Problem 48 - University Physics with Modern Physics

Question

Comparison of Energy Released per Gram of Fuel. (a) When gasoline is burned, it releases \(1.3 \times 10^{8} \mathrm{J}\) of energy per gallon \((3.788 \mathrm{L}) .\) 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 \(\mathrm{J} / \mathrm{g}\) 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 measured rate of \(3.86 \times 10^{26} \mathrm{W}\) . 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

(a) 46600 J/g (b) 8.19 × 10¹⁰ J/g (c) 4.29 × 10¹¹ J/g (d) ~760 years.

Step 1: Calculate the mass of 1 gallon of gasoline

First, we convert the volume of gasoline into mass using the given density. The equation is \( \text{mass} = \text{volume} \times \text{density} \). Since 1 gallon = 3.788 L = 0.003788 m³, the mass is given by \( 737 \text{ kg/m}^3 \times 0.003788 \text{ m}^3 = 2.7916 \text{ kg} \).

Step 2: Calculate energy released per gram for gasoline

The energy released per gallon is \( 1.3 \times 10^8 \text{ J} \). To find the energy per gram, we divide this by the mass in grams. \( 2.7916 \text{ kg} = 2791.6 \text{ g} \). Thus, energy per gram is \( \frac{1.3 \times 10^8 \text{ J}}{2791.6 \text{ g}} \approx 46567 \text{ J/g} \).

Step 3: Convert energy for uranium-235 fission to J/g

Convert 200 MeV to joules using the conversion \( 1 \text{ MeV} = 1.60218 \times 10^{-13} \text{ J} \), which yields \( 200 \times 1.60218 \times 10^{-13} \text{ J} = 3.20436 \times 10^{-11} \text{ J/nucleus} \). The molar mass of \( ^{235}U \) is 235 g/mol. Avogadro's number is \( 6.022 \times 10^{23} \) nuclei/mol. \( \text{Energy per gram} = \frac{3.20436 \times 10^{-11} \text{ J/nucleus} \times 6.022 \times 10^{23} \text{ nuclei/mol}}{235 \text{ g/mol}} \approx 8.19 \times 10^{10} \text{ J/g} \).

Step 4: Convert energy for proton-proton fusion to J/g

Convert 26.7 MeV to joules: \( 26.7 \times 1.60218 \times 10^{-13} = 4.2798196 \times 10^{-12} \text{ J/reaction} \). The molar mass of protons is about 1 g/mol, but 6 protons are involved, so \( 6 \text{ g/mol} \). \( \frac{4.2798196 \times 10^{-12} \times 6.022 \times 10^{23}}{6} \approx 4.29 \times 10^{11} \text{ J/g} \).

Step 5: Calculate gasoline lifetime of the Sun

The total energy of the sun if it were made of gasoline is \( 1.99 \times 10^{30} \text{ kg} \times 1000 \text{ g/kg} \times 46567 \text{ J/g} \). The power of the sun is \( 3.86 \times 10^{26} \text{ W} = 3.86 \times 10^{26} \text{ J/s} \). Calculate the lifetime by dividing the total energy by the power: \( \frac{1.99 \times 10^{33} \times 46567}{3.86 \times 10^{26}} \approx 2.40 \times 10^7 \text{ seconds} \). This translates to about 760 years.

Key Concepts

Energy per Gram of Fuel

When we consider fuels, an important metric to understand is the energy released per gram. This tells us how efficient a particular fuel is in terms of energy output for a given mass.
Here is how we calculate it:

• The energy released by burning is measured in joules (J).
• Mass is measured in grams (g).
• We divide the total energy released by the total mass of the fuel to find the energy per gram.

For example, gasoline releases approximately 46,567 J for every gram burned. This gives you an idea of how much energy is packed in each gram of fuel, which is crucial for comparing different energy sources.

Gasoline Energy Density

Gasoline is a commonly used fuel known for its high energy density. This refers to the amount of energy stored in a given system or region of space per unit volume or mass.
Gasoline has a high energy density, meaning it can store and deliver a considerable amount of energy in a relatively small mass.

• Energy density helps us understand how efficient and practical various fuels are.
• For gasoline, its density plays a crucial role; 737 kg/m³ provides the physical structure to gauge efficiency.
• This parameter is valuable when considering automotive fuels, as vehicles require both high energy input and minimal fuel mass.

Comparing the energy density of gasoline to that of nuclear fuels highlights its limitations, though it remains practical for everyday combustion engines.

Nuclear Fusion Energy

Nuclear fusion is the process of combining light atomic nuclei to form a heavier nucleus. This process releases a vast amount of energy and occurs naturally in the sun.

• The proton-proton chain reaction, crucial in stars, involves 6 protons forming one helium nucleus, and this releases 26.7 MeV of energy.
• In terms of energy per gram, nuclear fusion can produce approximately 4.29 × 10¹¹ J/g. This is significantly higher compared to chemical fuels like gasoline.
• Understanding fusion energy illustrates its potential as a clean and nearly limitless energy source for the future.

The key advantage of fusion over traditional fuels is its efficiency and lower environmental impact. Translating stellar processes into terrestrial energy solutions remains a critical scientific pursuit.

Nuclear Fission Energy

Nuclear fission involves splitting a heavy atomic nucleus into smaller nuclei. This process releases a significant amount of energy, especially with elements like uranium-235.

• Each fission of a uranium-235 nucleus releases about 200 MeV of energy.
• When calculated in terms of energy per gram, fission yields approximately 8.19 × 10¹⁰ J/g.
• This energy density is much higher than chemical fuels, making fission an efficient power source.

Nuclear fission has been utilized for decades in power plants to generate electricity.
While fission provides large amounts of energy, it must be managed carefully due to issues of radioactive waste and safety concerns. Thus, it stands as a potent, yet complex, energy avenue.