Ethanol (C2H5OH) and gasoline (assumed to be all octane, C8H18) are both used as automobile fuel. If gasoline is selling for $1.45/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 of ethanol are 0.7894 g/mL and -277.0 kJ/mol, respectively. 1 gal = 3.785 L.
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To determine the price of ethanol that would provide the same amount of heat per dollar as gasoline, we need to compare the energy content of ethanol and gasoline and consider their densities.
First, let's calculate the energy content of ethanol and gasoline:
1. Calculate the molar mass of octane (C8H18):
Molar mass of C8H18 = (12.01 g/mol * 8) + (1.01 g/mol * 18)
= 114.23 g/mol
2. Calculate the energy content of octane (ΔH°f):
Energy content of octane = -249.9 kJ/mol
3. Calculate the energy content per gram of octane:
Energy content per gram of octane = Energy content of octane / Molar mass of octane
= -249.9 kJ/mol / 114.23 g/mol
≈ -2.186 kJ/g
4. Calculate the energy content per liter of octane:
Energy content per liter of octane = Energy content per gram of octane * Density of octane
= -2.186 kJ/g * 0.7025 g/mL
= -1.534 kJ/mL
Now, let's perform the same calculations for ethanol:
1. Calculate the molar mass of ethanol (C2H5OH):
Molar mass of C2H5OH = (12.01 g/mol * 2) + (1.01 g/mol * 6) + (16.00 g/mol * 1)
= 46.07 g/mol
2. Calculate the energy content of ethanol (ΔH°f):
Energy content of ethanol = -277.0 kJ/mol
3. Calculate the energy content per gram of ethanol:
Energy content per gram of ethanol = Energy content of ethanol / Molar mass of ethanol
= -277.0 kJ/mol / 46.07 g/mol
≈ -6.017 kJ/g
4. Calculate the energy content per liter of ethanol:
Energy content per liter of ethanol = Energy content per gram of ethanol * Density of ethanol
= -6.017 kJ/g * 0.7894 g/mL
= -4.748 kJ/mL
Now, let's compare the energy content per liter between ethanol and octane:
Energy content per liter of ethanol = -4.748 kJ/mL
Energy content per liter of octane = -1.534 kJ/mL
Finally, we can calculate the price of ethanol per gallon:
Price of gasoline per gallon = $1.45
Price of ethanol per gallon = (Price of gasoline per gallon) * (Energy content per liter of ethanol) / (Energy content per liter of octane)
= $1.45 * (-4.748 kJ/mL) / (-1.534 kJ/mL)
≈ $4.49/gal
Therefore, in order for ethanol to provide the same amount of heat per dollar, its price would need to be approximately $4.49 per gallon.
To find the price of ethanol necessary to provide the same amount of heat per dollar as gasoline, we need to compare the heat energy released by each fuel.
First, we need to calculate the molar masses of octane (C8H18) and ethanol (C2H5OH).
The molar mass of octane (C8H18) can be calculated as follows:
8 * (12.01 g/mol) + 18 * (1.01 g/mol) = 114.23 g/mol
The molar mass of ethanol (C2H5OH) can be calculated as follows:
2 * (12.01 g/mol) + 6 * (1.01 g/mol) + 16.00 g/mol = 46.07 g/mol
Next, we need to calculate the heat energy released per volume for each fuel.
For octane (gasoline):
ΔH° = -249.9 kJ/mol
For ethanol:
ΔH° = -277.0 kJ/mol
Now, we can calculate the heat energy released per unit volume for each fuel.
For octane (gasoline):
Heat energy released per mole = -249.9 kJ/mol
Since 1 mole of octane (C8H18) has a molar mass of 114.23 g/mol, we can calculate the heat energy released per gram:
Heat energy released per gram = (-249.9 kJ/mol) / (114.23 g/mol) = -2.184 kJ/g
To convert from grams to liters, we need to know the density of octane:
Density of octane = 0.7025 g/mL
Since 1 gal = 3.785 L, we can calculate the heat energy released per gallon:
Heat energy released per gallon = (-2.184 kJ/g) * (0.7025 g/mL) * (3785 mL/1 gal) = -6167.4 kJ/gal
For ethanol:
Heat energy released per mole = -277.0 kJ/mol
Since 1 mole of ethanol (C2H5OH) has a molar mass of 46.07 g/mol, we can calculate the heat energy released per gram:
Heat energy released per gram = (-277.0 kJ/mol) / (46.07 g/mol) = -6.013 kJ/g
To convert from grams to liters, we need to know the density of ethanol:
Density of ethanol = 0.7894 g/mL
Since 1 gal = 3.785 L, we can calculate the heat energy released per gallon:
Heat energy released per gallon = (-6.013 kJ/g) * (0.7894 g/mL) * (3785 mL/1 gal) = -18080.3 kJ/gal
Now, we can compare the heat energy released per gallon for both fuels.
Let's assume the price of ethanol is x dollars/gal. To determine the price of ethanol that provides the same amount of heat per dollar as gasoline, we need to equate the heat energies released per gallon for both fuels.
-6167.4 kJ/gal (gasoline) = -18080.3 kJ/gal (ethanol)
Divide both sides of the equation by -18080.3 kJ/gal:
-6167.4 kJ/gal / -18080.3 kJ/gal = 1/x dollars/gal
Simplifying the equation:
x = (-18080.3 kJ/gal) / (-6167.4 kJ/gal)
x ≈ 2.93 dollars/gal
Therefore, the price of ethanol would have to be approximately $2.93/gal in order to provide the same amount of heat per dollar as gasoline.