Question

Ethanol (C2H5OH) and gasoline (assumed to be all octane, C8H18) are both used as automobile fuel. If gasoline is selling for $2.25/gal, what would the price of ethanol have to be in order to provide the same amount of heat per dollar? The density and ΔH f
of octane are 0.7025 g/mL and −249.9 kJ/mol and the density and ΔH f
of ethanol are 0.7894 g/mL and −277.0 kJ/mol, respectively. Assume that the products of combustion are CO2(g) and H2O(l). (1 gal = 3.785 L)

____ dollars/gal

Answer 1

Gasoline (C8H18):

mass of 1 gallon of gasoline = 1 gal x (3.785 L / 1 gal) x (1000 mL / 1L) x 0.7025 g/mL = 2658.96 g

molar mass of gasoline (C8H18) = 114.23 g/mol

Hence moles of gasoline present in 1 gallon of gasoline

= mass / molar mass = 2658.96 g / 114.23 g/mol = 23.277 mol

Hence total heat produced by 23.277 mol (1 gal) gasoline = 23.277 mol x (−249.9 kJ/mol ) = - 5817 kJ

Hence we get 5817 kJ of heat at a cost of $2.25/gal.

=> amount of heat produced by $1 gasoline = 5817 kJ / 2.25 = 2585.33 kJ

Now we expect the same for ethanol

Ethanol: mass of 1 gallon of ethanol = 1 gal x (3.785 L / 1 gal) x (1000 mL / 1L) x 0.7894 g/mL = 2987.88 g

molar mass of ethanol = 46.07 g/mol

moles of ethanol present in 1 gallon of ethanol = mass/molar mass = 2987.88 g / 46.07 g/mol = 64.855 mol

Our target is to get 2585.33 kJ heat at a cost of $1 ethanol.

Moles of ethanol that produces 2585.33 kJ heat = 2585.33 kJ / 277.0 kJ/mol = 9.33 mol ethanol

Hence the cost of 9.33 mol ethanol = $1

=> the cost of 1 gal ethanol (= 64.855 mol) = ($1 / 9.33 mol) x 64.855 mol = $ 6.95 / gal

Hence the price of ethanol is 6.95 dollars/gal (answer)