Mass and energy Revision Notes for AQA A-Level Physics

Mass and energy

Mass-energy equivalence

In 1905, Albert Einstein published groundbreaking research on the behavior of moving objects, from which he derived an equation that would revolutionize our understanding of mass and energy. This equation is known as the mass-energy equivalence equation:

$E = mc^2$

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Equation components:

$E$ is the energy (measured in joules, J)
$m$ is the mass (measured in kilograms, kg)
$c$ is the speed of light in a vacuum ( $3.00 \times 10^8 \, \text{m} \cdot \text{s}^{-1}$ )

Einstein determined that an object's mass represents the amount of energy it contains. Furthermore, if energy is added to or removed from an object, its mass will change accordingly. This principle has profound implications for nuclear physics, where mass and energy transformations occur during nuclear reactions and radioactive decay.

Mass changes in nuclear processes

When a nucleus undergoes radioactive decay, energy is released. This energy appears as kinetic energy of the decay products and sometimes as electromagnetic radiation in the form of gamma photons. According to Einstein's mass-energy relationship, the mass measured before the decay must exceed the total mass measured after the decay.

The mass difference (also called mass defect) represents the mass that has been converted into energy. This mass difference should equal the total kinetic energy and electromagnetic energy of the products, when converted using $E = mc^2$ .

For example, when a radioactive nucleus decays, the combined mass of all particles after the decay will be slightly less than the mass of the original nucleus. This "missing" mass has been transformed into the kinetic energy of the emitted particles and any gamma radiation produced.

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Calculating mass difference:

$\text{mass difference} = \text{total mass before decay} - \text{total mass after decay}$

The masses involved in nuclear decay calculations must be determined very accurately, typically requiring values with six to eight significant figures. This precision is necessary because the mass changes are extremely small compared to the total masses involved.

Atomic mass units

Nuclear and atomic masses are most conveniently expressed using atomic mass units (symbol: $u$ ). One atomic mass unit is defined as one-twelfth of the mass of a carbon-12 atom:

$1 \, \text{u} = \frac{1}{12} \times (\text{mass of carbon-12 atom})$

When expressed in SI units:

$1 \, \text{u} = 1.661 \times 10^{-27} \, \text{kg}$

Using atomic mass units, data can be provided for atomic and nuclear masses with an accuracy of six significant figures or better. For instance, masses might be expressed as 0.000010 u or with even greater precision.

Converting between atomic mass units and energy

Since $E = mc^2$ , atomic mass units can be directly converted to energy. The conversion factor between atomic mass units and electronvolts is:

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Key conversion factor:

$1 \, \text{u} \text{ is equivalent to } 931.5 \, \text{MeV}$

This conversion is extremely useful in nuclear physics calculations, as it allows us to quickly determine the energy equivalent of a mass difference without having to first convert to kilograms and then use $E = mc^2$ .

When particle masses are given in atomic mass units and energy changes are required in MeV, this direct conversion eliminates several calculation steps and reduces the chance of errors.

Nuclear binding energy

The binding energy of a nucleus is defined as the work required to completely separate all of its nucleons (protons and neutrons) from each other.

Separating nucleons requires energy input because it is necessary to overcome the strong nuclear force holding them together. The strong nuclear force binds the nucleons in the nucleus, and work must be done against this force to pull the nucleons apart. Since performing work transfers energy, and according to Einstein's principle transferring energy changes mass, the mass of the separated nucleons must be greater than the mass of the intact nucleus.

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Mass relationship:

$\text{mass of separated nucleons} > \text{mass of bound nucleus}$

The difference in mass (mass defect) multiplied by $c^2$ gives the binding energy of the nucleus.

Calculating binding energy

To determine the binding energy of a nucleus:

Calculate the total mass of all separated nucleons (sum of individual proton and neutron masses)
Determine the actual mass of the nucleus
Find the mass difference: mass of separated nucleons - mass of nucleus
Convert this mass difference to energy using $E = mc^2$ or the conversion factor $1 \, \text{u} = 931.5 \, \text{MeV}$

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Worked Example: Calculating Binding Energy

Consider a nucleus with mass 234.993 u. If the constituent nucleons have a combined mass of 236.909 u when separated, then:

Step 1: Calculate the mass difference

$\text{mass difference} = 236.909 \, \text{u} - 234.993 \, \text{u} = 1.916 \, \text{u}$

Step 2: Convert to energy using the conversion factor

$E = 1.916 \, \text{u} \times 931.5 \, \text{MeV/u} = 1785 \, \text{MeV}$

Result: The binding energy of this nucleus is 1785 MeV. This is the amount of energy that would need to be supplied to completely disassemble the nucleus into its individual nucleons.

Binding energy per nucleon

Nuclei with the greatest number of nucleons generally have the largest total binding energies. However, to compare the stability of different nuclei fairly, we need to consider how tightly bound each nucleon is on average. The binding energy per nucleon is defined as the total binding energy divided by the number of nucleons in the nucleus:

$\text{binding energy per nucleon} = \frac{\text{total binding energy}}{\text{nucleon number (A)}}$

where $A$ is the mass number (total number of protons and neutrons).

The binding energy per nucleon indicates how strongly each nucleon is held within the nucleus. Nuclei with higher binding energy per nucleon are more tightly bound and therefore more stable.

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The nucleus with the highest binding energy per nucleon is iron-56, making it the most tightly bound and most stable nucleus. When binding energy per nucleon is plotted against nucleon number for all nuclei, iron-56 appears at the peak of the graph. This has important implications for nuclear reactions.

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Key Points to Remember:

Mass and energy are equivalent: Einstein's equation $E = mc^2$ shows that mass can be converted to energy and vice versa.
Mass defect in nuclear reactions: The mass before a nuclear decay or reaction differs from the mass after, with the missing mass converted to energy.
Atomic mass units: $1 \, \text{u} = 1.661 \times 10^{-27} \, \text{kg}$ , and $1 \, \text{u}$ is equivalent to $931.5 \, \text{MeV}$ - this conversion simplifies nuclear energy calculations.
Binding energy represents nuclear stability: Higher binding energy means more work is needed to separate the nucleons, indicating a more stable nucleus.
Binding energy per nucleon allows comparison: Dividing total binding energy by nucleon number lets us compare stability across different nuclei, with iron-56 being the most stable.

Mass and energy (AQA A-Level Physics): Revision Notes

Mass and energy

Mass-energy equivalence

Mass changes in nuclear processes

Atomic mass units

Converting between atomic mass units and energy

Nuclear binding energy

Calculating binding energy

Binding energy per nucleon

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