Question

on a 10 day wilderness expedition youll need to heat 4.0 kg of water to the boiling point each day. the air temperature will average 25°c. you have available canisters of compressed propane (c₃h₈) fuel, which youll burn to heat the water. each canister has 75. g of propane in it. what is the minimum number of fuel canisters you must bring?
the standard heat of formation of propane at 25°c is - 103.8 kj/mol. youll probably find other helpful data in the aleks data resource.
canisters:

Explanation:

Step1: Calculate the heat needed to heat water each day

The specific heat capacity of water \(c = 4.18\,\text{J/g}\cdot^{\circ}\text{C}\), mass of water \(m = 4000\,\text{g}\), initial temperature \(T_1=25^{\circ}\text{C}\), final temperature \(T_2 = 100^{\circ}\text{C}\).
The heat formula is \(Q=mc\Delta T\).
\(\Delta T=T_2 - T_1=100 - 25=75^{\circ}\text{C}\)
\(Q=(4000\,\text{g})\times(4.18\,\text{J/g}\cdot^{\circ}\text{C})\times75^{\circ}\text{C}=1254000\,\text{J}=1254\,\text{kJ}\)
For 10 - day, \(Q_{total}=10\times1254\,\text{kJ}=12540\,\text{kJ}\)

Step2: Write the combustion reaction of propane

The combustion reaction of \(C_3H_8\): \(C_3H_8(g)+5O_2(g)
ightarrow3CO_2(g)+4H_2O(l)\)
Using the formula \(\Delta H_{rxn}=\sum n\Delta H_f^{\circ}(\text{products})-\sum n\Delta H_f^{\circ}(\text{reactants})\)
\(\Delta H_f^{\circ}(CO_2)=- 393.5\,\text{kJ/mol}\), \(\Delta H_f^{\circ}(H_2O(l))=-285.8\,\text{kJ/mol}\), \(\Delta H_f^{\circ}(C_3H_8)=-103.8\,\text{kJ/mol}\), \(\Delta H_f^{\circ}(O_2) = 0\)
\(\Delta H_{rxn}=[3\times(-393.5)+4\times(-285.8)]-(-103.8)\)
\(=(-1180.5-1143.2)+103.8=-2220\,\text{kJ/mol}\)

Step3: Calculate the moles of propane needed

From \(Q = n\times|\Delta H_{rxn}|\), \(n=\frac{Q_{total}}{|\Delta H_{rxn}|}\)
\(n=\frac{12540\,\text{kJ}}{2220\,\text{kJ/mol}} = 5.65\,\text{mol}\)

Step4: Calculate the mass of propane needed

The molar mass of \(C_3H_8\), \(M=(3\times12 + 8\times1)\,\text{g/mol}=44\,\text{g/mol}\)
\(m=n\times M=5.65\,\text{mol}\times44\,\text{g/mol}=248.6\,\text{g}\)

Step5: Calculate the number of canisters

Each canister has \(m_{canister}=75\,\text{g}\)
\(N=\frac{248.6\,\text{g}}{75\,\text{g}}\approx3.31\)

Answer:

\(4\)