The most common isotope of uranium, 92 238 U _{92}^{238}U , has atomic mass 238.050788 238.050788 u. Calculate (a) the mass defect; (b) the binding energy (in MeV); (c) the binding energy per nucleon.

Hi, everyone. In this practice problem, we're being asked to calculate or find the mass defect or DELTA M and a total binding energy E B and the binding energy per nucleon. For carbon 14, we have the concentration of carbon 14, 14, 6 C A radioactive isotope of carbon used to determine the age of organic materials where the atomic mass of carbon 14 is about 14.3241 U. We're being asked to first calculate the mass defect or delta M second, the total binding energy or E B and third, the binding energy per nucleon for carbon 14. The options given listed the different mass defect, total binding energy and binding energy per nucleon for carbon 14. So let's tackle this problem one by one. So for the first part, the mass defect or delta M is the difference between the mass of the nucleus and the combined mass of its constituent nucleons. So delta M essentially is going to equal to Z mh plus N M N minus A Z M. So in this uh equation here, the nucleus of carbon 14 will contain six protons and eight neutrons. So essentially the Z will be equals to six and the N will be equals to eight, we will substitute the numerical values so that we can obtain delta M to then be equal to first Z which is going to be 66 multiplied by 1.7825 U which is going to be mh which is a constant and then plus N which is going to be eight multiplied by M N which is going to be 1. U minus A Z M which is going to be the mass atomic mass of carbon 14 which is 14.3241 U calculating this delta M will be equals to 0.113029 U which will be the mass defect calculated. That will be the first part. So moving on to the second part, the second part ask the total binding energy or E B and we want to recall that the total binding energy or E B can be calculated by multiplying delta M with C squared C is the speed of light in vacuum. And C squared in mega E V per U will be equal to C squared will be equal to 931.5 mega E V divided by U. The binding energy can then be calculated. So E B will then be calculated by multiplying delta M which is 0.113029 U multiplied by C squared which is 931.5 mega E V divided by U calculating this, the binding energy will come up to a value of 100 and five mega E V. So lastly, we want to move on to the third part which is asking about the uh binding energy per nucleon of carbon 14 where uh this will essentially be asking E B divided by A. So for uh four, uh carbon 13 D A is going to be equals to 14. So in this case, E B divided by A is going to E equal to 100 and five mega E V divided by A which is 14. So in this case, that will give out our E B divided by A or the binding energy per nucleon of carbon 14 to be equal to 7.5 mega E V. So the binding energy per nucleon for the third part is 7.5 mega E V and the total binding energy E B is going to be 100 and five mega E V. And lastly, the delta M or the mass defect is going to be 0.113 oh 29 U. This will correspond to option A in our answer choices and option A will be the answer to this particular practice problem. So that will be it for this video. If you guys still have any sort of confusion. Please feel free to check out our adolescent videos on similar topics available on our website. But other than that, that will be it for this one. Thank you.

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Problem 6

Textbook Question

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} $.

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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.

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