Thermodynamics Formulas Sheet: Laws, Processes, and Units

Overview

This article gives you a working thermodynamics reference, not just a memorization list. The goal is to make the equations usable under exam pressure.

Thermodynamics problems usually become manageable when you sort them into a small number of categories:

• State relationships, such as the ideal gas law
• Energy accounting, especially the first law of thermodynamics
• Process equations, such as isothermal, isobaric, isochoric, and adiabatic changes
• Heat and temperature relations, including specific heat and latent heat
• Entropy and efficiency formulas for more advanced courses

Before the workflow, here is a compact formula sheet organized by topic.

Core state and gas formulas

• Ideal gas law: PV = nRT
• Combined gas law for fixed amount of gas: P₁V₁/T₁ = P₂V₂/T₂
• Density form of ideal gas law: P = ρR_specificT
• Moles from mass: n = m/M

Common symbols: P pressure, V volume, n amount of substance, R universal gas constant, T absolute temperature in kelvin, m mass, M molar mass.

First law and internal energy

• First law of thermodynamics: ΔU = Q − W if W is work done by the system
• Alternative sign convention: ΔU = Q + W if W is work done on the system
• Work for constant pressure: W = PΔV
• General boundary work: W = ∫P dV
• Internal energy change for ideal gas: ΔU = nC_VΔT

Always check which work sign convention your course uses. Many mistakes in thermodynamics come from correct algebra applied to the wrong sign rule.

Heat capacity and calorimetry

• Heat transfer: Q = mcΔT
• Molar form: Q = nCΔT
• Latent heat: Q = mL

Use these when a problem focuses on heating, cooling, or phase change rather than gas expansion alone.

Enthalpy formulas

• Definition of enthalpy: H = U + PV
• Change in enthalpy: ΔH = ΔU + Δ(PV)
• For an ideal gas: ΔH = nC_PΔT
• At constant pressure: Q_P = ΔH

Relationships between heat capacities

• Mayer's relation for an ideal gas: C_P − C_V = R
• Heat capacity ratio: γ = C_P/C_V

Common ideal gas process equations

• Isothermal process (T constant): PV = constant, ΔU = 0, and for a reversible ideal gas process W = nRT ln(V₂/V₁)
• Isochoric process (V constant): P/T = constant, W = 0, so Q = ΔU = nC_VΔT
• Isobaric process (P constant): V/T = constant, W = PΔV, and Q = nC_PΔT
• Adiabatic process (Q = 0): PV^γ = constant, TV^γ−1 = constant, and TP^(1−γ)/γ = constant

Entropy and efficiency formulas

• Entropy change definition for a reversible process: ΔS = ∫(δQ_rev/T)
• At constant temperature: ΔS = Q_rev/T
• Carnot efficiency: η = 1 − T_cold/T_hot
• Thermal efficiency: η = W_out/Q_in
• Coefficient of performance for a refrigerator: COP = Q_cold/W

Units you should keep consistent

• Temperature: kelvin, not degrees Celsius, in gas-law and efficiency equations
• Pressure: pascal, where 1 Pa = 1 N/m²
• Volume: cubic meters in SI work and gas-law calculations
• Energy, work, heat: joule
• Amount: mole
• Specific heat capacity: J kg⁻¹ K⁻¹
• Molar heat capacity: J mol⁻¹ K⁻¹

One practical habit: convert units before you choose the final formula. A correct equation with mixed units often produces an answer that looks reasonable but is wrong by a large factor.

Step-by-step workflow

Use this sequence whenever you face a thermodynamics problem. It turns a long list of formulas into a decision process.

1) Identify the system and the process

Ask what the system is: a gas in a piston, water being heated, a thermal engine, or an insulated container. Then identify the process type if the problem gives one.

• Isothermal: temperature constant
• Isochoric: volume constant
• Isobaric: pressure constant
• Adiabatic: no heat transfer
• Cyclic: final state equals initial state, so ΔU = 0 over the full cycle

If the process is not named directly, look for clues in the wording. “Rigid container” usually means constant volume. “Well insulated” often means adiabatic. “Slow expansion at constant temperature” points to an isothermal process.

2) List known and unknown quantities

Write down all given values with units: P, V, T, m, n, Q, W, C_P, C_V, and so on. Then mark what the question asks for.

This small step prevents formula hunting. If you know pressure, volume, and temperature, the ideal gas law is a likely tool. If you know mass, specific heat, and temperature change, use Q = mcΔT.

3) Convert to SI units early

This is especially important in thermodynamics because pressure-volume work naturally gives joules only when pressure is in pascals and volume is in cubic meters.

• 1 kPa = 1000 Pa
• 1 L = 10⁻³ m³
• T(K) = T(°C) + 273.15

Even if your course allows some non-SI shortcuts, SI consistency is the safest exam habit.

4) Choose the governing law before choosing a process shortcut

Start with the broad law, then simplify.

• For energy changes, begin with the first law
• For pressure-volume-temperature links, begin with the ideal gas law
• For heating or cooling materials, begin with heat capacity relations
• For engines and entropy, begin with second-law ideas

For example, in an adiabatic ideal gas expansion, first write ΔU = Q − W. Then apply Q = 0. Then, if needed, use ΔU = nC_VΔT to connect the energy change to temperature.

5) Apply the process condition

This is where many problems become short.

• Isothermal ideal gas: ΔU = 0
• Isochoric: W = 0
• Isobaric: W = PΔV
• Adiabatic: Q = 0

These conditions reduce the number of variables and often reveal the needed equation immediately.

6) Solve symbolically first if possible

Before substituting numbers, rearrange the equation algebraically. This helps you see whether the result should be positive or negative and reduces calculator errors.

For instance, in a constant-pressure expansion, if volume increases then ΔV > 0, so W = PΔV is positive when work is done by the gas. That sign should match your physical picture.

7) Check physical meaning at the end

After the arithmetic, ask whether the answer makes sense.

• Did an expanding gas do positive work under your sign convention?
• Did an insulated process keep Q = 0?
• Did an ideal gas law calculation use kelvin?
• Is the final pressure or temperature in a realistic range for the problem?

That final ten-second check catches a surprising number of mistakes.

A quick example workflow

Suppose a problem says an ideal gas in a cylinder expands at constant pressure. You are given n, ΔT, and C_P, and asked for heat added and work done.

1. Process type: isobaric
2. Needed heat formula: Q = nC_PΔT
3. Work relation: W = PΔV
4. Use ideal gas law at constant pressure to get PΔV = nRΔT
5. So W = nRΔT
6. Check with first law: ΔU = Q − W = nC_VΔT, consistent with C_P − C_V = R

This is the kind of connection that makes a formula sheet valuable: not isolated equations, but links between them.

Tools and handoffs

To keep this thermodynamics equation sheet useful over time, treat it as a working reference with a few supporting tools.

Build your sheet in layers

A strong study sheet often has three levels:

1. Core page: the equations you should know from memory or near-memory
2. Process page: special-case formulas for isothermal, isobaric, isochoric, and adiabatic changes
3. Check page: units, constants, sign conventions, and common mistakes

This structure is more useful than one crowded page of symbols.

Use handoffs between concepts

Thermodynamics often overlaps with broader physics topics. A few handoffs help:

• If you need a wider equation reference, see Physics Formulas List by Topic: Equations, Units, and When to Use Them.
• If you are comparing thermal systems with electrical analogies such as energy transfer and power, Ohm's Law Problems and Circuit Basics: Solved Questions for Beginners can help build intuition.
• If your course pairs thermodynamics with rotational mechanics in engineering contexts, Torque and Rotational Motion Formulas, Concepts, and Worked Problems is a useful adjacent reference.

These links matter because formula sheets work best when they sit inside a larger network of problem-solving tools.

Keep constants and assumptions visible

On the same page as your formulas, include:

• The value and units of R used in your course
• Your class sign convention for work
• A note that ideal gas equations use absolute temperature
• A reminder that many process equations assume an ideal gas and, in some cases, a reversible path

That last point is easy to miss. Some formulas, especially entropy and adiabatic relations, come with assumptions. If the assumptions are not met, the formula may not apply directly.

Use worked examples as handoff points

After every formula group, attach one solved problem type:

• One ideal gas law state-change problem
• One first-law energy accounting problem
• One constant-pressure or constant-volume heating problem
• One adiabatic compression or expansion problem
• One engine efficiency or entropy-change problem

This turns a passive formula sheet into an active revision tool. It also supports the kind of repeat practice that makes physics explained simply and clearly.

Quality checks

A formula sheet is only as good as its error control. Use these checks every time you revise or use it.

Check 1: Unit consistency

Make sure every equation is paired with the units it expects. For example, PV has units of energy in SI only when pressure is in pascals and volume in cubic meters.

Check 2: Sign convention clarity

Write your sign convention at the top of the page in plain language, such as:

Use ΔU = Q − W, where W is work done by the system.

Do not assume you will remember this during a timed test.

Check 3: Process assumptions

Label formulas that require special conditions:

• Ideal gas only
• Reversible process
• Constant pressure
• Constant volume
• No heat transfer

This prevents common overuse of formulas outside their intended setting.

Check 4: State versus path distinction

Remember that U, H, S, P, V, and T are state-related quantities, while heat and work depend on the path taken. Even if your course does not emphasize this language strongly, it helps organize your reasoning.

Check 5: Reasonableness of answer

Use simple sanity checks:

• If temperature rises for an ideal gas, internal energy should usually rise
• If volume is fixed, boundary work should be zero
• If the process is a full cycle, net internal energy change should be zero
• An efficiency should usually fall between 0 and 1 when written as a fraction

These checks make your thermodynamics formulas far more reliable in practice.

Common mistakes to flag on the sheet

• Using Celsius instead of kelvin in gas laws
• Mixing up C_P and C_V
• Forgetting whether W means work done by or on the system
• Using Q = mcΔT during a phase change, where Q = mL may be needed instead
• Applying adiabatic formulas without the needed assumptions

When to revisit

The most useful equation sheet is a living document. Revisit and update it whenever your course level changes or your problem types become more advanced.

Update it when your class moves topics

Add or revise formulas when you move from introductory heat and gas laws into:

• Entropy and the second law
• Heat engines and refrigerators
• Real gas corrections
• Enthalpy in chemistry-linked thermodynamics
• Engineering topics such as control volumes and steady-flow energy equations

At each stage, keep the same structure: concept, process, units, assumptions, and one worked example.

Refresh it after each exam set or homework block

If you repeatedly make the same mistake, your sheet should change. Add a warning note, a sign reminder, or a small example beside the formula that caused trouble. This is one of the simplest ways to improve exam revision notes over time.

Use a practical maintenance routine

1. Highlight the 10 to 15 formulas you use most often
2. Circle formulas that depend on assumptions you tend to forget
3. Add one line for units beside every major equation
4. Attach one short solved example per process type
5. Review the sheet before problem sets, not only before exams

If you want a broader cross-topic reference, pair this page with Physics Formulas List by Topic: Equations, Units, and When to Use Them. Then keep this article as your dedicated thermodynamics formulas and ideal gas process equations guide.

As a final action step, copy the sections that match your course into your own revision sheet today: core gas laws, first law, heat capacities, process equations, entropy and efficiency, then add your class sign convention at the top. That small setup will save time every time you return to thermodynamics.