Question

In: Mechanical Engineering

Prove the equivalence of Kelvin-Planck and Clausius statements.

Prove the equivalence of Kelvin-Planck and Clausius statements.

Solutions

Expert Solution

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 TH. Equivalent amount of work W(W=Q1) is performed.

Consider another device 2 operating as a cycle, which absorbs energy QL as heat from a low temperature thermal reservoir at TL and rejects energy QH (QH=QL+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 QL from the low temperature reservoir at TL to a high temperature reservoir at TH 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 TL to a high temperature thermal reservoir at TH . Consider another device 2, which absorbs heat Q1 from a high temperature reservoir at TH does work W and rejects energy Q as heat tot the low temperature reservoir at TL 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.

Thnak you !


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