Question

Atoms can bond to other atoms to make molecules. Atoms sometimes break these bonds and form them in a different way. For example, it is possible to take two molecules of hydrogen (H_2) and one molecule of oxygen (O_2), put them near one another, give them a little spark to help them start a reaction, and you will end up with two molecules of water (H_2O). Do you see what happened? Some of the H-H bonds and the O-O bonds broke and were replaced by O-H bonds. The process of forming or breaking bonds is called a chemical reaction. Chemical reactions come in a vast variety of types. But they typically involve energy changes that are, very roughly, 5 times 10^-19 J per molecule. We can use that number to estimate some useful quantities. Be sure to write your units in the answers below. Gasoline is made up of molecules with carbon and hydrogen (C and H). There are many kinds of molecules in a typical gasoline; but let's assume that all of them are in the form of C_8H_17. That substance has a mass of 18 times 10^-26 kg per molecule. If the mass density of gasoline is 720 kg/m^3, then how many molecules are in one gallon of gasoline? (1 m^3 = 264 gallons.) Please show your work and don't just look it up on the web. Here's a keep track of your units. At the end, the units should cancel out and your answer for the number of molecules should have no units. When you burn gasoline, you convert chemical energy to heat or work (to make your car move). If, when you burn gasoline, you convert 5 times 10^-19 J per molecule, how much energy can be obtained from one gallon of gasoline? If your car has a mass of 1,000 kg, then how much energy is needed to make your car go from rest to 65 mph (29 m/s)? (Think of this using kinetic energy. Ignore friction with the air or road.) How many gallons of gasoline are needed to make your car go from rest to 65 mph (29 m/s). (This is a rough estimate, neglecting the energy lost to air friction, heat, and other things.) If gasoline costs $2.20 per gallon, how much money does your answer to part (d) cost ?

Answer 1

a) 1 m³ = 264 gallons

1 gallon = (1/264) m³

Density = 720 kg/m³

So, this means that mass of 1 m³ = 720 kg

So, mass of (1/264) m³ = (720/264) = 2.73 kg

Mass of 1 molecule = 18*10^-26 kg

Number of molecules in 2.73 kg = 2.73/(18*10^-26) = 0.15*10²⁶ molecules = 15*10²⁴ molecules

1 gallon has 15*10²⁴ molecules.

b) 1 molecule of gasoline produces = 5*10^-19 J

1 gallon has 15*10²⁴ molecules.

So, 15*10²⁴ molecules will produce energy = 5*10^-19 J * 15*10²⁴ = 75*10⁵ J

1 gallon will produce 75*10⁵ J.

c) mass = m = 1000 kg

speed = 29 m/s

Kinetic energy = (1/2)mv² = (1/2)*1000*(29)² = 420500 J = 4.20*10⁵ J

d) 1 gallon will produce 75*10⁵ J.

Kinetic energy = 4.20*10⁵ J

So, gallons required to produce 4.20*10⁵ J = (4.20*10⁵)/(75*10⁵) = 0.056 gallons

Gallons required = 0.056 gallon

e) 1 gallon = $ 2.20

Cost of 0.056 gallon = 0.056*2.20 = $ 0.12

The gasoline will cost $ 0.12.