Thermodynamics: Laws and Processes

Chapter 14: The Laws of Thermodynamics

14.1 Zeroth Law of Thermodynamics

• If objects A and B are each in thermal equilibrium with object C, then A and B are in thermal equilibrium with each other.
• Temperature determines thermal equilibrium.

14.2 First Law of Thermodynamics

• Internal energy refers to all the energy of a system associated with its microscopic components from a reference frame at rest with respect to the system's center of mass.

First Law of Thermodynamics

• $\Delta E_{int} = Q + W$
  • $\Delta E_{int}$ represents the change in internal energy.
  • $Q$ is the energy transferred to the system as heat.
  • $W$ is the energy transferred to the system as work.

Work Done by the System

• $W = -\int_{V_i}^{V_f} P dV$
  • $P$ is the pressure exerted.
  • $V$ denotes the volume.

Types of Thermodynamic Processes

• Adiabatic: $Q = 0$, meaning no heat transfer occurs.
• Isobaric: The process occurs at constant pressure.
• Isovolumetric: The process occurs at constant volume, where $W = 0$.
• Isothermal: The process occurs at constant temperature.

Heat Engines

• Thermal Efficiency: $e = \frac{W_{eng}}{|Q_h|} = 1 - \frac{|Q_c|}{|Q_h|}$
  • $W_{eng}$ is the net work done by the engine.
  • $Q_h$ is the energy taken in from a hot reservoir.
  • $Q_c$ is the energy exhausted to a cold reservoir.

Heat Pumps and Refrigerators

• Transfers energy from a cold reservoir to a hot reservoir.

Coefficient of Performance (COP)

• $COP = \frac{|Q_c|}{W}$

14.3 Second Law of Thermodynamics

• Energy spontaneously flows from hot to cold objects.
• It is impossible to make a heat engine that only intakes energy as heat from a reservoir and performs an equal amount of work in a cycle.
• Constructing a cyclical machine that transfers energy by heat from one object to another at a higher temperature without work input is impossible.

Entropy

• Entropy measures a system's disorder.

Change in Entropy

• $\Delta S = \int_{i}^{f} \frac{dQ}{T}$
  • $S$ is the entropy.
  • $Q$ is the energy transferred as heat.
  • $T$ is the absolute temperature.

Entropy Statement of the Second Law of Thermodynamics

• The total entropy of an isolated system can increase, but never decrease: $\Delta S \ge 0$

Introduction to Probability

Terminology

Experiment

• Activity with an observable outcome.
  • Tossing a Coin
  • Drawing a Card
  • Rolling a Die

Sample Space

• Sample Space, denoted $\Omega$, is the set of all possible outcomes of an experiment
  • Toss a coin: $\Omega = {H, T}$
  • Draw a card from a deck: $\Omega = {A\heartsuit, 2\heartsuit,..., K\clubsuit }$
  • Roll a die: $\Omega = {1, 2, 3, 4, 5, 6}$

Event

• Event is a subset of the sample space
  • Toss a coin: $E = {H}$
  • Draw a card from a deck: $E = {draw a heart}$
  • Roll a die: $E = {roll an even number}$

Definition of Probability

• Given a sample space $\Omega$, a probability function $P$ assigns a real number to each event $E \subseteq \Omega$, denoted $P(E)$, satisfying the following axioms:

Implications

• $0 \le P(E) \le 1$
• $P(\Omega) = 1$
• If $E_1, E_2,...$ are mutually exclusive events, then $P(\bigcup_{i=1}^{\infty} E_i) = \sum_{i=1}^{\infty} P(E_i)$
• $P(\emptyset) = 0$
• If $E_1 \subseteq E_2$, then $P(E_1) \le P(E_2)$
• $P(E^c) = 1 - P(E)$
• $P(E_1 \cup E_2) = P(E_1) + P(E_2) - P(E_1 \cap E_2)$

Examples

Example 1

• Toss a coin twice. What is the probability of getting at least one head?
  • Sample space: $\Omega = {HH, HT, TH, TT}$
  • Event: $E = {at least one head} = {HH, HT, TH}$
  • Assuming a fair coin, each outcome is equally likely
    • $P(HH) = P(HT) = P(TH) = P(TT) = \frac{1}{4}$
    • $P(E) = P(HH) + P(HT) + P(TH) = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{3}{4}$

Example 2

• Roll a die twice. What is the probability that the sum of the two rolls is equal to 7?
  • Sample space: $\Omega = {(1,1), (1,2),..., (6,6)}$
  • Event: $E = {sum of two rolls is 7} = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}$
  • Assuming a fair die, each outcome is equally likely
    • $P(each outcome) = \frac{1}{36}$
    • $P(E) = P(1,6) + P(2,5) + P(3,4) + P(4,3) + P(5,2) + P(6,1) = \frac{1}{36} \cdot 6 = \frac{1}{6}$