Physics (9702) | Internal energy

A Level Physics Study Notes (9702): Chapter 16 – Thermodynamics: Internal Energy

Welcome to Thermodynamics! This chapter looks inside matter to understand where energy is actually stored and how it moves around. Don't worry if these concepts seem abstract—we will use plenty of analogies to make sure the ideas stick!

Understanding internal energy and the First Law of Thermodynamics is vital because it explains how engines work, how refrigerators cool things down, and essentially, why energy is conserved in heating and working processes.

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16.1 Internal Energy (U)

What is Internal Energy?

Imagine a container of gas, liquid, or solid. The tiny particles (atoms or molecules) inside are constantly moving and interacting. The Internal Energy (\(U\)) of the system is simply the total energy contained within it.

This energy is split into two main, randomly distributed components:

1. Kinetic Energy (KE): The energy due to the random motion of the molecules (moving, rotating, vibrating).
2. Potential Energy (PE): The energy stored in the forces (bonds) between the molecules. This is related to the separation and arrangement of the molecules.

The definition you must recall is:

Definition: Internal energy (\(U\)) is the sum of the randomly distributed kinetic and potential energies associated with the molecules of a system.

Focus on the Components: KE vs. PE

The relative importance of KE and PE depends entirely on the state (phase) of the substance:

• Ideal Gases: We often treat ideal gases as having no forces between molecules (except during instantaneous collisions). This means the potential energy is zero (or negligible). Therefore, the internal energy of an ideal gas is entirely kinetic energy.
• Real Gases, Liquids, and Solids: In these states, intermolecular forces are significant, so both KE and PE contribute substantially to the internal energy.

Internal Energy is a State Function

A crucial point for A Level Physics:

Internal energy is determined entirely by the state of the system. The "state" is defined by macroscopic variables like temperature, pressure, and volume.

Analogy: Internal energy is like the amount of money in your bank account right now. It doesn't matter *how* you got that money (by working, winning a lottery, or finding it on the street)—only the final balance matters. Similarly, the internal energy of a gas at a specific temperature and volume is always the same, regardless of the path taken to reach that state.

Relating Temperature and Internal Energy

The connection between temperature and internal energy is direct and important (Syllabus 16.1.2):

A rise in the temperature (\(T\)) of an object always corresponds to an increase in its internal energy (\(U\)).

Why? Because temperature is a measure of the average random kinetic energy of the molecules. If the temperature increases, the average speed of the molecules increases, and thus their total KE (and total internal energy \(U\)) must increase.

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16.2 The First Law of Thermodynamics

The First Law of Thermodynamics is fundamentally a statement of the Principle of Conservation of Energy applied to thermal processes. It tells us how the internal energy of a system changes when you heat it or do work on it.

The change in internal energy ($\Delta U$) of a system is equal to the energy supplied by heating ($q$) plus the work done on the system ($W$).

The Equation and Terminology (Syllabus 16.2.2)

The mathematical form of the First Law (as defined in the syllabus, using work done *on* the system) is:

$$ \Delta U = q + W $$

Where:

• \(\Delta U\): The increase in the internal energy of the system (measured in J).
• \(q\): The heating of the system (energy transferred to the system by heating, in J).
• \(W\): The work done on the system (energy transferred to the system by doing work, in J).

The All-Important Sign Convention

The sign of each term determines whether energy is entering or leaving the system. Mastering this convention is crucial for solving problems!

Quantity	Positive (\(+ve\))	Negative (\(-ve\))
\(\Delta U\)	Internal energy increases (usually temperature increases)	Internal energy decreases (usually temperature decreases)
\(q\) (Heating)	Heat energy is supplied to the system (System absorbs heat)	Heat energy is removed from the system (System loses heat)
\(W\) (Work Done)	Work is done on the system (e.g., Gas is compressed, volume decreases)	Work is done by the system (e.g., Gas expands, volume increases)

Memory Aid: Think of the system as an unhappy person. They only get happier (positive $\Delta U$) if you give them things (positive $q$ and positive $W$).

Work Done When Volume Changes (Syllabus 16.2.1)

Work is done when a force causes a displacement. For a gas in a piston, if the gas expands or is compressed, work is done.

If a gas changes volume by \(\Delta V\) while being held at a constant pressure (\(p\)), the magnitude of the work done is:

$$ W = p \Delta V $$

Where:

• \(p\) is the constant pressure (in Pa).
• \(\Delta V\) is the change in volume (in \(\text{m}^3\)).

Important Distinction: Work done on vs. Work done by

The syllabus requires you to understand the difference between work done by the gas and work done on the gas. This determines the sign of \(W\) in the First Law, \(\Delta U = q + W\).

1. Work Done BY the Gas (Expansion)

If the gas expands, \(\Delta V\) is positive. The gas is pushing out the piston. It uses up its internal energy to do this work on the surroundings.
In the First Law ($\Delta U = q + W$), $W$ must be negative.

2. Work Done ON the Gas (Compression)

If the gas is compressed, \(\Delta V\) is negative. The surroundings push the piston in, adding energy to the gas.
In the First Law ($\Delta U = q + W$), $W$ must be positive.

Common Mistake to Avoid: When using the formula \(W = p \Delta V\), remember that the sign of \(W\) used in the First Law must be decided based on whether the work was done on or by the system, regardless of the sign resulting from calculating \(\Delta V\). If you calculate $W_{\text{by gas}}$, you must use $W_{\text{on gas}} = -W_{\text{by gas}}$ in $\Delta U = q + W$.

Example: A gas expands, doing 100 J of work on the surroundings.
Work done by the gas = \(+100\text{ J}\).
Work done on the gas = \(-100\text{ J}\).
When applying $\Delta U = q + W$, you must use $W = -100\text{ J}$.

Step-by-Step Application of the First Law

Let's use the First Law ($\Delta U = q + W$) to analyze a simple scenario:

Scenario: A balloon containing air is heated by 500 J ($q$) and, in the process, it expands, doing 150 J of work on the surrounding atmosphere ($W_{\text{by gas}}$).

Step 1: Determine the sign and magnitude of \(q\)

Energy is supplied to the system (heating).
\(q = +500\text{ J}\)

Step 2: Determine the sign and magnitude of \(W\) (Work done ON the system)

The system did work on the surroundings (expansion).
\(W_{\text{by gas}} = +150\text{ J}\).
Therefore, the work done ON the system is negative:
\(W = -150\text{ J}\)

Step 3: Calculate the change in Internal Energy \(\Delta U\)

$$ \Delta U = q + W $$ $$ \Delta U = 500\text{ J} + (-150\text{ J}) $$ $$ \Delta U = +350\text{ J} $$

Conclusion: The internal energy of the air in the balloon increased by 350 J, meaning its temperature has risen.

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Key Takeaways for Thermodynamics

Key Concepts

• Internal energy (\(U\)) is the total random KE and PE of molecules.
• Temperature is directly related to the total KE component of internal energy.
• The First Law is energy conservation: \(\Delta U = q + W\).

Sign Convention Check

• \(\Delta U\) positive $\implies$ Temperature Rises.
• \(q\) positive $\implies$ Heat In (System heated).
• \(W\) positive $\implies$ Work In (System compressed, $V$ decreases).

Remember these rules, practice the sign conventions, and you'll be able to tackle any thermodynamics problem!