Change in U Formula: Internal Energy Physics

This calculator works out the change in internal energy (ΔU) of a system using the first law of thermodynamics. You enter the heat added to the system (Q) and the work done by the system (W), and you get the change in internal energy in joules. The calculator also shows the work done on the system, which is the negative of the work done by the system, for clarity.

Formula

The change in internal energy of a system is given by the first law of thermodynamics:

ΔU = Q − W

Where:

• ΔU = change in internal energy (joules, J)
• Q = net heat added to the system (joules, J). Positive when heat flows into the system, negative when heat leaves.
• W = work done by the system (joules, J). Positive when the system expands and does work on its surroundings, negative when the surroundings compress the system.

This sign convention is the most common in chemistry and physics textbooks. Some engineering texts use the convention ΔU = Q + W, where W is work done on the system. Both are correct as long as you stay consistent. This calculator uses the first convention (work done by the system).

Worked Example

Using the default values from the calculator: Q = 1500 J of heat is added to a gas, and the gas does W = 400 J of work by expanding against a piston.

1. Start with the formula: ΔU = Q − W
2. Substitute the values: ΔU = 1500 J − 400 J
3. Subtract: ΔU = 1100 J

The internal energy increases by 1100 J. The calculator also reports work done on the system as −400 J, since work done by the system is the negative of work done on it.

When to Use the First Law

The first law of thermodynamics is a statement of energy conservation. It applies to any thermodynamic process, whether the system is a gas in a cylinder, a chemical reaction in a beaker, or a steam turbine. You use it whenever you know two of the three quantities (heat, work, and internal energy change) and need the third.

Common situations include:

• Calculating the energy change of a gas during compression or expansion.
• Determining how much heat a chemical reaction absorbs or releases at constant volume (where W = 0).
• Analyzing engines and refrigerators where heat and work are measured at the boundaries.

Remember that internal energy is a state function: its change depends only on the initial and final states, not on the path taken. Heat and work, however, are path-dependent. The first law always links them, but you cannot calculate ΔU from Q and W alone unless you know the actual process.

Reading the Result

A positive ΔU means the system’s internal energy increased. This typically corresponds to a temperature rise for an ideal gas, or a phase change (like melting) where energy is stored in molecular bonds. A negative ΔU means the system lost internal energy, usually cooling down or condensing.

For an ideal gas, internal energy depends only on temperature. So if ΔU is positive, the gas temperature rises; if negative, it falls. For real substances, internal energy also includes potential energy between molecules, so temperature may not change during a phase transition even though ΔU changes.

The units are joules in the SI system. In engineering, you may see kilojoules (kJ), calories (cal), or British thermal units (BTU). To convert: 1 kJ = 1000 J, 1 cal = 4.184 J, 1 BTU ≈ 1055 J.

Common Mistakes and Misconceptions

The most frequent error is mixing up the sign convention for work. If you use work done on the system instead of work done by the system, you will get the wrong sign for ΔU. Always check which convention your textbook or problem uses.

Another common mistake is forgetting that Q is net heat. If heat flows both in and out, you must sum the heat transfers with their signs. Similarly, if multiple work interactions occur, sum them.

A widespread misconception is that ΔU always equals Q for a constant-volume process. That is true only when W = 0, which happens exactly at constant volume. At constant pressure, some heat is converted to work of expansion, so ΔU ≠ Q.

Rounding errors can also creep in when you convert units. Always carry extra digits during intermediate steps and round only at the end.

Limits of the Formula

The first law is universally valid, but the formula ΔU = Q − W assumes you correctly identify all heat and work interactions. In real systems, there may be other energy transfers like electrical work, magnetic work, or mass flow across boundaries. For a closed system (no mass transfer), the formula is complete. For an open system (like a pipe), you need the steady-flow energy equation, which includes enthalpy and kinetic/potential energy terms.

Also, the first law does not tell you how the process occurs, only that energy is conserved. It cannot predict whether a process will actually happen; that requires the second law of thermodynamics. For example, the first law allows heat to flow from cold to hot, but the second law forbids it.

Finally, the formula assumes a closed system. If mass enters or leaves, you must account for the energy carried by that mass. For most introductory problems, though, the closed-system assumption is valid.

Frequently Asked Questions

What is the difference between ΔU and ΔH?

ΔU is the change in internal energy, while ΔH (enthalpy change) is defined as ΔH = ΔU + PΔV (at constant pressure). Enthalpy includes the work done to make room for the system against external pressure. For reactions at constant pressure (common in chemistry), ΔH is easier to measure because it equals the heat exchanged. At constant volume, ΔH and ΔU differ by the PΔV term.

Why is work done by the system positive in this formula?

This is a convention. In physics and chemistry, it is common to define work from the system’s perspective: when the system expands, it pushes the surroundings, so it does positive work. This makes the first law read ΔU = Q − W. In engineering, the opposite convention (work done on the system positive) is often used, giving ΔU = Q + W. Both are valid; just be consistent.

Can ΔU be negative? What does that mean?

Yes. A negative ΔU means the system loses internal energy. This happens when the system does more work than the heat it absorbs, or when it releases more heat than the work done on it. For example, an expanding gas that does work without any heat input will cool down, so its internal energy decreases.

What if the system is not a gas?

The first law applies to any system—solid, liquid, gas, or plasma. For solids and liquids, work done by expansion is usually tiny and often neglected, so ΔU ≈ Q. But if the system undergoes a chemical reaction or phase change, internal energy changes even without heat exchange. The formula still holds.

How do I calculate Q or W if I only know ΔU?

You need additional information about the process. For example, if the process is adiabatic (Q = 0), then ΔU = −W. If it is isochoric (constant volume, W = 0), then ΔU = Q. For other processes, you need equations of state or measured heat/work values. The first law alone gives one equation with three unknowns, so you need two known quantities.

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in q | Heat added (Q) | J | 1500 | 0
in w | Work done by system (W) | J | 400 | 0
out du | Change in internal energy (ΔU) | J | q - w | 0
out won | Work done on system | J | -w | 0