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Calculators/First Law of Thermodynamics
Thermodynamics

First Law of Thermodynamics Calculator

Solve ΔU = Q − W for internal energy change, heat, or work. Supports joules, calories, and BTU.

Solve for

ΔU = Q − W
Solving for ΔU
Quick examples
Energy flow
SystemΔU = 300 JQ = 500 JW = 200 JRed = heat flow · Blue = work done by system
Internal Energy Change
300 J
ΔU
300 J
Q
500 J
W
200 J

First law equations

Internal energy change
ΔU = Q − W
Heat added
Q = ΔU + W
Work by system
W = Q − ΔU

The First Law of Thermodynamics Explained

The first law of thermodynamics is conservation of energy applied to heat and work: the change in a system's internal energy equals the heat it absorbs minus the work it does on its surroundings. Energy isn't created or destroyed — it just moves between forms.

Sign convention matters here more than in most equations: heat Q is positive flowing into the system, and work W is positive when the system does work on its surroundings (like a gas pushing a piston outward). Get a sign backwards and the whole answer flips.

What is the first law of thermodynamics?+
The first law states that the change in a system’s internal energy equals the heat added to it minus the work done by it: ΔU = Q − W. It’s a statement of conservation of energy applied to thermal systems — energy cannot be created or destroyed, only transferred as heat or work.
What is the sign convention used here?+
This calculator uses the standard physics convention: Q is positive when heat flows INTO the system, and W is positive when the system does work ON its surroundings (like gas expanding against a piston). Compression, where work is done ON the gas, is entered as a negative W.
What does it mean when ΔU is negative?+
A negative ΔU means the system’s internal energy decreased — it either released more heat than it gained, did more work than was done on it, or both. Internal energy is linked to temperature, so a drop usually (for an ideal gas) means the system cooled.
What is an adiabatic process?+
An adiabatic process is one with no heat exchange with the surroundings (Q = 0). The first law then simplifies to ΔU = −W: any work the system does comes entirely from its own internal energy, and vice versa. Rapid gas compressions and expansions are often approximated as adiabatic.
How does this relate to the ideal gas law?+
The first law is independent of what the substance is, but for an ideal gas, internal energy depends only on temperature, and work done during expansion/compression can be calculated from pressure and volume change (W = PΔV for constant pressure). See the Ideal Gas Law calculator for the PV = nRT side of the problem.
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First Law of Thermodynamics — The Complete Guide

Steam engines, refrigerators, car engines, and your own metabolism all run on the same rule: energy in minus energy out equals the change stored inside. That's the first law of thermodynamics, and it's really just the conservation of energy dressed up for thermal systems.

Formally: ΔU = Q − W, where ΔU is the change in internal energy, Q is heat added to the system, and W is work done by the system on its surroundings. No exceptions have ever been found — it's one of the most thoroughly tested laws in all of physics.

Worked Example — Gas Expansion

Problem: A gas absorbs 500 J of heat and does 200 J of work pushing a piston outward. Find the change in internal energy.

ΔU = Q − W = 500 − 200 = 300 J — the gas's internal energy rose by 300 J; not all the absorbed heat went into expansion work.

Worked Example — Adiabatic Compression

Problem: A gas is compressed adiabatically (Q = 0), and 300 J of work is done on it. Find ΔU.

Work done ON the system means W (work done BY the system) = −300 J

ΔU = Q − W = 0 − (−300) = +300 J — all the compression work went straight into internal energy, which is why pumping up a bike tyre makes the pump warm.

Common Special Cases

Isochoric (constant volume): No work is done (W = 0) since there's no volume change to push against, so ΔU = Q directly — all added heat goes into internal energy.

Isothermal (constant temperature, ideal gas): Internal energy depends only on temperature for an ideal gas, so ΔU = 0, which means Q = W — all heat added is converted directly to work.

Adiabatic (no heat exchange): Q = 0, so ΔU = −W — the system's internal energy changes only because of work done on or by it.

Physics Fundamentals

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