Problem 41 - College Physics

Question

Comparison of energy released per gram of fuel. (a) When gasoline is burned, it releases \(1.3 \times 10^{8}\) J per gallon \((3.788\) L) of energy. Given that the density of gasoline is \(737 \mathrm{kg} / \mathrm{m}^{3},\) express the quantity of energy released in \(\mathrm{J} / \mathrm{g}\) of fuel. (b) During fission, when a neutron is absorbed by a \(^{235} \mathrm{U}\) nucleus, about 200 \(\mathrm{MeV}\) of energy is released for each nucleus that undergoes fission. Express this quantity in \(\mathrm{J} / \mathrm{g}\) of fuel. (c) In the proton-proton chain that takes place in stars like our sun, the overall fusion reaction can be summarized as six protons fusing to form one 4 He nucleus with two leftover protons and the liberation of 26.7 \(\mathrm{MeV}\) of energy. The fuel is the six protons. Express the energy produced here in units of J/g of fuel. Notice the huge difference between the two forms of nuclear energy, on the one hand, and the chemical energy from gasoline, on the other. (d) Our sun produces energy at a meas- ured rate of \(3.92 \times 10^{26} \mathrm{W}\) . If its mass of \(1.99 \times 10^{30} \mathrm{kg}\) were all gasoline, how long could it last before consuming all its fuel? (Historical note: Before the discovery of nuclear fusion and the vast amounts of energy it releases, scientists were confused. They knew that the earth was at least many millions of years old, but could not explain how the sun could survive that long if its energy came from chemical burning.)

Step-by-Step Solution

Answer

1. Gasoline: ~46,554 J/g; 2. Uranium fission: ~8.2 x 10¹⁰ J/g; 3. Fusion: ~4.26 x 10¹¹ J/g; 4. Sun's gasoline: ~7,490 years.

Step 1: Calculate Energy Released Per Gram of Gasoline

First, convert the volume of gasoline to liters and then to cubic meters. 1 gallon equals 3.788 liters, which is 0.003788 cubic meters. With the density given as 737 kg/m³, determine the mass per gallon using the formula for mass: \[ \text{Mass} = \text{Density} \times \text{Volume} = 737 \, \text{kg/m}^3 \times 0.003788 \, \text{m}^3 = 2.7929 \, \text{kg} \]Convert this mass to grams (1 kg = 1000 g):\[ 2.7929 \, \text{kg} = 2792.9 \, \text{g} \]The energy per gallon is given as \(1.3 \times 10^8\) J. Therefore, the energy per gram of gasoline is:\[ \frac{1.3 \times 10^8 \, \text{J}}{2792.9 \, \text{g}} \approx 46554.11 \, \text{J/g} \]

Step 2: Convert Energy Released by Uranium Fission into Joules

Given that 200 MeV is released per nucleus during fission, first convert MeV to joules:\[ 1 \, \text{MeV} = 1.60218 \times 10^{-13} \, \text{J} \]So, 200 MeV is:\[ 200 \, \text{MeV} \times 1.60218 \times 10^{-13} \, \text{J/MeV} = 3.20436 \times 10^{-11} \, \text{J} \]Calculate the number of nuclei in 1 gram of \(^{235}\text{U}\):\[ \text{Number of nuclei} = \frac{1}{235} \times \text{Avogadro's number} = \frac{1}{235} \times 6.022 \times 10^{23} \approx 2.56 \times 10^{21} \]The energy per gram is then:\[ 3.20436 \times 10^{-11} \, \text{J/nucleus} \times 2.56 \times 10^{21} \text{nuclei/g} \approx 8.2 \times 10^{10} \, \text{J/g} \]

Step 3: Calculate Energy Released by Proton-Proton Fusion in Stars

In one fusion cycle using six protons, the energy released is 26.7 MeV. First, transform MeV to joules:\[ 26.7 \, \text{MeV} \times 1.60218 \times 10^{-13} \, \text{J/MeV} = 4.278631 \times 10^{-12} \, \text{J} \]Calculate mass of six protons: \[ \text{Mass of one proton} = 1.67262 \times 10^{-27} \, \text{kg} \Rightarrow \text{Mass of six protons} = 6 \times 1.67262 \times 10^{-27} \, \text{kg} = 1.003572 \times 10^{-26} \, \text{kg} \]Convert this mass to grams:\[ 1.003572 \times 10^{-26} \, \text{kg} = 1.003572 \times 10^{-23} \, \text{g} \]Energy per gram of protons is:\[ \frac{4.278631 \times 10^{-12} \, \text{J}}{1.003572 \times 10^{-23} \, \text{g}} \approx 4.26 \times 10^{11} \, \text{J/g} \]

Step 4: Calculate How Long Sun's Mass in Gasoline Would Last

Given the sun's total energy output as \(3.92 \times 10^{26}\) W, equivalent to J/s, determine total energy with sun's total mass:Calculate energy per kilogram for gasoline:\[ \text{Energy per kg} = 46554.11 \, \text{J/g} \times 1000 \, \text{g/kg} = 46554110 \, \text{J/kg} \]Determine total energy using full mass of sun:\[ \text{Total energy} = 1.99 \times 10^{30} \, \text{kg} \times 46554110 \, \text{J/kg} \approx 9.266 \times 10^{37} \, \text{J} \]Calculate duration available using the total energy output per second:\[ \frac{9.266 \times 10^{37} \, \text{J}}{3.92 \times 10^{26} \, \text{J/s}} \approx 2.36 \times 10^{11} \, \text{s} \Rightarrow \approx 7.49 \times 10^{3} \, \text{years} \]

Key Concepts

Energy Release Per Gram

Understanding energy release per gram helps us compare different fuels. It tells us how much energy is produced when a gram of fuel is consumed. The higher the energy release per gram, the more efficient the fuel is. This metric is crucial in selecting energy sources for various applications. For example, fuels with high energy release like uranium or fusion reactions are essential for large-scale power generation.

• Energy release per gram is a measure of efficiency.
• Fuels like uranium and fusion have much higher values than gasoline.
• Higher energy release means we need less fuel for the same energy output.

Learning about this concept shows why traditional fuels like gasoline are less efficient compared to nuclear options.

Gasoline Combustion

Gasoline combustion is a chemical reaction involving hydrocarbons and oxygen, releasing energy. When gasoline burns, it releases about 46,554 J/g. This much energy is relatively moderate, especially when compared to nuclear processes like fission. Despite this, gasoline's ease of use and availability make it a common choice for vehicles.

• Gasoline burning involves chemical reactions.
• Releases moderate energy: about 46,554 J/g.
• Widely used due to convenience, but less efficient than nuclear fuel.

So while gasoline is convenient, it is not the most energy dense fuel available.

Uranium Fission

Uranium fission is a nuclear reaction where a uranium-235 atom splits, releasing massive energy. This process emits about 82 billion J/g, vastly more than gasoline. The high energy release makes nuclear power feasible and extremely potent.

• This is a nuclear process involving splitting an atom.
• Releases about 82 billion J/g.
• Enables large-scale electricity generation with less fuel.

Uranium fission is pivotal for many power plants across the globe, providing a much more efficient energy source compared to chemical fuels.

Proton-Proton Fusion

Proton-proton fusion is a process occurring in stars where nuclei combine, releasing energy. This reaction in the sun releases about 426 billion J/g, the highest among common fuels. Fusion promises a nearly limitless energy source with minimal waste compared to fission.

• This is the process fueling stars, including our sun.
• Releases about 426 billion J/g.
• Seen as the next frontier for clean and abundant energy on Earth.

While fusion power is not yet commercially viable, its potential makes it an exciting area of research for future energy solutions.