Thermodynamics Sign Convention: q, w, ΔU and ΔH Explained for NEET and JEE

Two students solve the same gas-expansion problem. One gets +500 J, the other −500 J, and both are sure their arithmetic is perfect. The difference is a single sign convention, and in NEET and JEE Chemistry thermodynamics that one sign decides whether you score +4 or lose a mark.

Quick Summary

• The IUPAC convention used in NCERT is ΔU = q + w: heat absorbed and work done on the system are positive.

• For expansion against external pressure, w = −PextΔV, so the work is negative.

• Enthalpy and internal energy are linked by ΔH = ΔU + ΔngRT.

• Every named process (isothermal, adiabatic, isochoric, free expansion) just fixes one term to zero.

What Do q, w, ΔU and ΔH Mean?

Internal energy (U) is the total energy stored in a system. It is a state function, so ΔU depends only on the initial and final states. Heat (q) and work (w) are path functions: they describe how energy crossed the boundary, not what the system holds.

Enthalpy (H = U + PV) is the more useful quantity for reactions in open flasks, because at constant pressure the heat exchanged equals ΔH. At constant volume, the heat exchanged equals ΔU.

The Sign Convention That Ends the Confusion

Treat the system like a bank account. Anything flowing into the system is positive; anything flowing out is negative. Heat absorbed: q positive. Work done on the gas (compression): w positive. Work done by the gas (expansion): w negative.

The first law then reads ΔU = q + w. Older books write ΔU = q − W with W as work done by the system, so always check which form a question or book is using before substituting numbers.

Pro Tip: Before touching any numbers, write one line: "gas expands or compresses?" Expansion means w is negative, compression means w is positive. Students who drill this one-line habit until it is automatic, as we encourage in Sukriti NeetJee classes, almost never flip a sign under exam pressure.

Calculating Work in Each Process

For irreversible expansion against a constant external pressure, w = −Pext(V₂ − V₁). For a reversible isothermal process of an ideal gas, w = −2.303 nRT log(V₂/V₁). Both come out negative when the gas expands.

In free expansion (expansion into a vacuum) Pext = 0, so w = 0. This is a favourite trap: no pressure opposing means no work, even though the volume changes a lot.

What Each Named Process Fixes

• Isothermal (ΔT = 0): for an ideal gas, ΔU = 0, so q = −w.

• Adiabatic (q = 0): ΔU = w. An expanding gas cools because it spends its own internal energy.

• Isochoric (ΔV = 0): w = 0, so ΔU = qv.

• Isobaric (constant P): qp = ΔH.

Memorising this list is less useful than understanding it. Each process simply deletes one term from ΔU = q + w, and the rest follows.

Linking ΔH and ΔU for Reactions

For reactions involving gases, ΔH = ΔU + ΔngRT, where Δng is moles of gaseous products minus moles of gaseous reactants. If Δng = 0, then ΔH = ΔU. Only gases count; solids and liquids are ignored.

Watch units: if R = 8.314 J mol⁻¹ K⁻¹, your ΔU comes out in joules, while ΔH is often given in kJ. Convert before you add.

Heat Capacities: Cp, Cv and R

For one mole of an ideal gas, Cp − Cv = R. At constant volume, qv = nCvΔT, and for an ideal gas ΔU = nCvΔT holds for any process, not only constant-volume ones. Likewise ΔH = nCpΔT for any ideal-gas process.

Common Mistakes to Avoid

• Using ΔU = q − w when the question's w means work done on the system.

• Treating q as zero in an isothermal process; it is ΔU that is zero.

• Counting solids or liquids in Δng.

• Mixing J and kJ in the same equation.

Frequently Asked Questions

Is work done by the system positive or negative in NEET?

Under the IUPAC convention followed in NCERT, work done by the system (expansion) is negative and work done on the system (compression) is positive. This matches ΔU = q + w. Always confirm the convention if a question uses the older ΔU = q − W form.

Why is w zero in free expansion?

Work is calculated as −PextΔV. In a vacuum the external pressure is zero, so the product is zero regardless of how much the volume increases.

What is the difference between ΔH and ΔU?

ΔU is the heat exchanged at constant volume, while ΔH is the heat exchanged at constant pressure. They differ by ΔngRT, which is nonzero only when the number of gas moles changes in the reaction.

Is ΔU always zero in an isothermal process?

For an ideal gas, yes, because internal energy depends only on temperature. For real gases or for processes involving phase change or reaction, it need not be.

Is thermodynamics important for JEE Main and NEET?

Yes. It appears regularly in both exams as a mix of conceptual and numerical questions, and the same ideas support chemical equilibrium and electrochemistry. Confirm the current syllabus on the official NTA information bulletin.

Got a thermodynamics doubt, or a sign-convention trap that caught you out? Leave it in the comments, and share this post with a fellow aspirant who keeps losing marks to a minus sign.