The most common isotope of uranium, 92238U_{92}^{238}U, has atomic mass 238.050788238.050788 u. Calculate (a) the mass defect; (b) the binding energy (in MeV); (c) the binding energy per nucleon.

Ch 43: Nuclear Physics

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

All textbooks

Young & Freedman Calc 15th Edition

Ch 43: Nuclear Physics

Problem 6

Chapter 42, Problem 6

The most common isotope of uranium, $sub 92 to the 238th power U$ , has atomic mass $238.050788$ u. Calculate (a) the mass defect; (b) the binding energy (in MeV); (c) the binding energy per nucleon.

Verified step by step guidance

Step 1: Understand the problem. The mass defect is the difference between the mass of the nucleus and the sum of the masses of its individual protons and neutrons. The binding energy is the energy equivalent of the mass defect, and the binding energy per nucleon is the binding energy divided by the total number of nucleons.

Step 2: Calculate the total mass of the individual nucleons. The nucleus of 238 92U contains 92 protons and 238 - 92 = 146 neutrons. Use the mass of a proton (1.007276 u) and the mass of a neutron (1.008665 u) to compute the total mass of the nucleons: $ \text{Total mass of nucleons} = 92 \times 1.007276 + 146 \times 1.008665 $.

Step 3: Compute the mass defect. The mass defect is the difference between the total mass of the nucleons and the actual mass of the nucleus: $ \text{Mass defect} = \text{Total mass of nucleons} - \text{Atomic mass of uranium nucleus} $.

Step 4: Convert the mass defect to energy using Einstein's equation $ E = \Delta m c^2 $. Since 1 u corresponds to 931.5 MeV/c^2, the binding energy can be calculated as $ \text{Binding energy} = \text{Mass defect} \times 931.5 $.

Step 5: Calculate the binding energy per nucleon. Divide the total binding energy by the number of nucleons (238) to find $ \text{Binding energy per nucleon} = \frac{\text{Binding energy}}{238} $.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Mass Defect

The mass defect is the difference between the mass of an atomic nucleus and the sum of the masses of its individual protons and neutrons. This discrepancy arises because some mass is converted into energy when nucleons bind together, according to Einstein's equation E=mc². Understanding mass defect is crucial for calculating the binding energy of a nucleus.

Binding Energy

Binding energy is the energy required to disassemble a nucleus into its constituent protons and neutrons. It is a measure of the stability of a nucleus; a higher binding energy indicates a more stable nucleus. The binding energy can be calculated using the mass defect and is often expressed in mega-electronvolts (MeV).

Binding Energy per Nucleon

Binding energy per nucleon is the binding energy divided by the total number of nucleons (protons and neutrons) in the nucleus. This value provides insight into the stability of the nucleus relative to its size, allowing for comparisons between different isotopes. A higher binding energy per nucleon generally indicates a more stable nucleus.

Recommended video:

Guided course

04:10

Intro to Energy & Types of Energy

Related Practice

Textbook Question

What nuclide is produced in the following radioactive decays?

(a) $α$ decay of $sub 94 to the 239th power P u$

(b) $β^{-}$ decay of $sub 11 to the 24th power N a$

views

Textbook Question

How many protons and how many neutrons are there in a nucleus of the most common isotope of (a) silicon, $sub 14 to the 28th power S i$ ; (b) rubidium, $sub 37 to the 85th power R b$ ; (c) thallium, $sub 81 to the 205th power T l$ ?

views

Textbook Question

What particle (a particle, electron, or positron) is emitted in the following radioactive decays?

(a) $_{14}^{27} S i \to_{13}^{27} A l$

(b) $_{92}^{238} U \to_{90}^{234} T h$

views

Textbook Question

(a) Is the decay $n \to p + β^{-} + \overline{v_{e}}$ energetically possible? If not, explain why not. If so, calculate the total energy released.

(b) Is the decay $n \to p + β^{+} + \overline{v_{e}}$ energetically possible? If not, explain why not. If so, calculate the total energy released.

views

Textbook Question

Hydrogen atoms are placed in an external magnetic field. The protons can make transitions between states in which the nuclear spin component is parallel and antiparallel to the field by absorbing or emitting a photon. What magnetic-field magnitude is required for this transition to be induced by photons with frequency $22.7$ MHz?

views