2nd Law of Thermodynamics

Recap on Prerequisites

The 2nd Law of thermodynamics is all about the efficiency of heat transfer processes

Thermal Energy Reservoirs

Hypothetical body with a relatively large thermal energy capacity (mass x specific heat) that can absorb/supply finite amounts of heat without undergoing any change in temperature.

Source - supplies heat energy

Sink - absorbs heat energy

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Heat Engines

Converts heat to work

Energy Balance

From first law:

$$Q-W=\Delta U$$

The energy balance for heat engines is:

$$W_{net,out}=Q_H-Q_L$$

Thermal Efficiency

$$\text{Thermal Efficiency}=\frac{\text{Net work output}}{\text{Total heat input}}$$

So for Heat Engines:

$${\eta }_{th}=\frac{W_{net,out}}{Q_H}=1-\frac{Q_L}{Q_H}$$

Kelvin-Planck Statement

*It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work.

• Even theoretically perfect heat engines do not have an efficiency of 100%

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Reverse Heat Engines

Work is added to induce heat transfer from low temperatures to high temperatures

• E.g. Refrigerator / Heat pumps

Coefficient of performance

$$\text{Coefficient of Performance}=\frac{\text{Desired output}}{\text{Required input}}$$

For Refrigerator: Function - to cool refrigerated space

$$CO{P}_R=\frac{Q_L}{W_{net,in}}=\frac{Q_L}{Q_H-Q_L}=\frac{1}{Q_H/Q_L-1}$$

For Heat Pump: Function - to warm heated space

$$CO{P}_{HP}=\frac{Q_H}{W_{net,in}}=\frac{Q_H}{Q_H-Q_L}=\frac{1}{1-Q_L/Q_H}$$

Clausius Statement

It is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a lower-temperature body to a higher-temperature body.

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Perpetual Motion Machine

A device that contradicts either the 1st law or 2nd law of thermodynamics

Recap

1st Law - energy cannot be created or destroyed, only transformed from one form to another

2nd Law - the entropy, or disorder, of a closed system never decreases

PMM1: Infinite output with no inputs

PMM2: decreasing entropy

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Reversible & Irreversible processes

Reversible process - Can be reversed without leaving traces in the surrounding

• E.g frictionless pendulum

Irreversible process - opposite - Most processes in real life

Internally reversible - no irreversibilities occur within system boundaries (red box)

Externally reversible - no irreversibilities occur outside system boundaries (blue box)

Totally reversible - internally + externally reversible (magenta box)

Carnot cycle

A theoretical cycle that sets the limits for heat engines, refrigerator and heat pumps

Recap

(a) Reversible Isothermal Expansion (1-2, T_H = constant)

• Gas expands at constant temp while absorbing heat from energy source

(b) Reversible Adiabatic Expansion (2-3, T_H drops to T_L)

• Gas does work on surrounding and expands while its temp drops

(c) Reversible Isothermal Compression (3-4, T_L constant)

• Gas compression at constant temp while losing heat to energy sink

(d) Reversible Adiabatic Compression (4-1, T_L rises to T_H)

• Work done on gas to compress it and its temp rises

The reversed carnot cycle

The carnot cycle is a reversible cycle that nets the “Carnot refrigeration cycle”

Carnot Principles

1. The efficiency of an irreversible heat engine is always less than the efficiency of a reversible one operating between the same two reservoirs

$${\eta }_{\text{th},1,\text{irrev}}<{\eta }_{\text{th},2,\text{rev}}$$

2. The efficiencies of all reversible heat engines operating between the same two reservoirs are the same

$${\eta }_{\text{th},2,\text{rev}}={\eta }_{\text{th},3,\text{rev}}$$

Expand for Proof of Carnot Principles

Part 1 Part 2

Thermodynamic Temperature Scale

A temperature scale that is independent of the properties of substances that are used to measure temperature is called a thermodynamic temperature scale

• Thermal reservoirs are characterized only by their temperatures

$$T(K)=T{(}^{\circ }C)+273.15$$ $${\left(\frac{Q_H}{Q_L}\right)}_{\text{rev}}=\frac{T_H}{T_L}\text{or}{\left(\frac{Q_L}{Q_H}\right)}_{\text{rev}}=\frac{T_L}{T_H}$$

Derivation of the scale

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Carnot devices

AKA ideal / most efficient devices based on carnot principles

Carnot heat engine

Thermal efficiency of a Carnot heat engine

$${\eta }_{th,\text{rev}}=1-\frac{T_L}{T_H}$$

!! absolute temps only !!

Carnot refrigerator & heat pump

Coefficient of performance for a Carnot refrigerator:

$${\text{COP}}_{R,\text{rev}}=\frac{1}{T_H/T_L-1}$$

Coefficient of performance for a Carnot heat pump:

$${\text{COP}}_{HP,\text{rev}}=\frac{1}{1-T_L/T_H}$$

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References