Ch. 36 - The Special Theory of Relativity

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 36 - The Special Theory of Relativity

Problem 81

Chapter 35, Problem 81

How much energy would be required to break a helium nucleus into its constituents, two protons and two neutrons? The rest masses of a proton (including an electron), a neutron, and neutral helium are, respectively, 1.00783 u, 1.00867 u, and 4.00260 u. (This energy difference is called the total binding energy of the $sub 2 to the fourth power He$ nucleus.)

Verified step by step guidance

Step 1: Understand the concept of binding energy. The binding energy of a nucleus is the energy required to separate all its nucleons (protons and neutrons) into individual particles. It is calculated as the difference between the total mass of the individual nucleons and the mass of the nucleus, converted into energy using Einstein's equation, E = mc².

Step 2: Calculate the total mass of the individual nucleons. The helium nucleus consists of two protons and two neutrons. Using the given masses, the total mass of the nucleons is: $ m_{nucleons} = 2 \times 1.00783 \text{ u} + 2 \times 1.00867 \text{ u} $.

Step 3: Subtract the mass of the helium nucleus from the total mass of the nucleons to find the mass defect. The mass defect is given by: $ \Delta m = m_{nucleons} - m_{nucleus} $, where $ m_{nucleus} = 4.00260 \text{ u} $.

Step 4: Convert the mass defect into energy using Einstein's equation $ E = \Delta m \cdot c^2 $. To do this, first convert the mass defect from atomic mass units (u) to kilograms using the conversion factor $ 1 \text{ u} = 1.66054 \times 10^{-27} \text{ kg} $, and then multiply by $ c^2 $, where $ c = 3.00 \times 10^8 \text{ m/s} $.

Step 5: Express the binding energy in units of MeV. To convert the energy from joules to MeV, use the conversion factor $ 1 \text{ MeV} = 1.60218 \times 10^{-13} \text{ J} $. This will give you the total binding energy of the helium nucleus in MeV.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Binding Energy

Binding energy is the energy required to disassemble a nucleus into its individual protons and neutrons. It represents the stability of the nucleus; a higher binding energy indicates a more stable nucleus. In the context of helium, the binding energy quantifies how much energy must be supplied to overcome the attractive forces holding the nucleons together.

Mass-Energy Equivalence

Mass-energy equivalence, encapsulated in Einstein's equation E=mc², states that mass can be converted into energy and vice versa. In nuclear physics, the mass of a nucleus is often less than the sum of its constituent particles' masses due to the binding energy. This principle is crucial for calculating the energy required to break apart a nucleus, as it relates mass differences to energy.

Nuclear Mass Defect

The nuclear mass defect is the difference between the mass of a nucleus and the total mass of its individual protons and neutrons when they are free. This defect arises because some mass is converted into binding energy that holds the nucleus together. Understanding the mass defect is essential for calculating the binding energy and, consequently, the energy required to break the nucleus apart.

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