Thermodynamics Formula Sheet: Heat, Work, Internal Energy, and Efficiency

Overview

This formula sheet is designed as a durable thermodynamics reference rather than a one-time lesson. The goal is simple: help you identify which equation fits a given situation and avoid common mistakes with signs, units, and conditions.

At a basic level, thermodynamics tracks how energy moves and changes form in physical systems. In introductory physics, the most common quantities are:

• Heat Q: energy transferred because of a temperature difference
• Work W: energy transferred by a force acting through a distance or, in gases, by expansion or compression
• Internal energy U: microscopic energy stored in the system
• Temperature T: a measure connected to average microscopic kinetic energy
• Efficiency e or η: a measure of useful output compared with energy input

The central relation is the first law of thermodynamics. In one common sign convention:

ΔU = Q − W

Here, Q is positive when heat enters the system, and W is positive when the system does work on the surroundings. Some courses use a different sign convention for work, so always check your class notes or exam board formula sheet before substituting values.

Useful SI units:

• Energy, heat, and work: joules (J)
• Temperature: kelvin (K), though temperature changes satisfy ΔT in K = ΔT in °C
• Pressure: pascals (Pa)
• Volume: cubic meters (m³)
• Mass: kilograms (kg)

If you are revising broadly, this page pairs well with general equation references such as the GCSE Physics Equations List and Rearrangement Guide, the A-Level Physics Equations List with Definitions and Unit Checks, and the AP Physics Formula Sheet Guide: What Every Equation Means.

Core concepts

This section groups the main thermodynamics formulas by use case, with short notes on when each one applies.

1. Heat transfer and temperature change

For heating or cooling a substance without a phase change:

Q = mcΔT

• Q = heat transferred
• m = mass
• c = specific heat capacity
• ΔT = change in temperature

Use this when a material warms up or cools down but stays in the same phase. If ice melts or water boils, this equation alone is not enough.

For a temperature change rate based on power:

P = Q/t

or rearranged:

Q = Pt

This is useful in heater problems, electrical heating, and energy transfer over time. If electrical power is involved, it may connect naturally to circuit equations such as those in Ohm's Law and Basic Circuit Problems: Step-by-Step Practice Set.

2. Phase changes

For melting, freezing, boiling, or condensing at constant temperature:

Q = mL

• L may be latent heat of fusion or latent heat of vaporization

Use this when the substance changes phase and the temperature stays constant during the transition. In many exam questions, a full energy calculation combines both forms:

• Heating in one phase: Q = mcΔT
• Phase change: Q = mL
• Heating in the new phase: Q = mcΔT

3. First law of thermodynamics

The most important energy accounting relation is:

ΔU = Q − W

This is the formula to reach for when a problem asks how internal energy changes after heat transfer and work. It is especially useful in gas processes and engine cycles.

Typical interpretations:

• If heat enters and no work is done, internal energy increases
• If the system does work while receiving little heat, internal energy may decrease
• If a process is adiabatic, Q = 0, so ΔU = −W

For many ideal-gas situations, internal energy depends only on temperature. That means if the temperature does not change, then ΔU = 0 for an ideal gas.

4. Work done by a gas

For a constant-pressure process:

W = pΔV

• p = pressure
• ΔV = change in volume

This is one of the most common thermodynamics formulas in introductory courses. It works directly only when pressure is constant. If pressure changes, work is the area under the pressure-volume graph, which is often introduced conceptually even if calculus is not required.

Sign idea under the convention above:

• Expansion: ΔV > 0, so W > 0
• Compression: ΔV < 0, so W < 0

5. Ideal gas relations

The most common gas equation is:

pV = nRT

• n = number of moles
• R = ideal gas constant

This equation links pressure, volume, and temperature for an ideal gas. It is not the first law, but it often appears in the same problems because it helps you find a missing state variable before calculating work or internal energy changes.

Another useful particle form is:

pV = NkT

• N = number of particles
• k = Boltzmann constant

Use the molar form in chemistry-style questions and the particle form in microscopic or statistical interpretations.

6. Special thermodynamic processes

Several named processes appear again and again in worked physics problems:

Isothermal process (constant temperature)

• For an ideal gas, ΔU = 0
• So from the first law, Q = W

Isochoric process (constant volume)

• ΔV = 0
• So W = pΔV = 0
• Then ΔU = Q

Isobaric process (constant pressure)

• W = pΔV
• Then use ΔU = Q − W

Adiabatic process (no heat transfer)

• Q = 0
• So ΔU = −W

These short identities can save time in exams because they reduce the amount of substitution needed.

7. Internal energy of an ideal gas

In introductory treatments, internal energy for an ideal gas depends on temperature. A general form is:

ΔU = nC_VΔT

• C_V = molar heat capacity at constant volume

For specific idealized cases you may also see fixed forms such as:

• Monatomic ideal gas: U = (3/2)nRT
• So ΔU = (3/2)nRΔT

Only use these special forms when your course has introduced them or the gas type is stated clearly.

8. Heat capacities

It helps to distinguish between:

• Specific heat capacity c: energy per unit mass per unit temperature change
• Heat capacity C: energy required for the whole object to change by one degree

The relation between them is:

C = mc

Then another useful form is:

Q = CΔT

This is often neater than writing mass and specific heat separately when the total heat capacity of a system is already known.

9. Efficiency formulas

For any machine or process:

Efficiency = useful output energy / input energy

or in power form:

Efficiency = useful output power / input power

As a percentage:

Efficiency (%) = (useful output / input) × 100

For a heat engine:

η = W_out / Q_in

And since input heat either becomes useful work or is expelled as waste heat:

η = 1 − Q_out / Q_in

This is one of the most useful efficiency formula physics relations because it connects engine performance directly to energy flow.

Related terms

This section clarifies words that are often mixed up in thermodynamics questions.

Heat vs temperature

Heat is energy in transit due to a temperature difference. Temperature describes thermal state. A large object and a small object can be at the same temperature but contain very different amounts of internal energy.

Internal energy vs thermal energy

Internal energy is the total microscopic energy inside a system, including kinetic and potential contributions of particles. In many school-level contexts, thermal energy is used more loosely to describe energy associated with temperature. When precision matters, use the term your course defines.

State variable vs path-dependent quantity

Internal energy, pressure, volume, and temperature are state variables: they depend only on the current state. Heat and work depend on the process that got the system there. This explains why two paths between the same initial and final states can give different values of Q and W but the same ΔU.

Closed system and surroundings

A system is the part you analyze. The surroundings are everything else. Sign errors often happen when students forget whether work is done by the system or on the system.

Kelvin and Celsius

For gas-law formulas like pV = nRT, use temperature in kelvin. For temperature changes in Q = mcΔT, a change of 1 K is numerically the same as a change of 1 °C.

Power and rate of energy transfer

If the question includes time, it may be a power problem in disguise. Then use:

P = E/t or P = Q/t

This is common in heaters, refrigerators, and engines.

For lab-based work, it also helps to handle units and uncertainty carefully. See Physics Lab Report Checklist: Sections, Graphs, Uncertainty, and Common Mistakes and Significant Figures Rules in Physics: How to Round, Multiply, and Report Results.

Practical use cases

Here is how to choose the right thermodynamics formula in common situations.

Use case 1: Heating a substance with no phase change

If a metal block, water sample, or gas changes temperature without melting or boiling, start with:

Q = mcΔT

Checklist:

• Mass in kg if using SI units
• Specific heat capacity in J kg⁻¹ K⁻¹
• Temperature difference, not final temperature alone

Use case 2: Melting or boiling

If the temperature stays constant during a phase change, use:

Q = mL

Then add any heating or cooling before or after the transition using Q = mcΔT.

Use case 3: Gas expands at constant pressure

Find work with:

W = pΔV

Then combine with the first law:

ΔU = Q − W

This is a standard pattern in piston problems.

Use case 4: Engine efficiency

If the question gives heat input and useful work output, use:

η = W_out / Q_in

If it gives waste heat instead of work directly, use:

η = 1 − Q_out / Q_in

Always convert percentage answers only at the final step unless the question requests a decimal.

Use case 5: Ideal gas state change

If pressure, volume, moles, and temperature are related, begin with:

pV = nRT

Then decide whether the process is isothermal, isochoric, isobaric, or adiabatic. That classification often tells you instantly whether Q, W, or ΔU simplifies.

Use case 6: Exam problem solving sequence

For many thermodynamics questions, this sequence works well:

1. Define the system clearly
2. List known quantities with units
3. Identify the process type, if any
4. Choose the equation that matches the process
5. Check sign convention before substitution
6. Convert temperatures to kelvin if using gas laws
7. Review whether the answer is physically sensible

For broader revision planning, students often find it useful to cross-check thermodynamics against larger syllabuses such as the IB Physics Revision Guide: Topic-by-Topic Formula and Concept Checklist.

When to revisit

Come back to this formula sheet whenever the inputs or assumptions in your problem change. Thermodynamics is less about memorizing a long list and more about selecting the right relation for the right condition.

Revisit this page when:

• You switch between calorimetry, gas laws, and heat engine questions
• You are unsure whether to use Q = mcΔT or Q = mL
• You need to check the first law sign convention
• You are moving from school-level physics to introductory engineering thermodynamics
• You are writing lab reports and need unit consistency or careful reporting
• You want a quick pre-exam scan of the most reused thermodynamics formulas

A practical way to use this as a living reference is to build your own short summary under four headings: heat, work, internal energy, and efficiency. Under each heading, keep one main equation, one unit reminder, and one warning about when the formula does not apply. That single page becomes much easier to review than a chapter of notes.

Finally, if you are revising across topics, thermodynamics makes more sense when connected to the wider physics picture: energy transfer, graphs, measurement, and equation fluency. Reference pages like this work best when you revisit them actively, not passively. Before closing the page, pick one equation and ask: what physical situation would make me use this, what assumptions does it require, and what would the sign of my answer mean?