# Mass and Energy

## Mass and Energy Revision

**Mass and Energy**

E=mc^2 is an equation most people have heard of but many may not know it’s meaning or relevance. In this section we look at this equation and the link between **energy** and **mass**.

**Mass-Energy Equivalence**

While experimenting on his famous theory of **relativity**, Einstein proposed **mass** and **energy** can be considered **equivalent** and are **interchangeable**.

This idea is represented by the equation:

E=mc^2

- E=
**energy**in joules \text{(J)} - m=
**mass**in kilograms \text{(kg)} - c= the
**speed of light**(=3\times 10^8 \text{ ms}^{-1})

The most useful practice from this is the ability to **convert mass to energy**. For a small amount of **mass**, a huge amount of **energy** can be produced due to the c^2 in the equation.

Because of the production of huge amounts of **energy**, the **mass-energy equivalence** can be put to use in nuclear weapons and **nuclear power**, **nuclear fusion** in the sun and high energy collisions in **particle accelerators**.

**Atomic Mass Unit**

The **atomic mass unit (u)** is often used in nuclear physics instead of dealing with incredibly small masses in \text{(kg)}. The a.m.u is equal to \dfrac{1}{12}th of the mass of a carbon-12 atom. The conversion you need to use is: (this is given on your data sheet)

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

**Example:** An electron has a mass of 5.5 \times 10^{-4} \text{ u}. Convert this mass to kilograms.

**[1 mark]**

**Mass Defect**

When scientists measured the **mass** of the **nucleus of an atom as a whole** and then they measured the **mass of the nucleus separated into its constituents **and compared the two, they found that the mass of a nucleus as a whole was **always less than **the mass of its constituents. They named the difference in mass the **mass defect**.

The diagram below shows a representation of carbon-12 as a whole **nucleus** on the left and separated into **protons** and **neutrons** on the right. If measured, the mass of the **nucleus** on the left will **always be less than** the mass of the **protons and neutrons** on the right. The difference is the mass defect of carbon-12.

The **mass defect **(\Delta m)** **can be calculated using the equation:

\Delta m= Z m_p + (A-Z) m_n - m_{\text{total}}

- \Delta m= the
**mass defect**in kilograms \text{kg}) - Z= the
**number of protons** - m_p= the
**mass of a proton**(=1.67 \times 10^{-27} \text{ kg}=1.00728 \text{ u}) - A= the
**nucleon number** - m_n= the
**mass of a neutron**(= 1.67 \times 10^{-27} \text{ kg} = 1.00867 \text{ u}) - m_{\text{total}}= the
**mass of the nucleus**as a whole

This equation can be simplified to:

\text{mass defect}=\text{total mass of protons} + \text{total mass of neutrons} - \text{total mass of the nucleus as a whole}

**Example:** The mass of iron-56 is 55.845 \text{ u}. Calculate the mass defect of iron-56. Give your answer in kilograms.

**[3 marks]**

Find A and Z for iron-56:

Z=26 (from datasheet)

A=\textcolor{7cb447}{56}-26=30Substitute into the mass defect equation:

\begin{aligned} \bold{\Delta m} &= \bold{Z m_p + (A-Z) m_n - m_{\text{total}}} \\ &= (26 \times 1.00728)+(30\times 1.00867) - \textcolor{ffad05}{55.845} \\ &= 0.6043 \text{ u} \\ &= \bold{0.06043 \times 1.661 \times 10 {-27}} \\ &= \bold{1.0039 \times 10^{-27}} \textbf{ kg} \end{aligned}

**Binding Energy**

**Binding energy **is the energy needed to separate a **nucleus** into its components. The mass of the components is always greater than the mass of the nucleus as an energy input is needed to separate the nucleus into its components, and **mass and energy are interchangeable**. Therefore the input of energy needed to separate the components becomes the gain in mass.

This also means that when a nucleus is formed, the equal but opposite amount of energy is released. The amount of energy can be calculated using the equation:

E=\Delta m c^2

- E=
**energy**in joules \text{(J)} - \Delta m=
**change in mass**in kilograms \text{(kg)} - c= the
**speed of light**(=3 \times 10^8 \text{ms}^{-1})

**Example:** Using the previous example, calculate the binding energy per nucleon (\dfrac{E}{A})of iron-56. \Delta m = 1.0039 \times 10^{-27} \text{ kg}.

**[2 marks]**

## Mass and Energy Example Questions

**Question 1:** What is the mass defect and how is it calculated?

**[2 marks]**

The **difference in mass between the nucleus of an atom and the mass of its constituents when separated**.

where:

- \Delta m is the mass defect
- Z is the number of protons
- m_p is the mass of a proton
- A is the nucleon number
- m_n is the mass of a neutron
- m_{\text{total}} is the mass of the nucleus as a whole

**Question 2:** The mass of carbon-12 is 12.011 \text{ u}. Calculate the mass defect of carbon-12. (Mass of proton =1.00728 \text{ u} and mass of a neutron =1.00867 \text{ u}

**[3 marks]**

**Question 3:** Define the binding energy of a nucleus.

**[1 mark]**

Binding energy is the **energy needed to separate a nucleus into its components**.