Problem Statement
Explain why a perpetual motion machine of the second kind is impossible, and state the Kelvin-Planck and Clausius formulations of the second law.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: Kelvin-Planck: It is impossible to construct a heat engine that, operating in a cycle, produces no effect other than the absorption of heat from a single reservoir and the performance of an equivalent amount of work.
Step 2 — Apply the relevant physical law or equation: Clausius: It is impossible to construct a device that, operating in a cycle, produces no effect other than the transfer of heat from a cooler to a hotter body.
Step 3 — Solve algebraically for the unknown: Equivalence: Violating one implies violating the other.
Step 4 — Substitute numerical values with units: Perpetual motion II: A machine that converts heat from a single reservoir entirely to work (100% efficient heat engine). This violates Kelvin-Planck and the second law. The ocean contains vast thermal energy, but we cannot run a ship’s engine by cooling the seawater without a cold reservoir.
Worked Calculation
$$\text{Numerical result} = \text{given expression substituted with values}$$
$$\boxed{\text{See solution}}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
Kelvin-Planck: It is impossible to construct a heat engine that, operating in a cycle, produces no effect other than the absorption of heat from a single reservoir and the performance of an equivalent amount of work.
Clausius: It is impossible to construct a device that, operating in a cycle, produces no effect other than the transfer of heat from a cooler to a hotter body.
Equivalence: Violating one implies violating the other.
Perpetual motion II: A machine that converts heat from a single reservoir entirely to work (100% efficient heat engine). This violates Kelvin-Planck and the second law. The ocean contains vast thermal energy, but we cannot run a ship’s engine by cooling the seawater without a cold reservoir.
Answer
$$\boxed{\text{See solution}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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