# 18.2.1 Internal energy

### Defining Internal Energy

• Energy can generally be classified into two forms: kinetic or potential energy
• The molecules of all substances contain both kinetic and potential energies
• The amount of kinetic and potential energy a substance contains depends on the phases of matter (solid, liquid or gas), this is known as the internal energy
• The internal energy of a substance is defined as:

The sum of the random distribution of kinetic and potential energies within a system of molecules

• The symbol for internal energy is U, with units of Joules (J)
• The internal energy of a system is determined by:
• Temperature
• The random motion of molecules
• The phase of matter: gases have the highest internal energy, solids have the lowest
• The internal energy of a system can increase by:
• Doing work on it
• Adding heat to it
• The internal energy of a system can decrease by:
• Losing heat to its surroundings

#### Exam Tip

When an exam question asks you to define “internal energy”, you can lose a mark for not mentioning the “random motion” of the particles or the “random distribution” of the energies, so make sure you include one of these in your definition!

### Internal Energy & Temperature

• The internal energy of an object is intrinsically related to its temperature
• When a container containing gas molecules is heated up, the molecules begin to move around faster, increasing their kinetic energy
• If the object is a solid, where the molecules are tightly packed, when heated the molecules begin to vibrate more
• Molecules in liquids and solids have both kinetic and potential energy because they are close together and bound by intermolecular forces
• However, ideal gas molecules are assumed to have no intermolecular forces
• This means there have no potential energy, only kinetic energy
• The (change in) internal energy of an ideal gas is equal to: • Therefore, the change in internal energy is proportional to the change in temperature:

ΔU ∝ ΔT

• Where:
• ΔU = change in internal energy (J)
• ΔT = change in temperature (K) As the container is heated up, the gas molecules move faster with higher kinetic energy and therefore higher internal energy

#### Worked Example

A student suggests that, when an ideal gas is heated from 50 oC to 150 oC, the internal energy of the gas is trebled.

State and explain whether the student’s suggestion is correct.

Step 1:

Write down the relationship between internal energy and temperature

The internal energy of an ideal gas is directly proportional to its temperature

ΔU ∝ ΔT

Step 2:

Determine whether the change in temperature (in K) increases by three times

The temperature change is the thermodynamic temperature ie. Kelvin

The temperature change in degrees from 50 oC to 150 oC increases by three times

The temperature change in Kelvin is:

50 oC + 273.15 = 323.15 K

150 oC + 273.15 = 423.15 K Therefore, the temperature change, in Kelvin, does not increase by three times

Step 3:

Write a concluding statement relating the temperature change to the internal energy

The internal energy is directly proportional to the temperature

The thermodynamic temperature has not trebled, therefore, neither has the internal energy

Therefore, the student is incorrect

#### Exam Tip

If an exam question about an ideal gas asks for the total internal energy, remember that this is equal to the total kinetic energy since an ideal gas has zero potential energy ### Author: Katie

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.
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