Question

Prove the equivalence of Kelvin-Planck and Clausius statements.

Answer 1

Kelvin-Planck Statement: - It is impossible to develop a device which exchanges energy with a single reservoir and produces net work output,

The concept of such a device is called PMM-2 and is impossible..

Heat engine with single thermal reservoir is not possible.

For a 1-T engine the thermal efficiency h=W/Q=1. No heat engine can have efficiency equal to unity.

Clausius statements:- It is impossible to evelop a cyclic device which transfers heat from lower temperature to higher temperature without any external enegry input.

EQUIVALENCE OF THE TWO STATEMENTS:

Violation of kelvin plank statement leads to the violation of clausius statement and vice versa. Hence these two statement are called parallel statement of 2nd law of thermodynamics.

To prove that violation of the Kelvin-Planck Statement leads to a violation of the Clausius Statement, let us assume that Kelvin-Planck statement is incorrect.

Consider a cyclically working device 1, which absorbs energy Q₁ as heat from a thermal reservoir at T_H. Equivalent amount of work W(W=Q₁) is performed.

Consider another device 2 operating as a cycle, which absorbs energy Q_L as heat from a low temperature thermal reservoir at T_L and rejects energy Q_H (Q_H=Q_L+W).

Such a device does not violate Clausius statement.

If the two devices are now combined, the combined device (enclosed by the dotted boundary) transfers heat Q_L from the low temperature reservoir at T_L to a high temperature reservoir at T_H with out receiving any aid from an external agent, which is the violation of the Clausius statement.

Likewise,

let us assume that the Clausius statement is incorrect. So we have a device 1, cyclically working transferring heat Q from a low temperature reservoir at T_L to a high temperature thermal reservoir at T_H . Consider another device 2, which absorbs heat Q₁ from a high temperature reservoir at T_H does work W and rejects energy Q as heat tot the low temperature reservoir at T_L as shown in figure.

If the two devices are combined (shown in figure by a dotted enclosure), then the combined device receives energy (Q₁-Q) as heat from a thermal reservoir and delivers equivalent work (W=Q₁-Q) in violation of the Kelvin-Planck statement.

Therefore violation of Clausius statement leads to the violation of the Kelvin-Planck statement. Hence, these two statements are equivalent.

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