \(\Delta H_{combustion}=- 2658\space kJ/mol\) (exothermic), \(Q_{total}=34.48\times2658\times10^{3}\space J\approx9.17\times10^{7}\space J\)

Step6: Calculate the number of balloons

The number of balloons \(N=\frac{Q_{total}}{Q}\). \(N=\frac{9.17\times 10^{7}}{8.25\times 10^{5}}\approx111\)

Answer:

\(111\)