Gases: Properties and Gas Laws

The Properties of Gases

• Gases are compressible, form homogeneous mixtures, and uniformly occupy containers.

Pressure

• Pressure ($P$) equals force ($F$) per unit area ($A$): $P = \dfrac{F}{A}$
• Pascal (Pa) is the SI unit for pressure (N/m²).
• 1 standard atmosphere equals 1 atm, which is 760 mm Hg, 760 torr, or 1.01325 x 10⁵ Pa.

Gas Laws

Boyle's Law

• At constant temperature, the volume of a fixed amount of gas is inversely proportional to its pressure: $V \propto \dfrac{1}{P}$
• $P_1V_1 = P_2V_2$

Charles's Law

• At constant pressure, the volume of a fixed amount of gas is directly proportional to its absolute temperature: $V \propto T$
• $\dfrac{V_1}{T_1} = \dfrac{V_2}{T_2}$

Avogadro's Law

• The volume of gas at fixed temperature and pressure is directly proportional to the amount of gas: $V \propto n$
• $V_1/n_1 = V_2/n_2$

Ideal-Gas Equation

• $PV = nRT$
• $R = 0.08206 \dfrac{L \cdot atm}{mol \cdot K} = 8.314 \dfrac{J}{mol \cdot K}$

Gas Densities and Molar Mass

• $d = \dfrac{m}{V} = \dfrac{PM}{RT}$
• $M = \dfrac{dRT}{P}$

Dalton's Law of Partial Pressures

• Each gas in a mixture exerts the same pressure as it would if it were alone.
• $P_t = P_1 + P_2 + P_3 +...$
• $P_i = X_iP_t$
• $X_i = \dfrac{n_i}{n_t}$

Kinetic-Molecular Theory

• Gases have lots of molecules in continuous, random motion.
• Total volume of molecules is negligible compared to total volume.
• Attractive and repulsive forces between molecules are negligible.
• Energy can transfer between molecules during collisions. At constant temperature, average kinetic energy remains the same.
• Average kinetic energy of molecules is proportional to absolute temperature.
• $\epsilon = \dfrac{1}{2}mv^2$
• $\epsilon \propto T$

Root-Mean-Square Speed

• $u = \sqrt{\dfrac{3RT}{M}}$

Graham's Law of Effusion

• $\dfrac{r_1}{r_2} = \sqrt{\dfrac{M_2}{M_1}}$

Real Gases Deviate

• Real gases diverge from ideal behavior at high pressure.
• $(P + \dfrac{an^2}{V^2})(V - nb) = nRT$

Floating Point Numbers

Scientific Notation

• Components include sign, mantissa, base, and exponent.
• For example, $-6.022 \times 10^{23}$.

Floating Point Representation

• Floating point representation includes sign, mantissa, base (2), and exponent.

IEEE 754 Floating Point Standard

• Single precision uses 32 bits.
• Double precision uses 64 bits.

Sign Bit

• Uses 1 bit to denote the sign.
• 0 means positive.
• 1 means negative.

Mantissa

• Mantissa is the significand.
• Normalized to be in the form 1.xxxx.
• The 1. is implicit and not stored.
• This implicit 1 is also called the "hidden bit".

Mantissa: Single Precision

• Uses 23 bits.
• $2^{-1} + 2^{-2} + 2^{-3} + \dots + 2^{-23} = 0.99999988079$
• Approximately 7 decimal digits

Mantissa: Double Precision

• Uses 52 bits.
• $2^{-1} + 2^{-2} + 2^{-3} + \dots + 2^{-52} = 0.999999999999999777955$
• Approximately 16 decimal digits.

Exponent

• Biased exponent.
• The actual exponent equals the stored exponent minus the bias.

Exponent: Single Precision

• Uses 8 bits.
• Bias is 127.
• Range is -126 to 127.

Exponent: Double Precision

• Uses 11 bits.
• Bias is 1023.
• Range is -1022 to 1023.

Example: Single Precision

• 0.0 in single precision floating point format: 0 00000000 00000000000000000000000

Lab 7: Functions

Task 1: addition(x, y)

• Function addition(x, y) takes two numbers, x and y, as input and returns their sum.

Task 2: average(numbers)

• Function average(numbers) takes a list of numbers and returns the average.

Task 3: is_even(number)

• Function is_even(number) takes an integer and returns True if even, False otherwise.

Task 4: reverse_string(s)

• Function reverse_string(s) takes a string s and returns the reversed string.

Task 5: factorial(n)

• Function factorial(n) takes an integer n and returns the factorial of that number.
• Factorial of a non-negative integer $n$, denoted by $n!$, is the product of all positive integers less than or equal to $n$.

Task 6: is_palindrome(s)

• Function is_palindrome(s) takes a string s and returns True if palindrome, False otherwise.
• Palindrome: A word, phrase, number, or sequence which reads the same backward or forward.

Task 7: count_vowels(s)

• Function count_vowels(s) takes a string s and returns number of vowels (a, e, i, o, u).

Lecture 14: Mechanism Design without Money

Introduction

• Mechanism design without money refers to scenarios where payments are infeasible or undesired.

Social Choice

Preliminaries

• $A$ is the set of alternatives.
• Agents’ set: $N = {1, 2,..., n}$.
• Valuation of agent $i$: $v_i : A \rightarrow \mathbb{R}$.
• Valuation profile: $v = (v_1,..., v_n)$.
• Social choice function: $f : V \rightarrow A$, where $V$ is the set of all valuation profiles.
• $f(v)$ is chosen alternative by the mechanism for valuation profile $v$.
• Social welfare: $SW(a, v) = \sum_{i \in N} v_i(a)$, where $a \in A$.

Example 1: Facility Location

• Build a facility at some location.
• Set of alternatives $A = \mathbb{R}$.
• Set of agents $N = {1, 2,..., n}$.
• Agent $i$ has valuation $v_i(a) = -|x_i - a|$, $x_i$ is location of agent $i$.
• The optimal location is the median of the agents' locations.

Axioms

• Pareto efficiency (PE): There is no outcome $a'$ that will improve any agent, without making another worse off compared to outcome $a$.
• Strategy-proofness (SP):
  • A social choice function $f$ is strategy-proof for all agents $i$, for all valuation profiles $v$.
  • For all valuations $v_i'$, $v_i(f(v)) \geq v_i(f(v_i', v_{-i}))$.
  • $v_{-i}$ is valuation profile for all agents except $i$.
  • No agent can benefit by misreporting their valuation, no matter what others do.
• Dictatorship:
  • Social choice function $f$ is a dictatorship when there exists agent $i$ such that $f(v) = \arg \max_{a \in A} v_i(a)$ for all valuation profiles $v$.
  • One agent always gets their most preferred outcome, regardless of others.

Impossibility Result

• Theorem 1. [Gibbard-Satterthwaite] If $|A| \geq 3$, any social choice function that is strategy-proof and onto is a dictatorship.

Circumventing the Impossibility

Single-Peaked Preferences

• Valuation function $v_i$ is single-peaked given alternative $x_i \in A$.
• For all $a, b \in A$, if $a < b \leq x_i$ or $a > b \geq x_i$, then $v_i(b) > v_i(a)$.
• Agent's preference decreases as alternative moves away from their most preferred.

Median Voter Theorem

• Theorem 2. Given all agents have single-peaked preferences, then median voter rule is strategy-proof and Pareto efficient.

Restricted Domain

• Domain of valuation profiles is restricted to circumvent the impossibility.

Condorcet Winner

• An alternative that beats every other alternative in pairwise election.
• $a \in A$ is Condorcet winner given for all $b \in A \setminus {a}$, $|{i \in N : v_i(a) > v_i(b)}| > \frac{n}{2}$.
• Preferred by majority to every other alternative.

Example 2: Electing a President

• Set of alternatives is $A = {\text{Trump, Biden, Other}}$.
• Set of agents is $N = {1, 2,..., n}$.

The Fourier Transform and its Applications

Introduction

• Decomposes a function into its constituent frequencies.
• Has applications in signal processing, image analysis, data compression.

Definition

• The Fourier Transform of a function $f(t)$ is: $F(\omega) = \int_{-\infty}^{\infty} f(t)e^{-j\omega t} dt$
  • $F(\omega)$ = frequency-domain representation of $f(t)$
  • $\omega$ = angular frequency
  • $j$ = imaginary unit
• The inverse Fourier Transform is: $f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega)e^{j\omega t} d\omega$

Properties

• Linearity: $F{af(t) + bg(t)} = aF(\omega) + bG(\omega)$
• Time Shifting: $F{f(t - t_0)} = e^{-j\omega t_0}F(\omega)$
• Frequency Shifting: $F{e^{j\omega_0 t}f(t)} = F(\omega - \omega_0)$
• Scaling: $F{f(at)} = \frac{1}{|a|}F(\frac{\omega}{a})$
• Convolution: $F{f(t) * g(t)} = F(\omega)G(\omega)$

Applications

Signal Processing

• Filtering: Removes unwanted frequencies from a signal.
• Spectral Analysis: Analyzes the frequency content of a signal.

Image Analysis

• Image Compression: Uses Discrete Cosine Transform (DCT) like in JPEG.
• Edge Detection: Detects edges in an image.

Data Compression

• Audio Compression: Uses Modified Discrete Cosine Transform (MDCT) in MP3.
• Video Compression: Uses Discrete Cosine Transform (DCT) in MPEG.