ethanol(C2H5OH)and gasoline(assumed to be octane,C8H18) are both used as automobile fuel.if gasoline is selling for $2.20/gal,what would the price of ethanol have to be in order to provide the same amount of heat per dollar?The density and ΔHf of octane are 0.7025g/ml and -249.9kJ/mol,respectively and of ethanol are 0.7894g/ml and -277.0 kJ/mol,respectively(1 gal =3.785L)
To compare the prices of ethanol and gasoline in terms of the amount of heat generated per dollar, we need to calculate the heat generated per unit volume and then divide it by the price per unit volume for each fuel.
1. Calculate the heat generated per unit volume for each fuel:
- For gasoline (octane):
- Find the molar mass of octane (C8H18):
Molar mass of C = 12.01 g/mol
Molar mass of H = 1.008 g/mol
Total molar mass of octane = (8 * 12.01) + (18 * 1.008) = 114.22 g/mol
- Calculate the heat generated per mole of octane:
ΔHf of octane = -249.9 kJ/mol
- Calculate the heat generated per ml:
Heat generated per ml of octane = ΔHf of octane / molar mass of octane
- Convert ml to L (1 gal = 3.785 L)
Heat generated per L of octane = (Heat generated per ml of octane) * (1/3.785)
- For ethanol (C2H5OH):
- Find the molar mass of ethanol:
Molar mass of C = 12.01 g/mol
Molar mass of H = 1.008 g/mol
Molar mass of O = 16.00 g/mol
Total molar mass of ethanol = (2 * 12.01) + (6 * 1.008) + 16.00 = 46.07 g/mol
- Calculate the heat generated per mole of ethanol:
ΔHf of ethanol = -277.0 kJ/mol
- Calculate the heat generated per ml:
Heat generated per ml of ethanol = ΔHf of ethanol / molar mass of ethanol
- Convert ml to L (1 gal = 3.785 L)
Heat generated per L of ethanol = (Heat generated per ml of ethanol) * (1/3.785)
2. Calculate the cost per unit volume for each fuel:
- Given: Gasoline price = $2.20/gal
- Convert to L (1 gal = 3.785 L)
Gasoline price per L = Gasoline price / 3.785
- The cost per L of ethanol is unknown (let's call it X).
3. Calculate the amount of heat per dollar for each fuel:
- For gasoline, the heat per dollar = Heat generated per L of octane / Gasoline price per L
- For ethanol, the heat per dollar = Heat generated per L of ethanol / Ethanol price per L (X)
4. Set up the equation to find X (the price of ethanol per L):
Heat per dollar of gasoline = Heat per dollar of ethanol
Heat generated per L of octane / Gasoline price per L = Heat generated per L of ethanol / Ethanol price per L
5. Solve for X (the price of ethanol per L):
X = (Heat generated per L of ethanol / Heat generated per L of octane) * Gasoline price per L
Plug in the given values to calculate the price of ethanol per L:
X = (Heat generated per L of ethanol) * (Gasoline price per L) / (Heat generated per L of octane)
Note: 1 kJ = 1000 J, 1 L = 1000 mL