Class 12 > Unit # 17:Second Law of Thermodynamics > Clausius Statement and Working of Refrigerators

Class 12 Physics • Thermodynamics

Clausius Statement & Working of a Refrigerator

Detailed explanation of the Clausius statement of the Second Law of Thermodynamics, mechanical working of refrigerators, and step-by-step derivation of the Coefficient of Performance (COP).

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Q. State the Clausius Statement of the 2nd law of thermodynamics and describe the working of a refrigerator. Also, derive an expression for its coefficient of performance (COP).

1. Clausius Statement of 2nd Law of Thermodynamics

Statement: According to Clausius:

"It is impossible to cause heat to flow from a cold body to a hot body without the expenditure of energy."

Explanation: Naturally, heat flows spontaneously from a higher temperature body to a lower temperature body. Transferring energy in the opposite direction—from a cold reservoir to a hot reservoir—is an unnatural process. To achieve this, external work must be performed on the system. It is a proven physical fact that no machine or refrigerator can ever operate continuously without an external energy supply.

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2. Refrigerator & Its Working Principle

Definition: A refrigerator (or heat pump) is a thermodynamic device that transfers heat energy from a low-temperature reservoir (T_c) to a high-temperature reservoir (T_h) by spending external mechanical work (W).

Example: An air conditioner works on the exact same principle—it extracts heat energy (Q_c) from a cool room interior and discharges heat (Q_h) to the warmer outside air.

Working Mechanism

In practice, a refrigerator operates through a closed cycle using a circulating fluid (refrigerant) passing through two sets of metal heat-exchanger coils:

1. Internal Coils (Evaporator): Located inside the cooling compartment. The refrigerant fluid enters these coils at low pressure and cold temperature. It absorbs heat energy (Q_c) from the food items or interior environment, causing the refrigerant to evaporate into a warm gas.
2. Compressor: Driven by an electric motor, the compressor consumes external work (W) to compress the warm fluid gas into a high-pressure, high-temperature fluid.
3. External Coils (Condenser): Located behind or underneath the unit. As the hot, high-pressure fluid passes through these coils, it releases its stored heat energy (Q_h) to the warmer surroundings (room air) and condenses back into a liquid state to repeat the cycle.

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3. Coefficient of Performance (COP)

The effectiveness or efficiency of a refrigerator is expressed by a dimensionless quantity known as the Coefficient of Performance (COP), represented as K or COP.

Definition: The Coefficient of Performance is defined as the ratio of the heat energy removed from the cold reservoir (Q_c) to the external work done (W) by the compressor to extract that heat.

Mathematically:

COP = Heat removed from cold reservoirWork done by compressor = Q_cW

Derivation of Formula

From the First Law of Thermodynamics, the total heat energy ejected into the hot reservoir (Q_h) equals the sum of heat extracted from the cold reservoir (Q_c) plus the work done (W):

Q_h = Q_c + W   &implies;   W = Q_h − Q_c

Substituting W into the COP expression gives:

COP = Q_cQ_h − Q_c

In terms of absolute temperatures of the cold reservoir (T_c) and hot reservoir (T_h), this can also be expressed as:

COP = T_cT_h − T_c

Key Point for Board Exams: Unlike heat engines (whose thermal efficiency η is always less than 100% or 1), the Coefficient of Performance of a refrigerator can be greater than 1 (or > 100%). A typical domestic refrigerator operates with a COP between 5 and 6.