Mass-Energy Conversion In U-235 Fission A Detailed Solution

The realm of nuclear physics unveils the profound relationship between mass and energy, a concept elegantly encapsulated by Einstein's famous equation, E=mc². This equation serves as the cornerstone for understanding the immense energy released during nuclear reactions, such as the fission of Uranium-235 (U-235). In this article, we will delve into a fission reaction of U-235 where a significant amount of energy is released, specifically $9.20 imes 10^{11}$ kJ. Our primary objective is to determine the amount of mass that was converted into energy during this reaction. To achieve this, we will employ Einstein's equation and the given speed of light, $c = 3 imes 10^8$ m/s. This exploration will not only provide a numerical answer but also enhance our comprehension of the fundamental principles governing nuclear reactions and mass-energy equivalence.

At the heart of understanding the energy release in nuclear fission lies Albert Einstein's groundbreaking equation, E = mc². This equation reveals the intrinsic relationship between energy (E) and mass (m), where mass can be converted into energy and vice versa. The constant 'c' represents the speed of light in a vacuum, approximately $3 imes 10^8$ meters per second, which underscores the immense scale of energy that can be derived from a small amount of mass. In nuclear reactions, such as fission, a tiny fraction of the mass of the atomic nuclei is converted into a substantial amount of energy, explaining the potency of nuclear power and weapons. Understanding this equation is crucial for grasping the magnitude of energy released in nuclear processes and the underlying principle of mass-energy conservation. The equation signifies that mass and energy are interchangeable; a small amount of mass can yield a tremendous amount of energy due to the large value of the speed of light squared. This principle is not only fundamental to nuclear physics but also has far-reaching implications in various fields of science and technology, influencing our understanding of the universe and its energetic phenomena. Einstein’s equation has revolutionized our approach to energy generation and utilization, making it possible to harness nuclear power while also highlighting the potential dangers associated with uncontrolled nuclear reactions. It is a testament to the profound interconnectedness of mass and energy, shaping our understanding of the fundamental laws of nature.

The specific scenario we are addressing involves the fission reaction of Uranium-235 (U-235), a widely used isotope in nuclear applications. During this reaction, a considerable amount of energy is released, quantified as $9.20 imes 10^{11}$ kJ. The key question we aim to answer is: What amount of mass, measured in kilograms, was converted into energy during this fission process? This problem requires us to apply the principles of mass-energy equivalence as defined by Einstein's equation. We are provided with the energy released and the speed of light, which are the essential pieces of information needed to calculate the mass converted. This problem is not only a numerical exercise but also a practical application of theoretical physics to a real-world nuclear reaction. By solving this, we gain insight into the sheer magnitude of mass conversion in nuclear reactions and the efficiency of these processes in releasing energy. The calculation will demonstrate how a relatively small amount of mass can result in a substantial energy output, highlighting the power of nuclear fission and the importance of understanding and managing such reactions for both energy production and safety considerations. It underscores the significance of precise measurements and calculations in nuclear physics to ensure accurate assessments of energy release and the associated mass changes.

To determine the amount of mass converted into energy during the U-235 fission reaction, we will follow a step-by-step approach using Einstein's equation, E = mc². This structured method will ensure clarity and accuracy in our calculations. First and foremost, it's essential to ensure that all units are consistent. The energy is given in kilojoules (kJ), but to use it with the speed of light (m/s) in Einstein's equation, we need to convert it to joules (J). Since 1 kJ equals 1000 J, we will multiply the given energy value by 1000 to make this conversion. Secondly, we will rearrange Einstein's equation to solve for mass (m). The equation E = mc² can be rearranged to m = E/c². This algebraic manipulation allows us to directly calculate the mass converted from the given energy and the known speed of light. Thirdly, we will substitute the known values into the rearranged equation. We will plug in the converted energy value (in joules) for E and the speed of light ($3 imes 10^8$ m/s) for c. It's crucial to square the speed of light in the denominator, as indicated by the equation. Fourthly, we will perform the calculation to obtain the mass. This involves dividing the energy by the square of the speed of light. The result will give us the mass in kilograms, which is the standard unit for mass in the International System of Units (SI). Finally, we will present the result in scientific notation to maintain clarity and conciseness, especially since the value is expected to be very small. By following these methodical steps, we can confidently arrive at the correct answer, providing a clear understanding of the mass-energy conversion in this nuclear reaction.

Step 1: Convert Energy from kJ to J

The initial step in solving this problem involves converting the energy released from kilojoules (kJ) to joules (J). This conversion is crucial because the standard unit of energy in the International System of Units (SI) is the joule, and using consistent units is essential for accurate calculations with Einstein's equation. Since 1 kJ is equivalent to 1000 J, we need to multiply the given energy value by 1000 to perform this conversion. The energy released in the fission reaction is given as $9.20 imes 10^{11}$ kJ. To convert this to joules, we perform the following calculation: $9.20 imes 10^{11} ext{ kJ} imes 1000 ext{ J/kJ} = 9.20 imes 10^{14} ext{ J}$. This conversion is a straightforward application of the metric system, but it is a critical step to ensure that our subsequent calculations are accurate. The result, $9.20 imes 10^{14}$ J, represents the amount of energy released in the fission reaction in terms of joules, which can now be directly used in Einstein's mass-energy equivalence equation. This conversion highlights the importance of unit consistency in scientific calculations, and it sets the stage for the next steps in determining the mass converted into energy during the nuclear reaction. It underscores the need for meticulous attention to detail when dealing with physical quantities and their units.

Step 2: Rearrange Einstein's Equation to Solve for Mass

The second step in our solution process involves rearranging Einstein's famous equation, E = mc², to isolate mass (m) as the subject of the formula. This algebraic manipulation is necessary to directly calculate the mass converted into energy, given the energy released (E) and the speed of light (c). The original equation, E = mc², expresses the relationship between energy, mass, and the speed of light, but it is not in a form that directly provides the mass value. To solve for mass, we need to rearrange the equation. We can achieve this by dividing both sides of the equation by c², which results in the equation m = E/c². This rearranged equation now expresses mass (m) in terms of energy (E) and the speed of light (c), allowing us to directly calculate the mass converted. The process of rearranging the equation is a fundamental algebraic technique used in physics and other sciences to solve for specific variables in equations. In this case, it allows us to apply Einstein's principle of mass-energy equivalence in a practical way, enabling us to determine the amount of mass converted into energy during the U-235 fission reaction. This step is crucial for the subsequent substitution of values and the final calculation of the mass converted.

Step 3: Substitute Known Values into the Equation

In this step, we substitute the known values into the rearranged equation m = E/c². We have already determined the energy released in the fission reaction to be $9.20 imes 10^{14}$ J, and we know the speed of light, c, is $3 imes 10^8$ m/s. These values are the key inputs for calculating the mass converted. Substituting these values into the equation, we get: m = ($9.20 imes 10^{14} ext{ J}$) / ($3 imes 10^8 ext{ m/s}$)^2. This substitution step is a straightforward but critical application of the information we have gathered and the equation we have derived. It sets up the final calculation that will give us the answer to our problem. The accuracy of this step is paramount, as any error in substitution will propagate through the rest of the calculation. It is essential to ensure that the values are correctly placed in the equation and that the units are consistent, as we have already addressed in the previous steps. This meticulous approach ensures that our final result will be a reliable and accurate representation of the mass converted during the U-235 fission reaction.

Step 4: Perform the Calculation

The penultimate step in our solution is to perform the calculation using the substituted values in the equation m = ($9.20 imes 10^{14} ext{ J}$) / ($3 imes 10^8 ext{ m/s}$)^2. This calculation will yield the mass converted into energy during the U-235 fission reaction. First, we need to square the speed of light: ($3 imes 10^8 ext{ m/s}$)^2 = $9 imes 10^{16} ext{ m}^2/ ext{s}^2$. Next, we divide the energy by this value: m = ($9.20 imes 10^{14} ext{ J}$) / ($9 imes 10^{16} ext{ m}^2/ ext{s}^2$). Performing this division gives us m ≈ $0.01022 ext{ kg}$. This result represents the mass in kilograms that was converted into energy during the fission reaction. The calculation demonstrates the immense amount of energy that can be released from a relatively small amount of mass, a key principle of nuclear physics. It underscores the power of Einstein's mass-energy equivalence and its implications for nuclear reactions. This step is the culmination of our efforts, providing a numerical answer to the problem we set out to solve. The final step will involve expressing this result in a clear and concise manner, often using scientific notation to represent the magnitude of the value.

Step 5: Express the Result

The final step in our solution is to express the result in a clear and concise manner, typically using scientific notation. We have calculated the mass converted into energy during the U-235 fission reaction to be approximately 0.01022 kg. To express this value in scientific notation, we write it as $1.022 imes 10^{-2} ext{ kg}$. This notation is particularly useful for very small or very large numbers, as it provides a compact and easily interpretable representation. In this case, $1.022 imes 10^{-2} ext{ kg}$ clearly indicates that the mass converted is a small fraction of a kilogram. This final expression of the result is crucial for communicating our findings effectively. Scientific notation not only simplifies the representation of the value but also highlights its magnitude, making it easier to compare with other quantities and understand its significance in the context of the nuclear reaction. The result, $1.022 imes 10^{-2} ext{ kg}$, is the answer to our problem, representing the amount of mass converted into energy during the U-235 fission reaction, and it completes our step-by-step solution process. This concludes our detailed exploration of the mass-energy conversion in nuclear fission.

Based on our calculations, the amount of mass converted into energy during the fission reaction of U-235, with a release of $9.20 imes 10^{11}$ kJ of energy, is approximately $1.02 imes 10^{-2}$ kg. This answer aligns with option A, making it the correct choice. This result underscores the immense energy that can be derived from a relatively small amount of mass, as described by Einstein's mass-energy equivalence. The conversion of just a fraction of a kilogram of mass into energy resulted in a substantial release of $9.20 imes 10^{11}$ kJ, highlighting the potency of nuclear reactions. This conclusion not only answers the specific problem posed but also reinforces the fundamental principles of nuclear physics and the practical implications of mass-energy conversion in nuclear processes. It demonstrates the power of applying theoretical concepts to real-world scenarios, providing a tangible understanding of the energy released in nuclear fission and the associated mass changes.

In conclusion, we have successfully determined the amount of mass converted into energy during the fission reaction of U-235, where $9.20 imes 10^{11}$ kJ of energy was released. By applying Einstein's mass-energy equivalence equation, E = mc², and following a step-by-step approach, we calculated the mass converted to be approximately $1.02 imes 10^{-2}$ kg. This calculation not only provides a numerical answer but also reinforces the fundamental relationship between mass and energy in nuclear reactions. The immense energy released from a small amount of mass highlights the power of nuclear fission and the importance of understanding the principles governing these reactions. Throughout this article, we have emphasized the significance of unit consistency, algebraic manipulation, and accurate substitution of values in solving physics problems. The process of converting energy units, rearranging equations, and performing calculations underscores the practical application of theoretical concepts in nuclear physics. Furthermore, this exploration has deepened our understanding of Einstein's groundbreaking equation and its implications for energy production and nuclear processes. The result obtained, $1.02 imes 10^{-2}$ kg, serves as a concrete example of the mass-energy conversion principle and its relevance in nuclear science and technology.