Integrate a temperature-dependent state-energy change

Declare the distribution or system

For a fixed-volume system with no nonmechanical work or phase transition, dU=n Cv(T)dT. If heat capacity varies with temperature, the finite energy change is an integral, not one arbitrarily chosen endpoint capacity times deltaT. The packet has n=2 mol, Ti=300 K and molar Cv(T)=A+BT, with A=20 J/(mol K).

Derive the integral or response

Integrating gives delta U=n[A(Tf−Ti)+B(Tf²−Ti²)/2]. The factor B has units J/(mol K²), so each term has molar energy units before multiplying by n. At fixed volume with no other work, heat into the system equals this internal-energy increase. That equality would not automatically hold for a volume-changing path.

Audit limits and evidence

The initial-capacity approximation is n(A+BTi)(Tf−Ti). Subtracting it from the exact integral gives nB(deltaT)²/2, positive for positive B. Using the arithmetic mean of initial and final capacities is exact for this linear law, but that trapezoid shortcut is not exact for every nonlinear capacity.

Predict a changed premise

Split a temperature interval at any intermediate temperature and integrate both pieces: the sum must match the original state difference. Reversing the endpoints changes the sign. These operations test state-function consistency; they do not prove that the supplied constitutive law describes an actual substance outside its stated range.

Worked original case

For Tf=500 K and B=0.02 J/(mol K²), the integrated change is 11200 J. Freezing capacity at 300 K predicts 10400 J, missing 800 J. The final molar capacity is 30 J/(mol K).

Investigation

Derive the correction, compare a split interval with the direct integral, and state why constant volume is necessary for the simple Q=delta U account in this packet.

Changed-case practice

Try the changed case before feedback

Use Tf=400 K and B=0.01 J/(mol K²), with the same two moles. Find delta U in joules.

Numerical result in the requested unitsCompare this result

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and synthetic cases checked 31 October 2026 against the primary teaching reference. No source exercises or graphics are reproduced. These bounded CPU calculations do not claim actual gas handling, pressure-vessel work, thermal exposure or practical measurement. Public feedback is practice, not independently verified reasoning; retain original work, critique, revision and a newly assigned case in the course evidence workspace.