Question 43.7: Uranium versus Fossil Fuel Figure 43.18 shows pellets of enr......
Uranium versus Fossil Fuel
Figure 43.18 shows pellets of enriched uranium used in a nuclear power plant. Estimate the total energy released by just one of the pellets. In the Check and Think step, compare your answer to energy released by same mass of fossil fuel such as coal or gasoline. Hint: See Table 9.1 (page 265).
| TABLE 9.1 Energy content in joules. | |||
| Astronomical sources | Joules (J) | Human needs and everyday phenomena | Joules (J) |
| Typical supernova (exploding star) | 10^{43} | 1000-MW power station in 1 year | 10^{16} |
| Milky Way’s radiation in 1 s | 10^{38} | 1 lb of uranium-235 | 3.7 \times 10^{13} |
| Sun’s radiation in 1 year | 10^{34} | Energy to put space shuttle into orbit | 10^{13} |
| Crossing Atlantic Ocean in a jet plane | 10^{12} | ||
| 1 U.S. gallon of gasoline | 1.3 \times 10^8 | ||
| 1 lb of coal | 1.6 \times 10^7 | ||
| Two-ton truck traveling at highway speed | 10^6 | ||
| Terrestrial sources | Joules (J) | 1 lb of TNT | 10^6 |
| Earth’s rotation | 10^{29} | 1 can lighter fluid | 10^6 |
| Volcanic detonation | 10^{19} | 1 candy bar, order of french fries, or slice of pizza | 10^6 |
| Largest recorded earthquake | 10^{18} | 1 tablespoon sugar, 1 apple, or 5 crackers | 10^5 |
| San Francisco earthquake of 1906 | 10^{17} | 1 AA battery | 10^3 |
| Tornado or thunderstorm | 10^{15} | Major league pitch | 10^2 |
| Lightning flash | 10^{10} | Striking a typewriter key | 10^{-2} |
| Adapted from Howard Keller, “Energy Yield of Various Sources,” Physics Teacher 30:455 (1992). | |||
INTERPRET and ANTICIPATE
We can use the photo to estimate the mass of single pellet. Once we know the mass of a whole pellet, we’ll estimate the mass of uranium-235. From this we can estimate the number of uranium-235 nuclei. We know that each fission reaction releases about 200 MeV per nuclei. We’ll assume all the nuclei undergo fission to find the total energy released.
SOLVE
The pellets look fairly light. Perhaps each one is the mass of a few quarters. From Appendix B, we find that the mass of a single quarter is about 6 g. So let’s assume a single pellet is about 20 g. We’ll also assume that it has been enriched so that the abundance of uranium-235 is about 3%. We’ll assume the pellet is comprised of uranium isotopes of roughly the same mass. (That is not really true. The uranium isotopes have slightly different masses and the pellets are not pure uranium, but this is an estimate and such details don’t make a huge difference.)
Next we need the number of uranium-235 nuclei. The molar mass of uranium-235 is 235 g, which means 1 mole of uranium-235 has a mass of 235 g. We calculate the number of nuclei using Avogadro’s number N_A found on the back inside cover.
\begin{aligned}& N=m_{235} \frac{N_{ A }}{235 g } \\& N=(0.6 g ) \frac{6.02 \times 10^{23} \text { nuclei }}{235 g } \\& N=1.5 \times 10^{21} \text { nuclei }\end{aligned}Assume every uranium-235 nuclei undergoes a fission reaction and the each reaction releases 200 MeV of energy.
\begin{aligned}E_{\text {released }} & =(200 MeV / \text { nuclei }) N \\& =(200 MeV / \text { nuclei })\left(1.5 \times 10^{21}\right) \\E_{\text {released }} & =3 \times 10^{23} MeV =5 \times 10^{10} J\end{aligned}CHECK and THINK
First, to check our answer we can use the fact that Table 9.1 says that 1 lb of uranium-235 contains 3.7 \times 10^{13} J .1 lb is the weight of 453 g. So, according to Table 9.1, our 0.6 g of uranium-235 contains 0.6 g \left(\frac{3.7 \times 10^{13} J }{453 g }\right)=5 \times 10^{10} J exactly as we estimated. Next, Table 9.1 says that 1 lb of coal contains 1.6 \times 10^7 J. So 20 g of coal contains 20 g \left(\frac{1.6 \times 10^7 J }{453 g }\right)=7 \times 10^5 J. Our results are amazing; a pellet of uranium releases tens of thousands of times more energy than an equal amount of coal. If you would like, you can repeat this comparison for gasoline, which you will also find in Table 9.1. The results are similar; fission releases millions of times more energy than burning fossil fuel.