Second Law and Entropy

Overview

The second law of thermodynamics establishes the direction of natural processes and introduces entropy, a measure of disorder in a system.

Statements of the Second Law

Clausius Statement

Heat cannot spontaneously flow from a colder body to a hotter body.

Kelvin-Planck Statement

No cyclic process can convert heat completely into work without other effects.

Entropy Statement

The total entropy of an isolated system never decreases.

$$\Delta S_{\text{total}} \geq 0$$

Entropy

Definition

For a reversible process:

$$dS = \frac{dQ_{\text{rev}}}{T}$$ $$\Delta S = \int \frac{dQ_{\text{rev}}}{T}$$

• Unit: J/K
• State function (path independent)

For Isothermal Process

$$\Delta S = \frac{Q}{T}$$

For Temperature Change

$$\Delta S = mc \ln\left(\frac{T_2}{T_1}\right) = nC \ln\left(\frac{T_2}{T_1}\right)$$

For Ideal Gas

$$\Delta S = nC_v \ln\left(\frac{T_2}{T_1}\right) + nR \ln\left(\frac{V_2}{V_1}\right)$$

Or:

$$\Delta S = nC_p \ln\left(\frac{T_2}{T_1}\right) - nR \ln\left(\frac{P_2}{P_1}\right)$$

Heat Engines

Efficiency

$$\eta = \frac{W}{Q_H} = \frac{Q_H - Q_C}{Q_H} = 1 - \frac{Q_C}{Q_H}$$

Where:

• $Q_H$ = heat absorbed from hot reservoir
• $Q_C$ = heat rejected to cold reservoir
• $W$ = net work done

Carnot Engine (Maximum Efficiency)

$$\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}$$

No real engine can exceed Carnot efficiency!

Carnot Cycle

1. Isothermal expansion at $T_H$
2. Adiabatic expansion ($T_H \to T_C$)
3. Isothermal compression at $T_C$
4. Adiabatic compression ($T_C \to T_H$)

Refrigerators and Heat Pumps

Coefficient of Performance (COP)

Refrigerator:

$$\text{COP}_{\text{ref}} = \frac{Q_C}{W} = \frac{Q_C}{Q_H - Q_C}$$

Heat Pump:

$$\text{COP}_{\text{hp}} = \frac{Q_H}{W} = \frac{Q_H}{Q_H - Q_C}$$

Carnot COP

$$\text{COP}_{\text{ref,Carnot}} = \frac{T_C}{T_H - T_C}$$ $$\text{COP}_{\text{hp,Carnot}} = \frac{T_H}{T_H - T_C}$$

Note: $\text{COP}_{\text{hp}} = \text{COP}_{\text{ref}} + 1$

Entropy Changes

Heat Transfer Between Reservoirs

When heat $Q$ flows from $T_H$ to $T_C$:

$$\Delta S_{\text{total}} = \frac{Q}{T_C} - \frac{Q}{T_H} > 0$$

Irreversible Process

$$\Delta S_{\text{total}} > 0 \quad \text{(entropy increases)}$$

Reversible Process

$$\Delta S_{\text{total}} = 0 \quad \text{(entropy unchanged)}$$

For the Universe

Natural processes always increase total entropy.

Statistical Interpretation

Boltzmann Entropy

$$S = k_B \ln(\Omega)$$

Where:

• $k_B$ = Boltzmann constant = $1.38 \times 10^{-23}$ J/K
• $\Omega$ = number of microstates

Higher entropy = more disorder = more possible arrangements

Examples

Example 1: Carnot Efficiency

A Carnot engine operates between 600 K and 300 K.

$$\eta = 1 - \frac{T_C}{T_H} = 1 - \frac{300}{600} = 0.5 = 50\%$$

Example 2: Real Engine

An engine takes in 1000 J of heat and does 300 J of work.

$$\eta = \frac{W}{Q_H} = \frac{300}{1000} = 30\%$$ $$Q_C = Q_H - W = 700 \text{ J}$$

Example 3: Entropy of Melting

1 kg of ice melts at 0°C. Find entropy change.

$$\Delta S = \frac{Q}{T} = \frac{mL_f}{T} = \frac{1 \times 334000}{273} = 1223 \text{ J/K}$$

Example 4: Mixing Water

1 kg water at 80°C mixes with 1 kg water at 20°C.

Final temperature = 50°C = 323 K

$$\Delta S_{\text{hot}} = mc \ln\left(\frac{T_f}{T_i}\right) = 1 \times 4186 \times \ln\left(\frac{323}{353}\right) = -373 \text{ J/K}$$ $$\Delta S_{\text{cold}} = mc \ln\left(\frac{T_f}{T_i}\right) = 1 \times 4186 \times \ln\left(\frac{323}{293}\right) = +417 \text{ J/K}$$ $$\Delta S_{\text{total}} = -373 + 417 = +44 \text{ J/K} > 0 \checkmark$$

Example 5: Refrigerator

A refrigerator removes 400 J from food at 5°C, rejecting heat at 35°C.

Carnot COP:

$$\text{COP} = \frac{T_C}{T_H - T_C} = \frac{278}{308 - 278} = 9.3$$

Minimum work required:

$$W_{\min} = \frac{Q_C}{\text{COP}} = \frac{400}{9.3} = 43 \text{ J}$$

Example 6: Heat Pump

How much heat can a heat pump deliver using 1 kJ of work between 0°C and 20°C?

$$\text{COP}_{\text{hp}} = \frac{T_H}{T_H - T_C} = \frac{293}{20} = 14.65$$ $$Q_H = \text{COP} \times W = 14.65 \times 1000 = 14{,}650 \text{ J}$$

Practical Implications

• No perpetual motion machines (second kind)
• All real engines have $\eta < \eta_{\text{Carnot}}$
• Waste heat is unavoidable
• Energy quality decreases in all processes
• Heat death of the universe (maximum entropy)