Welcome to Thermodynamics: The First Law!

Hello future physicist! This chapter is all about energy—specifically, how energy moves in and out of a system (like a gas trapped in a container). The First Law of Thermodynamics is essentially the law of conservation of energy applied to thermal systems. It’s fundamental, powerful, and simpler than it sounds once you nail the sign conventions!

Don't worry if this seems tricky at first; we will break it down into three simple components that govern the energy of any system.

16.1 Quick Review: Internal Energy (\(\Delta U\))

Before diving into the First Law, we need to understand the 'U' part: Internal Energy.

What is Internal Energy?

The internal energy (\(U\)) of a system is the total energy stored inside it due to the random motion and positions of its molecules. It is the sum of two types of energy:

  • Random Kinetic Energy (KE): Associated with the random movement (translation, rotation, vibration) of the molecules. This component depends directly on the temperature of the gas.
  • Random Potential Energy (PE): Associated with the forces (bonds) between the molecules. For an ideal gas, this is often considered zero, but for real substances, it's important.

Key Takeaway:

If the temperature of an object increases, the average KE of its molecules increases, and therefore its Internal Energy (\(U\)) increases.

ⓘ Essential Definition

Internal energy is determined entirely by the state of the system (its temperature, pressure, and volume). It does not matter how the system reached that state, only what the final state is.

Quick Review Summary: Rise in T \(\Rightarrow\) Increase in KE \(\Rightarrow\) Increase in \(\Delta U\).

16.2 The Three Ways to Change Internal Energy

The First Law states that you can change a system's internal energy (\(\Delta U\)) in two ways: by transferring energy as heat (\(q\)), or by doing work (\(W\)).

A. Energy Transferred by Heating (\(q\))

When there is a temperature difference between the system and its surroundings, energy is transferred as heat. This quantity is represented by \(q\).

Sign Convention for \(q\):
  • Positive \(q\) (i.e., \(q > 0\)): Energy is transferred TO the system by heating (the system is heated). This increases internal energy.
  • Negative \(q\) (i.e., \(q < 0\)): Energy is transferred FROM the system by heating (the system cools down). This decreases internal energy.

Analogy: Think of \(q\) as placing a hot cup of coffee (your system) on a stove (surroundings). The stove transfers energy to the coffee, so \(q\) is positive.

B. Work Done (\(W\))

Work is done when a force causes a change in volume of the system, usually by moving a boundary like a piston.

Calculating Work Done at Constant Pressure

If a gas changes volume at a constant pressure (\(p\)), the work done (\(W\)) is given by:

$$W = p \Delta V$$

Where:

  • \(p\) is the constant pressure (in Pa)
  • \(\Delta V\) is the change in volume (in \(\text{m}^3\))
The Crucial Distinction: Work Done ON vs. BY the Gas

In Physics (9702), the First Law is defined using work done ON the system. This is where most students get confused, so pay close attention!

The work term \(W\) in the formula \(\Delta U = q + W\) represents the work done ON the system.

Common Mistake Alert!

There are two conventions in thermodynamics. Some textbooks use \(\Delta U = q - W\), where \(W\) is work done BY the system. For Cambridge A Level Physics (9702), you MUST use the convention where the sign of \(W\) is positive if work is done ON the system.

The official equation is: \(\Delta U = q + W\).

Sign Convention for \(W\) (Work Done ON the System):
  • Work Done ON the gas (Compression): If the surrounding environment pushes the piston IN, the gas volume decreases (\(\Delta V < 0\)). The environment does work ON the gas. \(W\) is positive. This increases internal energy.
  • Work Done BY the gas (Expansion): If the gas pushes the piston OUT, the gas volume increases (\(\Delta V > 0\)). The gas does work ON the environment. \(W\) is negative (since work done ON the gas is negative). This decreases internal energy.

Step-by-Step for Work Calculations:

  1. Calculate the magnitude of work: \(|W| = p |\Delta V|\).
  2. Determine the sign based on volume change:
    • If Volume increases (expansion), the gas is doing the work. \(W\) is negative.
    • If Volume decreases (compression), work is being done on the gas. \(W\) is positive.

Did you know? In the formula \(W = p\Delta V\), if \(\Delta V\) is positive (expansion), then \(W\) calculated using this formula is positive. If we are using the convention \(W\) = work done ON the gas, we must manually flip the sign because expansion means the work done ON the gas is negative.

16.3 The First Law of Thermodynamics

The First Law brings together the internal energy change (\(\Delta U\)), the heat transferred (\(q\)), and the work done (\(W\)).

The Core Equation

The First Law of Thermodynamics is expressed mathematically as:

$$ \Delta U = q + W $$

Where:

  • \(\Delta U\): The increase in internal energy of the system. (Units: Joules, J)
  • \(q\): The energy transferred TO the system by heating (heat supplied). (Units: Joules, J)
  • \(W\): The work done ON the system (by compression). (Units: Joules, J)

This equation is a simple statement of energy conservation: The total change in the energy stored in the system (\(\Delta U\)) must be equal to the total energy that has crossed the system boundary as heat (\(q\)) or work (\(W\)).

The Bank Account Analogy

If you think of the gas's internal energy (\(\Delta U\)) as a bank account:

  • Deposits (\(q\) positive): Adding heat to the system.
  • Withdrawals (\(q\) negative): System loses heat.
  • Deposits (\(W\) positive): Compressing the system (someone did work on your account).
  • Withdrawals (\(W\) negative): Expanding the system (the gas did work on the surroundings, so its internal energy decreases).

Your total change in your account balance (\(\Delta U\)) is the sum of deposits and withdrawals via heat and work.

Step-by-Step Guide to Applying the First Law

When solving problems, follow these steps to ensure you get the signs correct:

  1. Identify the Goal: Are you calculating \(\Delta U\), \(q\), or \(W\)?
  2. Check \(q\) sign: Is heat added (positive) or removed (negative)?
  3. Check \(W\) sign:
    • If the gas volume is compressed (decreases), \(W\) is positive.
    • If the gas volume expands (increases), \(W\) is negative.
    • Calculate: Substitute the values (with correct signs!) into \(\Delta U = q + W\).
Summary of Sign Conventions for \(\Delta U = q + W\)
Term Process Resulting Sign Effect on Internal Energy
\(q\) Heating/Energy transferred TO the system Positive (\(+\)) Increases \(U\)
\(q\) Cooling/Energy transferred FROM the system Negative (\(-\)) Decreases \(U\)
\(W\) Work done ON the system (Compression) Positive (\(+\)) Increases \(U\)
\(W\) Work done BY the system (Expansion) Negative (\(-\)) Decreases \(U\)

💡 Key Takeaway: The First Law

The First Law is the conservation of energy for a thermodynamic system. Remember the precise definitions:

  • \(\Delta U\): Increase in Internal Energy.
  • \(q\): Heat transferred TO the system.
  • \(W\): Work done ON the system.

Getting the signs correct is half the battle!