The First Law of Thermodynamics
The first law tracks how a thermodynamic system’s internal energy changes when energy enters or leaves by heating/cooling and when work is done on or by the system; P–V diagrams reveal the work associated with volume change.
The First Law of Thermodynamics: quick answer
The first law tracks how a thermodynamic system’s internal energy changes when energy enters or leaves by heating/cooling and when work is done on or by the system; P–V diagrams reveal the work associated with volume change. This is the energy-accounting center of Unit 9. Internal energy is a state function; work and heating are path-dependent transfers. The same initial and final states can therefore have the same ΔU but different work and heating depending on the P–V path.
State the work sign convention before using the first law; distinguish work done on the gas from work done by the gas, and apply U = (3/2)nRT only to the ideal monatomic-gas model.
What you should be able to explain and do
Use these as retrieval prompts before looking at notes.
Calculate internal energy for an ideal monatomic gas.
Apply ΔU = Q + W_on with a stated sign convention.
Determine work sign and magnitude from volume change and P–V area.
Compare isothermal, isobaric, isovolumetric and adiabatic processes.
Key terms you need for this lesson
Definitions stay next to the physics they support so unfamiliar language does not break the reasoning chain.
internal energy
internal energy is the sum of microscopic kinetic energy and microscopic potential energy inside the chosen system.
work done on a gas
work done on a gas is energy transferred mechanically into a gas by an external force as the system boundary moves.
state function
state function is a property whose change depends only on the initial and final states, not on the path between them.
P–V diagram
P–V diagram is a graph of pressure versus volume used to represent thermodynamic states and processes.
isothermal process
isothermal process is a process at constant temperature.
adiabatic process
adiabatic process is a process with no energy transferred into or out of the system by thermal processes.
Know what each relationship means and when it applies
The relationship, variables, condition and physical interpretation belong together.
| Relationship / model | Variables or meaning | When to use it | Physical anchor |
|---|---|---|---|
U = (3/2)nRT = (3/2)Nk_B T | U internal energy of an ideal monatomic gas (J); n amount (mol); R gas constant; N particle count; k_B Boltzmann constant; T absolute temperature (K) | ideal monatomic gas whose internal energy is translational kinetic energy | For a fixed amount of ideal monatomic gas, doubling T doubles U. |
ΔU = Q + W_on | ΔU internal-energy change (J); Q energy transferred into the system by heating (J, negative for cooling); W_on work done on the system (J) | closed-system energy bookkeeping with this sign convention stated explicitly | A compressed gas can gain internal energy even if no heating occurs because W_on is positive. |
W_on = −P_ext ΔV; |W| = area under a P–V path | W_on work done on gas (J); P_ext constant or average external pressure (Pa); ΔV = V_f−V_i (m³); P–V area has units Pa·m³ = J | constant/average external pressure for W_on = −P_extΔV; area reasoning for quasistatic P–V paths | Expansion has ΔV>0 so W_on<0; compression has ΔV<0 so W_on>0. |
What a strong diagram or graph must show
Translate the same physical model across words, diagrams, graphs and equations.
P–V diagram
area under the path gives the magnitude of boundary work; expansion/compression sets sign.
Energy-flow diagram
Q and W cross the system boundary; U is an energy state of the system.
State-vs-path distinction
ΔU depends only on endpoints, while Q and W depend on the process path.
Process labels
isovolumetric ΔV=0; isothermal ΔT=0; isobaric ΔP=0; adiabatic Q=0.
Change one variable and watch the model respond
Use the tool to make a prediction first, then move the control and explain why the output changes.
First-law sign and P–V work calculator
This tool uses ΔU=Q+W_on and W_on=−PΔV for a constant-pressure process.
Two examples with the model choice shown
The goal is to make the reasoning pathway visible, not just the final number.
Constant-pressure expansion
Situation: A gas expands at 150 kPa from 0.020 m³ to 0.032 m³ while 2.5 kJ of energy enters by heating. Find W_on and ΔU.
Reasoning: ΔV=0.012 m³. W_on=−PΔV=−150000×0.012=−1800 J. Then ΔU=Q+W_on=2500−1800.
W_on = −1.8 kJ and ΔU = +0.70 kJ.
Rigid-container heating
Situation: An ideal monatomic gas in a rigid container gains 900 J by heating.
Reasoning: Rigid means ΔV=0, so boundary work is zero. The first law becomes ΔU=Q.
ΔU = +900 J; the temperature rises because U for an ideal monatomic gas is proportional to T.
Recognize the physics before reaching for an equation
Use the cue to choose the model, connect it to nearby ideas, then reject any method whose assumptions do not match the situation.
What should trigger Topic 9.4?
Use the first law whenever the system boundary includes heating, compression/expansion work, or a P–V process.
How this idea fits the course
Topic 9.2 supplies state variables; Topic 9.3 supplies thermal-transfer direction; Topic 9.6 adds the directionality constraint of the second law.
Scope check before solving
State the work sign convention before using the first law; distinguish work done on the gas from work done by the gas, and apply U = (3/2)nRT only to the ideal monatomic-gas model.
Turn this topic into measurable evidence
A strong investigation varies the intended cause, controls competing causes, repeats measurements and uses a graph whose features have physical meaning.
Investigation target
Compare several P–V paths between chosen states, determine signed work from graph area, and use the first law to infer internal-energy or heating changes.
Reasoning errors to eliminate
If one of these appears in your work, repair the model before doing more arithmetic.
Six things you should be able to do before moving on
Use these as closed-note retrieval prompts.
- I can define the core quantities and distinguish closely related terms.
- I can choose the governing model before substituting values.
- I can explain each variable, unit and sign convention.
- I can translate the situation into an appropriate graph or diagram.
- I can predict a factor/direction change before calculating.
- I can design or critique an experiment and use evidence to justify a claim.
Practice is kept on its dedicated pages
Finish the lesson first. Then open the separate topic bank when you want exam-style practice without replacing the teaching content above.
42 multiple-choice questions
Use the exact-topic MCQ bank to test model selection, representations, calculations and boundaries after learning the concept.
Open 42 Topic MCQs4 multipart free-response tasks
Use the exact-topic FRQ set for mathematical, representation, experimental and qualitative/quantitative reasoning.
Open 4 Topic FRQsTopic 9.4 FAQs
What sign convention is used here?
ΔU = Q + W_on. Heating into the gas is positive Q; work done on the gas is positive W_on. Expansion therefore gives negative W_on.
What does P–V area represent?
The area under a P-versus-V path has units Pa·m³ = J and gives the magnitude of boundary work for the process; the sign depends on whether you use work on or work by the gas.
What happens in an isovolumetric process?
Volume is constant, so boundary work from PΔV is zero.
What happens in an isothermal ideal-monatomic-gas process?
Temperature is constant, so internal energy is constant. The first law then requires heating and work transfers to balance.
What happens in an adiabatic process?
No energy is transferred by thermal processes, so Q=0; internal-energy change comes from work transfer.
Does the ideal-gas internal-energy formula apply to every gas?
Use U=(3/2)nRT for the ideal monatomic-gas model described in this course, not indiscriminately for any molecular gas.