Mass Defect: Videos & Practice Problems

Mass defect is a crucial concept in nuclear chemistry, representing the difference between the predicted mass of an isotope—calculated from the sum of its subatomic particles' masses—and its actual nuclear mass. Subatomic particles, namely neutrons, protons, and electrons, have masses of 1.00866 AMU, 1.00727 AMU, and 0.00055 AMU, respectively, with the bulk of an atom's mass residing in the nucleus due to the larger masses of neutrons and protons. The first law of thermodynamics states that energy cannot be created or destroyed, only transformed. This principle applies to mass defect as the mass lost during isotope formation is converted into energy, a concept mathematically expressed by Einstein's famous equation:

$E=m{c}^2$

This equation allows for the conversion between mass and energy, indicating that the predicted mass will always be greater than the nuclear mass because of this energy conversion. If the nuclear mass is not provided, it can be calculated using the nuclear mass equation: nuclear mass equals the mass number minus the number of electrons times their individual atomic mass units (0.00055 AMU for electrons). This understanding of mass defect and energy conversion is fundamental in comprehending nuclear reactions and their associated energies.

Calculating Predicted Mass

To understand mass defect, it's essential to first determine the predicted mass of an element, which accounts for the total mass of its subatomic particles: neutrons, protons, and electrons. The concept of atomic mass units (amu) is crucial here, with one atomic mass unit defined as approximately \(1.66 \times 10^{-27}\) kilograms. This conversion allows us to express the masses of subatomic particles in relative terms.

The masses of the subatomic particles are as follows: neutrons have a mass of approximately 1.00866 amu, protons weigh about 1.00727 amu, and electrons, being the lightest, have a mass of only 0.00055 amu. Notably, neutrons are slightly heavier than protons, while both are significantly more massive than electrons. This disparity in mass indicates that the majority of an atom's mass is concentrated in the nucleus, where protons and neutrons are located.

To calculate the predicted mass of an atom, one must sum the contributions of all its subatomic particles based on their respective quantities. For example, if an atom has a certain number of protons (Z) and neutrons (N), the predicted mass can be calculated using the formula:

Predicted Mass (amu) = (Z × mass of proton) + (N × mass of neutron) + (number of electrons × mass of electron)

This calculation provides a foundational understanding of how the mass of an atom is derived from its constituent particles, setting the stage for further exploration of mass defect and binding energy in nuclear physics.

Mass Defect Example

To calculate the predicted mass of the helium-4 isotope, we start by identifying its components based on its atomic structure. Helium has a mass number of 4 and an atomic number of 2, indicating that it contains 2 protons. The number of neutrons can be determined by subtracting the atomic number from the mass number:

Number of Neutrons = Mass Number - Atomic Number = 4 - 2 = 2

Since helium-4 is an isotope and not an ion, it has an equal number of protons and electrons, which is also 2. Now, we can calculate the predicted mass using the relative masses of the subatomic particles:

Predicted Mass = (Number of Protons × Mass of Proton) + (Number of Neutrons × Mass of Neutron) + (Number of Electrons × Mass of Electron)

Substituting the values:

Predicted Mass = (2 × 1.00727 \, \text{amu}) + (2 × 1.00866 \, \text{amu}) + (2 × 0.00055 \, \text{amu})

Calculating each term:

Predicted Mass = 2.01454 \, \text{amu} + 2.01732 \, \text{amu} + 0.00110 \, \text{amu}

Adding these together gives:

Predicted Mass = 4.03296 \, \text{amu}

Thus, the predicted mass of the helium-4 isotope is approximately 4.03296 amu.

Energy to Mass Conversion

The first law of thermodynamics states that energy cannot be created or destroyed, only transformed. This principle is crucial when discussing the mass defect in nuclear physics, particularly in the formation of isotopes. For instance, when two neutrons, two protons, and two electrons combine to form a helium-4 isotope, their predicted total mass is 4.03296 atomic mass units (amu). However, during this combination, a portion of the mass is lost, known as the mass defect. This loss occurs because the binding energy released during the formation of the nucleus results in a lower actual mass than the predicted mass.

The relationship between the predicted mass, mass defect, and nuclear mass can be expressed mathematically. The equation can be summarized as:

Predicted Mass - Mass Defect = Nuclear Mass + Energy

Here, the nuclear mass is the actual mass of the isotope as listed on the periodic table. The energy released during the formation of the nucleus can also be absorbed if the isotope is broken down into its constituent particles, demonstrating the reversible nature of this process.

To quantify the relationship between mass and energy, Einstein's famous equation, E = mc^2, plays a pivotal role. This equation indicates that mass and energy are interchangeable; knowing one allows for the calculation of the other. In the context of isotopes, the predicted mass is always greater than the nuclear mass due to the mass defect, which is converted into energy during the formation of the nucleus.

In summary, when subatomic particles combine to form an isotope, not all of the mass is retained; some is transformed into energy. This energy can be reintroduced to break the isotope back into its original particles, highlighting the interconnectedness of mass and energy in nuclear reactions.

Energy to Mass Conversion Example

To determine the mass defect of calcium-42, we start with its atomic mass, which is 41.958618 AMU. The mass defect is calculated by finding the difference between the predicted mass of the nucleus and the actual atomic mass. The predicted mass is derived from the total mass of its constituent subatomic particles: protons and neutrons.

Calcium-42 has a mass number of 42 and an atomic number of 20, indicating it has 20 protons. Since it is neutral, it also has 20 electrons. The number of neutrons can be calculated by subtracting the number of protons from the mass number: 42 - 20 = 22 neutrons.

Next, we calculate the predicted mass by multiplying the number of each type of particle by their respective atomic mass units (AMU). The masses are approximately:

• Proton: 1.007276 AMU
• Neutron: 1.008665 AMU

Thus, the predicted mass is calculated as follows:

Predicted mass = (20 protons × 1.007276 AMU) + (22 neutrons × 1.008665 AMU) = 42.34692 AMU.

Now, we can find the mass defect (m) using the equation:

Predicted mass - Atomic mass = Mass defect

Substituting the values gives us:

42.34692 AMU - 41.958618 AMU = 0.388302 AMU.

To convert the mass defect from AMU to kilograms, we use the conversion factor where 1 AMU is equal to \(1.66 \times 10^{-27}\) kilograms:

Mass defect in kilograms = \(0.388302 \, \text{AMU} \times 1.66 \times 10^{-27} \, \text{kg/AMU} = 6.4458 \times 10^{-28} \, \text{kg}.\)

Therefore, the mass defect of calcium-42 is approximately \(6.4458 \times 10^{-28}\) kilograms.

Calculating Mass Defect

When the nuclear mass of an isotope is not provided, calculating the mass defect requires a more detailed approach. In such cases, the nuclear mass can be determined by considering the mass number of the isotope and the mass of its electrons. The relationship can be expressed with the following equation:

Nuclear Mass = Mass Number - (Number of Electrons × Mass of One Electron)

Here, the mass of one electron is approximately 0.00055 atomic mass units (amu). This equation is essential for calculating the nuclear mass when it is not directly given in a problem. By applying this formula, you can effectively find the nuclear mass, which is crucial for further calculations involving mass defect and nuclear stability.

Calculating Mass Defect Example

To calculate the mass defect for the isotope oxygen-16, we first need to determine the predicted mass based on the number of subatomic particles. Oxygen has an atomic number of 8, indicating it has 8 protons and, since it is neutral, also 8 electrons. The mass number of oxygen-16 tells us it has 16 total nucleons, which means it has 8 neutrons (16 - 8 = 8).

Next, we calculate the predicted mass by summing the masses of the protons, neutrons, and electrons. The atomic mass units (amu) for each particle are as follows: protons have a mass of approximately 1.00727 amu, neutrons about 1.00866 amu, and electrons roughly 0.00055 amu. Therefore, the predicted mass can be calculated as:

Predicted Mass = (Number of Protons × Mass of Proton) + (Number of Neutrons × Mass of Neutron) + (Number of Electrons × Mass of Electron)

Substituting the values:

Predicted Mass = (8 × 1.00727 \text{ amu}) + (8 × 1.00866 \text{ amu}) + (8 × 0.00055 \text{ amu})

Predicted Mass = 8.05816 \text{ amu} + 8.06928 \text{ amu} + 0.0044 \text{ amu} = 16.13184 \text{ amu}

Now, we find the nuclear mass by subtracting the combined mass of the electrons from the mass number. The mass number for oxygen-16 is 16, and the total mass of the electrons is:

Total Mass of Electrons = Number of Electrons × Mass of Electron = 8 × 0.00055 \text{ amu} = 0.0044 \text{ amu}

Nuclear Mass = Mass Number - Total Mass of Electrons = 16 - 0.0044 = 15.9956 \text{ amu}

Finally, we can determine the mass defect using the formula:

Mass Defect = Predicted Mass - Nuclear Mass

Substituting the values:

Mass Defect = 16.13184 \text{ amu} - 15.9956 \text{ amu} = 0.13624 \text{ amu}

Thus, the mass defect for the isotope oxygen-16 is approximately 0.13624 amu.

Calculate the mass defect (in mg) for the following isotope. (1 neutron = 1.00866 amu, 1 proton = 1.00727 amu, & 1 electron = 0.00055 amu). 
¹⁴C

• A. 3.1789 x 10^–22 mg
• B. 1.9837 x 10^–22 mg
• C. 1.0294 x 10^–20 mg
• D. 1.5123 x 10^–21 mg

Answer: (B)

Here’s what students ask on this topic:

How do you calculate mass defect?

Mass defect is the difference between the calculated mass of an atom's constituent protons, neutrons, and electrons and its actual mass as measured experimentally. To calculate mass defect, you follow these steps:

1. Determine the number of protons and neutrons in the nucleus of the atom. You can find this information on the periodic table of elements, where the atomic number represents the number of protons and the difference between the mass number and the atomic number gives you the number of neutrons.
2. Calculate the total mass of the protons, neutrons, and electrons if they were separate. Use the standard atomic mass units: a proton has a mass of approximately 1.007276 amu, a neutron has a mass of approximately 1.008665 amu, and an electron has a mass of approximately 0.0005486 amu.
3. Multiply the number of each particle by its respective mass to get the total mass for each type of particle, then sum these values to get the total calculated mass.
4. Obtain the actual atomic mass of the atom (isotope) from the periodic table or a reliable scientific resource.
5. Subtract the actual mass from the total calculated mass of the individual nucleons and electrons. This difference is the mass defect.

$\text{Mass Defect}=\left(\text{Total Calculated Mass}\right)$

What action results in the mass defect?

The mass defect is a result of the binding energy that holds the nucleus together in an atom. When protons and neutrons, which are the nucleons, come together to form a nucleus, they are bound by the strong nuclear force. This force is incredibly powerful, and it works to overcome the repulsion between the positively charged protons.

To maintain this binding, a certain amount of energy is required, known as the binding energy. According to Einstein's famous equation, $E^2=\mathrm{mc}{2}^{}$, energy and mass are interchangeable. This means that the binding energy has an equivalent mass. When nucleons bind together, the energy that's converted to bind them results in a loss of mass. This loss is what we refer to as the mass defect.

The mass defect is not a loss in the usual sense; the 'missing' mass has been converted into binding energy to hold the nucleus together. This is why the mass of a nucleus is always less than the sum of the masses of its individual protons and neutrons.

What does the mass defect measure?

The mass defect measures the difference between the sum of the individual masses of protons, neutrons, and electrons that make up an atom's nucleus and the actual mass of the nucleus itself. This discrepancy arises because some of the mass is converted into binding energy, which holds the nucleus together, according to the mass-energy equivalence principle expressed by Einstein's famous equation:

E² = mc²

When nucleons (protons and neutrons) bind together to form a nucleus, the mass defect is a manifestation of the energy released during this process. The larger the mass defect, the more stable the nucleus is, as it indicates a greater amount of binding energy. This concept is crucial in nuclear physics and has practical applications in areas such as nuclear energy and nuclear weapons development.