Question

on a 12 - day wilderness expedition youll need to heat 3.0 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 50. 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.

Explanation:

Step1: Calculate the total mass of water to be heated

The mass of water to be heated per day is \(m = 3.0\space kg\), and the expedition is \(12\) days. So the total mass of water \(M=3.0\space kg\times12 = 36\space kg=36000\space g\)

Step2: Calculate the heat required to heat the water

The specific - heat capacity of water \(c = 4.18\space J/(g\cdot^{\circ}C)\), the initial temperature \(T_1 = 25^{\circ}C\) and the boiling point \(T_2=100^{\circ}C\). The heat formula is \(Q = Mc\Delta T\), where \(\Delta T=T_2 - T_1\)

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Step3: Calculate the heat of combustion of propane

The standard heat of formation of propane \(\Delta H_f=- 103.8\space kJ/mol\). The combustion reaction of propane is \(C_3H_8(g)+5O_2(g)\to3CO_2(g)+4H_2O(l)\)
The heat of combustion \(\Delta H_{comb}\) can be calculated using \(\Delta H_{comb}=\sum n\Delta H_f(products)-\sum m\Delta H_f(reactants)\)
\(\Delta H_f(CO_2)=-393.5\space kJ/mol\), \(\Delta H_f(H_2O(l))=-285.8\space kJ/mol\), \(\Delta H_f(O_2) = 0\space kJ/mol\), \(\Delta H_f(C_3H_8)=-103.8\space kJ/mol\)

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The molar mass of propane \(M_{C_3H_8}=(3\times12 + 8\times1)\space g/mol=44\space g/mol\)

Step4: Calculate the mass of propane needed

Let the mass of propane be \(m_{propane}\). The heat from propane combustion is used to heat water, so \(Q = n\Delta H_{comb}\), and \(n=\frac{m_{propane}}{M_{C_3H_8}}\)

$$m_{propane}=\frac{Q\times M_{C_3H_8}}{\vert\Delta H_{comb}\vert}$$

Substitute \(Q = 1.1286\times 10^{4}\space kJ\), \(M_{C_3H_8}=44\space g/mol\), \(\vert\Delta H_{comb}\vert = 2220\space kJ/mol\)

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Step5: Calculate the number of canisters

Each canister has \(m_{canister}=50\space g\) of propane. The number of canisters \(N=\frac{m_{propane}}{m_{canister}}\)

$$N=\frac{223.7\space g}{50\space g}\approx4.47$$

Since we can't have a fraction of a canister, we need to round up.

Answer:

\(5\)