Second Law – Choose the Correct Formal Statements Select the correct formulation(s) of the second law of thermodynamics. The Kelvin–Planck engine statement and the Clausius heat-pump statement are both valid and equivalent expressions.

Introduction / Context:
The second law can be stated in multiple, equivalent ways. Two classical forms are the Kelvin–Planck statement (about the impossibility of 100% heat-to-work conversion in a cyclic engine) and the Clausius statement (about the impossibility of heat flowing spontaneously from cold to hot without external aid).

Given Data / Assumptions:

• Cyclic devices interacting with thermal reservoirs.
• Macroscopic classical thermodynamics; no statistical exceptions considered.
• Standard sign conventions for heat and work.

Concept / Approach:

Statement (a) is the Kelvin–Planck form: no engine can completely convert heat from a single reservoir to work. Statement (b) is the Clausius form: heat cannot, of itself, flow from colder to hotter bodies. These two are logically equivalent; violating one enables violation of the other. Statement (c) is not a correct second-law statement; it resembles an inaccurate interpretation of the mechanical equivalent of heat and ignores the second-law efficiency limits.

Step-by-Step Solution:

Verification / Alternative check:

Textbook proofs show that a hypothetical device violating Kelvin–Planck can be used to construct a device violating Clausius, and vice versa, establishing equivalence.

Why Other Options Are Wrong:

Option (e) includes (c), which is false. Picking only (a) or only (b) ignores the presence of the other correct statement.

Common Pitfalls:

Confusing first-law energy conversion statements with second-law limitations; assuming (c) is a conservation statement when it actually misstates second-law restrictions.

Final Answer:

Both (a) and (b) are correct second-law statements.