6. Clausius Statement Verifier — Theory
The Clausius statement of the second law says:
It is impossible for a device operating in a cycle to have as its sole effect the transfer of heat
from a colder body to a hotter body.
In other words, a refrigerator moving heat cold → hot must require work input.
Reservoir entropy argument
Consider two thermal reservoirs:
hot at temperature \(T_h\) and cold at \(T_c\) with \(T_h > T_c\).
Suppose a cyclic device removes heat \(Q_c\) from the cold reservoir and rejects heat \(Q_h\) to the hot reservoir.
If the device is cyclic, its internal state returns to the start; the entropy change we track is for the reservoirs.
The second law requires \(\Delta S_{\text{univ}} \ge 0\) for any real process.
The reversible limit is \(\Delta S_{\text{univ}}=0\).
Energy balance and the “no-work” claim
A refrigerator-like device satisfies the first-law energy balance (magnitudes):
If someone claims the device needs no work (\(W=0\)), then \(Q_h = Q_c\). Plugging into the entropy balance gives:
Therefore, the “cold → hot with no work” claim would force \(\Delta S_{\text{univ}}<0\), which violates the second law.
That is exactly the Clausius statement.
Minimum reversible work and Carnot COP
Setting \(\Delta S_{\text{univ}}=0\) gives the minimum (reversible) work required to move \(Q_c\) from cold to hot:
Real refrigerators require \(W > W_{\min}\), giving \(\Delta S_{\text{univ}}>0\).
What the calculator verifies
- Computes \(Q_h=Q_c+W\).
- Computes \(\Delta S_{\text{hot}}=Q_h/T_h\), \(\Delta S_{\text{cold}}=-Q_c/T_c\), and \(\Delta S_{\text{univ}}\).
- If \(\Delta S_{\text{univ}}<0\): flags the claim as impossible (Clausius violation).
- Shows \(W_{\min}\) and \(\mathrm{COP}_{\text{Carnot}}\) for comparison (optional).
