The First Law | AP Physics 2 Unit 1 Study Guide

Getting Started

We will investigate a confined quantity of gas, such as the air trapped inside a cylinder with a movable piston. At this microscopic and macroscopic scale, we are concerned with the total energy contained within the gas and how that energy changes. The core question is: How do heating, cooling, compressing, and expanding a gas affect its internal energy, and how can we account for these changes using a fundamental conservation law?

What You Should Be Able to Do

By the end of this chapter, you should be able to:

Define the internal energy of a system in terms of the motion and configuration of its constituent particles.
Calculate the change in internal energy for an ideal monatomic gas given a change in temperature.
Describe how heat transfer and work are two distinct mechanisms for changing a system's internal energy.
Apply the first law of thermodynamics to relate the change in internal energy, heat transferred, and work done on a system.
Interpret pressure-volume (PV) diagrams to determine the work done on or by a gas during a thermodynamic process.

Key Concepts & Mechanisms

System & Preconditions

In thermodynamics, our system is the collection of objects we are studying—most often, an ideal gas confined within a container (e.g., a cylinder with a piston). Everything outside this boundary is considered the surroundings. We make a key idealization: the gas is an ideal monatomic gas. This means we treat the gas particles as point masses with no volume, whose only form of energy is translational kinetic energy. We assume there are no intermolecular forces, so there is no potential energy associated with the configuration of the atoms.

Key Steps / Relations

Defining Internal Energy (U): The internal energy (U) of a system is the sum of all kinetic and potential energies of its constituent particles. For our ideal monatomic gas, this simplifies greatly. Since there are no intermolecular forces, the potential energy is zero. The internal energy is simply the sum of the translational kinetic energies of all the atoms. This total energy is directly proportional to the absolute temperature of the gas.
- Symbol: $U$
- SI Unit: Joules (J)
- Equation: $U = \frac{3}{2} n RT = \frac{3}{2} N k_{B} T$
  - Where $n$ is the number of moles, $R$ is the ideal gas constant (8.31 J/mol·K), $T$ is the absolute temperature (K), $N$ is the number of atoms, and $k_{B}$ is the Boltzmann constant (1.38 × 10⁻²³ J/K).
Changing Internal Energy: Heat (Q): The first way to change a system's internal energy is through heat (Q). Heat is defined as the transfer of energy between a system and its surroundings due to a temperature difference.
- Sign Convention: If heat flows into the system (heating), $Q$ is positive. If heat flows out of the system (cooling), $Q$ is negative.
- Symbol: $Q$
- SI Unit: Joules (J)
Changing Internal Energy: Work (W): The second way to change a system's internal energy is through work (W). Work is the transfer of energy that occurs when a force acts through a displacement. In thermodynamics, this typically involves the surroundings exerting a force on the system's boundary (like a piston), causing a change in volume.
- Sign Convention: If work is done on the system (e.g., compression), $W$ is positive. If work is done by the system on its surroundings (e.g., expansion), $W$ is negative.
- Symbol: $W$
- SI Unit: Joules (J)
Calculating Work Done on a Gas: When an external pressure changes the volume of a gas, work is done. For a constant external pressure $P$ , the work done on the gas as its volume changes by $Δ V$ is:
- Equation: $W = - P Δ V$
  - Where $P$ is the external pressure (Pascals, Pa) and $Δ V = V_{f ina l} - V_{ini t ia l}$ is the change in volume (m³).
  - The negative sign is crucial. If the gas is compressed, $Δ V$ is negative, making $W$ positive (work done on the gas). If the gas expands, $Δ V$ is positive, making $W$ negative (work done by the gas).
The First Law of Thermodynamics: This law is a restatement of the principle of conservation of energy for a thermodynamic system. It states that the change in a system's internal energy ( $Δ U$ ) is equal to the net heat added to the system ( $Q$ ) plus the net work done on the system ( $W$ ).
- Equation: $Δ U = Q + W$
- This equation is the central accounting tool for energy in thermal systems. Any change in the system's internal energy must be fully accounted for by energy that crossed the boundary as either heat or work.

Outputs & Effects

If $Δ U$ is positive, the internal energy of the system has increased, which for an ideal gas means its temperature has increased. This can happen if heat is added ( $Q > 0$ ), work is done on the system ( $W > 0$ ), or both.
If $Δ U$ is negative, the internal energy and temperature have decreased. This occurs if the system loses heat ( $Q < 0$ ), does work on its surroundings ( $W < 0$ ), or both.
If $Δ U$ is zero, the internal energy and temperature are constant. This implies that any heat added to the system is balanced by the work the system does on its surroundings (i.e., $Q = - W$ ).

Regulation & Limits

The equations presented here are most accurate for ideal gases and for processes that happen slowly enough for the system to remain in thermal equilibrium (quasi-static processes). When analyzing processes graphically using Pressure-Volume (PV) diagrams, the work done on the system is the negative of the area under the curve. For a process that moves from a larger volume to a smaller volume (leftward on the graph), the area is considered positive work. For an expansion (rightward on the graph), the area represents negative work.

Key Models & Diagrams

Thermodynamic processes describe how a system moves from one state to another. We can represent these processes on a PV diagram and analyze them with the First Law.

Process Type	PV Diagram Feature	Work Done ( $W = - P Δ V$ )	First Law ( $Δ U = Q + W$ )
Isobaric (Constant Pressure)	Horizontal Line	$W = - P Δ V$ (Non-zero if $Δ V \neq = 0$ )	$Δ U = Q - P Δ V$
Isochoric (Constant Volume)	Vertical Line	$W = 0$ (since $Δ V = 0$ )	$Δ U = Q$
Isothermal (Constant Temperature)	Hyperbolic Curve ( $P \propto 1/ V$ )	$W \neq = 0$ (Area under curve)	$0 = Q + W$ , so $Q = - W$
Adiabatic (No Heat Transfer)	Steep Hyperbolic Curve	$W \neq = 0$ (Area under curve)	$Δ U = W$

Key Components & Evidence

Internal Energy (U): The total microscopic energy of a system's particles (kinetic + potential). For an ideal gas, it's just the total kinetic energy and depends only on temperature. Unit: Joules (J).
Heat (Q): Energy transferred due to a temperature difference. It is a process, not a property of a system. Unit: Joules (J).
Work (W): Energy transferred when a force causes displacement, such as a change in the system's volume. Unit: Joules (J).
Pressure (P): The force per unit area exerted by the gas on the container walls. Unit: Pascals (Pa) or N/m².
Volume (V): The space occupied by the system. Unit: cubic meters (m³).
Temperature (T): A measure of the average kinetic energy of the particles in the system. Must be in Kelvin (K) for thermodynamic equations.
First Law of Thermodynamics: The fundamental law of energy conservation for thermal systems: $Δ U = Q + W$ .
PV Diagram: A graph of pressure versus volume for a thermodynamic system. The path on the diagram represents a process, and the area under the path represents the magnitude of the work done.
Ideal Monatomic Gas: A simplified model of a gas where particles are point masses that only have translational kinetic energy and do not interact.
Conservation of Energy: The core principle stating that energy cannot be created or destroyed, only transferred or transformed. The First Law is an application of this principle.

Skill Snapshots

Causation:
1. Compressing a gas ( $W > 0$ ) without allowing any heat to escape ( $Q = 0$ ) causes its internal energy and temperature to increase ( $Δ U > 0$ ).
2. Adding heat to a gas ( $Q > 0$ ) while holding its volume constant ( $W = 0$ ) causes its internal energy and temperature to increase ( $Δ U > 0$ ).
3. Allowing a gas to expand and do work on its surroundings ( $W < 0$ ) while keeping its temperature constant ( $Δ U = 0$ ) requires that heat be added to the system ( $Q > 0$ ).
Comparison:
1. In an isochoric process, no work is done because the volume is constant, whereas in an isobaric process, work is directly proportional to the change in volume.
2. An isothermal expansion and an adiabatic expansion both result in the gas doing work, but the temperature drops during the adiabatic process while it remains constant during the isothermal one.
3. Positive work ( $W > 0$ ) and positive heat ( $Q > 0$ ) are different processes, but both serve to increase the system's internal energy.
Change Over Time (CCOT):
- Baseline: A system of gas is in an initial state defined by pressure $P_{1}$ , volume $V_{1}$ , and temperature $T_{1}$ .
- Change 1: The gas is heated at constant pressure, causing it to expand to a new volume $V_{2} > V_{1}$ . The gas does work on the surroundings ( $W < 0$ ), but the heat added ( $Q > 0$ ) is greater, resulting in an increase in internal energy ( $Δ U > 0$ ) and a final temperature $T_{2} > T_{1}$ .
- Change 2: From this new state, the gas is cooled at a constant volume back to its original temperature $T_{1}$ . No work is done ( $W = 0$ ), and since heat is removed ( $Q < 0$ ), the internal energy decreases ( $Δ U < 0$ ).
- Continuity: Throughout any process or combination of processes, the total energy of the system plus its surroundings remains constant.

Common Misconceptions & Clarifications

Misconception: Heat and temperature are the same thing.
- Clarification: Temperature is a property of a system—a measure of the average kinetic energy of its particles. Heat is the transfer of energy due to a temperature difference. A system has temperature and internal energy; it does not "have" heat.
Misconception: If the temperature of a system doesn't change, no heat has been added.
- Clarification: In an isothermal (constant temperature) process, $Δ U = 0$ . According to the First Law, this means $Q = - W$ . If the system expands and does work ( $W < 0$ ), an equal amount of heat must be added ( $Q > 0$ ) to keep the temperature from dropping.
Misconception: The sign convention for work is always the same.
- Clarification: In physics, the convention $Δ U = Q + W$ defines $W$ as work done on the system. This means compression is positive work. In some chemistry contexts, the law is written as $Δ U = Q - W$ , where $W$ is work done by the system (expansion is positive work). It is critical to know which convention is being used.
Misconception: A positive $Q$ always means the temperature goes up.
- Clarification: Adding heat ( $Q > 0$ ) will increase the internal energy unless the system does an equal or greater amount of work on its surroundings. For example, during the phase change of boiling water, heat is continuously added, but the temperature remains constant at 100°C because the energy is used to do work against the atmosphere as steam expands.

One-Paragraph Summary

The first law of thermodynamics is a powerful statement of energy conservation tailored for thermal systems. It dictates that a system's internal energy ( $U$ ), which for an ideal gas is a direct measure of its temperature, can only be changed by two mechanisms: transferring energy as heat ( $Q$ ) or as work ( $W$ ). The governing equation, $Δ U = Q + W$ , provides a complete accounting of these energy transfers, where $Q$ is positive for heating and $W$ is positive for work done on the system (compression). We can visualize these processes on pressure-volume (PV) diagrams, where paths represent state changes and the area under the path quantifies the work done. By applying this law to specific processes like isobaric, isochoric, isothermal, and adiabatic changes, we can predict how a system's state variables will evolve.

The First Law of Thermodynamics - AP Physics 2: Algebra-Based Study Guide