Chapter 4 Notes: The Second Law of Thermodynamics

Chapter 4: The Second Law of Thermodynamics

Background

• The first law of thermodynamics describes the conservation of energy, but it doesn't fully explain how things behave in terms of direction or spontaneity.

• Example: Heat flows from hot coffee to the surroundings, but not spontaneously from the surroundings to cold coffee.

• Example: Sliding a box across the floor converts mechanical energy into heat due to friction (1st law), but the reverse (box spontaneously sliding as the floor cools) never happens.

• In nature, energy conservation has a certain directionality and conditions, which the Second Law of Thermodynamics addresses.

Kelvin-Plank Statement and Heat Engine

• The second law of thermodynamics can be expressed through the Kelvin-Plank and Clausius statements.

• Kelvin-Plank statement: It's impossible to create a heat engine that operates in a cycle and solely absorbs energy from a reservoir to perform an equal amount of work.

• A heat engine transforms heat partially into work, utilizing a working substance (e.g., gasoline and air in a car engine, water in a steam engine).

• Most engines use a cyclic process where the working substance returns to the same state at periodic intervals.

Heat Engine Equations

• $W = Q<em>H - Q</em>C$ where:

  • $W$ is the work done by the engine

  • $Q_H$ is the heat absorbed from the hot reservoir

  • $Q_C$ is the heat rejected to the cold reservoir

• $Q<em>H = Q</em>C + W$

• Thermal efficiency ($\eta$) is the ratio of work done to heat input: $\eta = \frac{W}{Q_{in}}$.

• Therefore, \eta < 1 (efficiency is always less than 1).

Problem Example

• An engine gains 150 J of heat from fuel and does 120 J of work per cycle.

  • Thermal efficiency: $\eta = \frac{120 \text{ J}}{150 \text{ J}} = 0.8$ or 80%.

Reversible and Irreversible Processes

• Reversible process: A system can return to its initial conditions along the exact same path on a PV diagram, with every point on the path being an equilibrium state.

• Irreversible process: A process that does not meet the requirements of a reversible process.

• All natural processes are irreversible.

Carnot Engine

• Carnot engine: A theoretical heat engine where all processes are reversible.

• Designed by Sadi Carnot, it represents the most efficient possible engine.

• It's an ideal, theoretical engine used to measure the efficiency of real engines.

Carnot Engine Cycle

• The Carnot Cycle consists of:

  • A-B: Isothermal expansion at temperature $T_h$

    • Heat absorbed: q > 0

    • Work done: $-w = nRT<em>h \ln(V</em>B/V_A)$

  • B-C: Adiabatic expansion (q=0)

  • C-D: Isothermal compression at temperature $T_c$

    • Heat released: q < 0

  • D-A: Adiabatic compression (q=0)

• Carnot engine efficiency:

  • $\eta<em>c = 1 - \frac{T</em>c}{T_h}$

Example: Steam Engine Efficiency

• A steam engine operates with a boiler at 800 K and a cold reservoir (outside air) at 300 K.

  • Maximum thermal efficiency: $\eta_c = 1 - \frac{300 \text{ K}}{800 \text{ K}} = 1 - 0.375 = 0.625$ or 62.5%.

Other Heat Engine Cycles

• Otto Cycle: Used in gasoline engines.

  • Involves intake, compression, combustion, power stroke, exhaust, and heat rejection stages.

  • $\eta = 1 - \frac{T<em>5 - T</em>6}{T<em>3 - T</em>2}$ (Ideal Otto cycle efficiency)

• Diesel Cycle:

  • Similar to Otto cycle but with different combustion process.

Clausius Statement and Heat Pump

• Clausius statement: It's impossible to create a cyclical machine with the sole effect of continuously transferring energy from a colder object to a hotter object without external work input.

• In simpler terms, heat doesn't spontaneously flow from cold to hot.

• A heat pump or refrigerator transfers heat from a cold to a hot object.

Heat Pump and Refrigerator

• Heat pump: Transfers heat from a cold reservoir to a hot reservoir, requiring work.

• Refrigerator: Operates on a similar principle, cooling an enclosed space.

• Refrigerator System Components:

  • Compressor

  • Condenser

  • Metering device (Expansion valve)

  • Evaporator

Heat Pump Performance

• The effectiveness of a heat pump is measured by the coefficient of performance (COP).

• In heating mode: $\text{COP} = \frac{\text{Energy transferred to hot reservoir}}{\text{Work required}}$.

Entropy

• Entropy provides another statement of the Second Law of Thermodynamics.

• First defined by Clausius in 1865 on a macroscopic scale.

Entropy: Definition

• Entropy is a measure of a system's ability to do useful work. As a system loses this ability, its entropy increases.

• Entropy is a measure of disorder. Systems naturally move toward greater disorder. The more order, the less entropy.

Entropy and Its Relation to Temperature and Heat

• Entropy is related to temperature and heat.

• Change in entropy ($\Delta S$) for a reversible process at constant temperature:

  • $\Delta S = \frac{Q}{T}$, where T is in Kelvins.

  • Units of entropy: Joules per Kelvin (J/K).

• Total change in entropy for a cycle: 0

Entropy Changes in Different Processes

• Adiabatic process: $\Delta S = 0$

• Isothermal process:

  • $\Delta S = nR \ln\left(\frac{V<em>2}{V</em>1}\right)$

• Isochoric process:

  • $\Delta S = nC<em>v \ln\left(\frac{T</em>2}{T_1}\right)$

Entropy and Natural Processes

• The direction of any process is toward an increase in entropy.

  • The entropy of an isolated system never decreases; it always increases for natural processes.

• Entropy is a function of state: Each state has a particular entropy value, and the change in entropy depends only on the initial and final states (like internal energy in an ideal gas).

• An unnatural process is one never observed (e.g., water at room temperature spontaneously turning into ice), while a natural process is commonly observed.