Imagine that your car is powered by a fusion engine in which the following reaction occurs.
3*deuterium--->helium + hydrogen + neutron
The masses are (2.0141 u), (4.0026 u), (1.0078 u) and (1.0087 u). The engine uses 8.1 10-6 kg of deuterium fuel. If one gallon of gasoline produces 2.1 109 J of energy, how many gallons of gasoline would have to be burned to equal the energy released by all the deuterium fuel?
To find the number of gallons of gasoline that would have to be burned to equal the energy released by all the deuterium fuel, we need to calculate the energy released by the fusion reaction and then convert it to the equivalent energy produced by burning gasoline.
Step 1: Calculate the energy released by the fusion reaction.
Given:
Mass of deuterium (m) = 8.1 * 10^(-6) kg
Mass of helium (mh) = 4.0026 u = 4.0026 * 1.6605 * 10^(-27) kg
Mass of hydrogen (mh2) = 1.0078 u = 1.0078 * 1.6605 * 10^(-27) kg
Mass of neutron (mn) = 1.0087 u = 1.0087 * 1.6605 * 10^(-27) kg
Mass change (Δm) = (3 * mh + mh2 + mn) - (3 * m)
Δm = (3 * 4.0026 * 1.6605 * 10^(-27) + 1.0078 * 1.6605 * 10^(-27) + 1.0087 * 1.6605 * 10^(-27)) - (3 * 8.1 * 10^(-6))
Δm = (11.98425 * 10^(-27) + 1.65945 * 10^(-27)) - (24.3 * 10^(-6))
Δm = (13.6437 - 0.0000243) * 10^(-27)
Δm = 13.6437 * 10^(-27) kg
Energy released (E) = Δm * (speed of light)^2
E = 13.6437 * 10^(-27) * (3 * 10^8)^2
E = 13.6437 * 10^(-27) * 9 * 10^16
E = 1.22793 * 10^(-10) J
Step 2: Convert the energy to equivalent gasoline energy.
Given:
Energy produced by burning one gallon of gasoline (Eg) = 2.1 * 10^9 J
Number of gallons of gasoline (ng) needed = E / Eg
ng = (1.22793 * 10^(-10)) / (2.1 * 10^9)
ng = 0.05852 * 10^(-10) / 10^9
ng = 0.05852 * 10^(-19) / 10^9
ng = 5.852 * 10^(-20) / 10^9
ng = 5.852 * 10^(-29) gallons
Therefore, approximately 5.852 * 10^(-29) gallons of gasoline would need to be burned to equal the energy released by all the deuterium fuel.