The change in internal energy,
Δ
𝐸
, of a closed system is equal to the heat flow,
𝑞
, for a process only if which state variable is constant?

Question

The change in internal energy, 
Δ
𝐸
, of a closed system is equal to the heat flow, 
𝑞
, for a process only if which state variable is constant?

BlackTom AI · Accepted Answer

For a closed system, the first law of thermodynamics gives ΔE = q + w, where ΔE is the change in internal energy, q is heat added to the system, and w is work done on the system.

1) Relevant concepts
- First law: ΔE = q + w.
- For a closed system the common form of mechanical work is pressure–volume (PV) work: w_PV = −∫P_ext dV (sign convention: work done by the system is negative).
- If no work is performed (w = 0), then ΔE = q.

2) Step-by-step reasoning
- Start with the first law: ΔE = q + w.
- To have ΔE equal to q, the work term must be zero: w = 0.
- For typical closed-system mechanical work, w = w_PV = −∫P_ext dV. This equals −P_ext ΔV for constant external pressure, and in general is zero only when ΔV = 0.
- Therefore, ΔV = 0 (constant volume) guarantees w_PV = 0 and hence ΔE = q.

3) Intermediate steps / algebra
- ΔE = q + w
- Require ΔE = q → q = q + w → w = 0.
- For PV work: w = −∫P_ext dV. If ΔV = 0, then w = −∫P_ext·0 = 0 → ΔE = q.

4) Why the correct answer is correct
- Constant volume eliminates PV work, so all energy transfer is via heat. In a rigid (constant-volume) closed vessel, measured heat q_v equals the change in internal energy: ΔE = q_v.

5) Why other state variables are incorrect (brief)
- Constant pressure: volume can still change, so PV work may be nonzero; ΔE ≠ q in general (at constant pressure q = ΔH for processes with only PV work).
- Constant temperature (isothermal): ΔE for an ideal gas may be zero, but that does not imply ΔE = q unless w = 0; typically q = −w for isothermal ideal-gas processes, so ΔE = 0.
- Constant enthalpy or other variables: these do not guarantee w = 0 and so do not ensure ΔE = q.

6) Conclusion
- Because zero PV work (w = 0) is achieved when volume is constant, the required state variable is Volume. This matches the provided answer: Volume.

The change in internal energy, Δ 𝐸 , of a closed system is equal to the heat flow, 𝑞 , for a process only if which state variable is constant?单项选择题

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