Questions: Types of Work: Mechanical PdV and Beyond

Question 1

A lithium-ion battery discharges completely at constant volume, converting stored chemical energy into electrical energy in an external circuit. Which work term in the first law accounts for this energy transfer?

• A. PdV work — all thermodynamic work transfers occur through volume changes
• B. Electrical work (voltage × charge transferred), because the dominant energy transfer mechanism is charge moving through a potential difference
• C. Surface work γdA, because ion transport across the electrode interface changes interfacial area
• D. No work term — the first law only applies to systems that exchange heat

Answer: (B)

At constant volume, PdV = 0 — no mechanical expansion work occurs. The energy leaving the battery is electrical work: charge q moves through potential difference V, giving W_elec = V·q (or in differential form, V dq). This is the appropriate work term for this system. The first law dU = đQ − đW_total remains valid; you simply use the correct work term. Option A is the common misconception that PdV is the only work type — it applies to gas systems but not batteries. Option C (surface work) is real but negligible here.

Question 2

A gas expands against external pressure, doing 100 J of work on the surroundings. A student uses the physics convention dU = đQ − đW (work done *by* system is positive). Their textbook uses the engineering convention dU = đQ + đW (work done *on* system is positive). Which statement is correct?

• A. Both conventions give đW = +100 J and agree on ΔU
• B. The student has đW = +100 J (work done by system); the textbook has đW = −100 J (work done on system is negative for expansion). Both give the same ΔU.
• C. The conventions give different values of ΔU, so one must be wrong
• D. The engineering convention always assigns positive work to expansion; the physics convention assigns negative work

Answer: (B)

Both conventions are self-consistent and give the same ΔU — they just define đW with opposite signs. Physics convention: đW = +100 J (the gas does work), so ΔU = Q − 100. Engineering convention: đW = −100 J (work *on* system during expansion is negative, since the system is doing the pushing), so ΔU = Q + (−100) = Q − 100. Identical result. The danger is mixing conventions: never use the physics đW formula with the engineering sign for đW, or vice versa.

Question 3

Surface tension work (γdA) is negligible in most physical systems and can safely be ignored when applying the first law.

• T. True
• F. False

Answer: False

Surface tension work is negligible at macroscopic scales where PdV dominates, but it is significant at small scales — for example, in living cells where membrane surface tension affects thermodynamics, in soap films, and in microfluidic systems. The general rule is that which work terms matter depends on the system and scale. Assuming PdV is always dominant ignores important physics at small scales and in specialized systems.

Question 4

Every work term in thermodynamics has the mathematical structure of an intensive variable multiplied by the differential of an extensive variable.

• T. True
• F. False

Answer: True

This pattern is universal: PdV (pressure × dvolume), V dq (voltage × dcharge), γ dA (surface tension × darea), μ₀ H dM (magnetic field × dmagnetization). Intensive variables are size-independent (pressure is the same whether you have a little or a lot of gas); extensive variables scale with the amount of substance. This structure is not a coincidence — it reflects the thermodynamic definition of work as a generalized force (intensive) times a generalized displacement (differential of the conjugate extensive variable).

Question 5

Why does applying the first law correctly require identifying which work terms are relevant to a given system, rather than always defaulting to dU = đQ − PdV?

Model answer: dU = đQ − PdV is only correct for a closed system where mechanical expansion/compression is the sole work mechanism. When a system transfers energy through other channels — electrical work in a battery, surface work in a membrane, magnetic work in a ferromagnet — those terms must be included in đW_total. Omitting a relevant work term breaks the energy balance: the equation will not close correctly, and calculated ΔU will be wrong. Including irrelevant terms is harmless (they evaluate to zero for the specific process), but missing relevant ones produces systematically incorrect thermodynamic analysis.

The first law's generality is its strength: dU = đQ − đW_total holds for any closed system in any process. The skill is identifying đW_total correctly for the system at hand — which requires understanding what energy transfer mechanisms are physically active. This is why thermodynamics problems begin by characterizing the system and process, not by writing equations.