Alkanes and IUPAC Nomenclature

Alcanes

• Alcanes are saturated hydrocarbons containing only single carbon-carbon and carbon-hydrogen bonds.
• Their general formula is $C_nH_{2n+2}$, where $n$ indicates the number of carbon atoms.

IUPAC Nomenclature for Alcanes

• Identify the longest continuous chain of carbon atoms, which determines the base name of the alkane.
• Number the main chain starting closest to the first substituent.
• Alkyl groups are named by replacing the "-ane" suffix of the corresponding alkane with "-yl" (e.g., methyl, ethyl, propyl).
• The complete name should include the position number of each substituent, followed by the substituent name and the main chain name; use prefixes like "di-", "tri-", "tetra-" for multiple identical substituents, indicating each position.

Examples of Alcanes

• $CH_4$ is methane.
• $CH_3CH_3$ is ethane.
• $CH_3CH_2CH_3$ is propane.
• $CH_3(CH_2)_2CH_3$ is butane.
• $CH_3(CH_2)_3CH_3$ is pentane.
• $CH_3(CH_2)_4CH_3$ is hexane.
• $CH_3(CH_2)_5CH_3$ is heptane.
• $CH_3(CH_2)_6CH_3$ is octane.
• $CH_3(CH_2)_7CH_3$ is nonane.
• $CH_3(CH_2)_8CH_3$ is decane.

Physical Properties of Alcanes

• Boiling points increase with the number of carbon atoms due to increasing Van der Waals forces.
• Alcanes are nonpolar, making them insoluble in water but soluble in organic solvents.

Reactions with Halogens

• Alcanes react with halogens (chlorine, bromine) in the presence of light or heat via a radical substitution mechanism.

Mechanism of Radical Halogenation

• Initiation: Light or heat breaks halogen molecules into free radicals: $X_2 \xrightarrow{light \ or \ heat} 2X \cdot$
• Propagation: A halogen radical abstracts a hydrogen atom from the alkane, creating an alkyl radical and a hydrogen halide; the alkyl radical then reacts with another halogen molecule to form a haloalkane and regenerate a halogen radical:
  • $R-H + X \cdot \rightarrow R \cdot + HX$
  • $R \cdot + X_2 \rightarrow R-X + X \cdot$
• Termination: Free radicals combine to form stable products:
  • $X \cdot + X \cdot \rightarrow X_2$
  • $R \cdot + X \cdot \rightarrow RX$
  • $R \cdot + R \cdot \rightarrow R-R$

Reactivity and Selectivity in Halogenation

• Reactivity order among halogens: Fluorine > Chlorine > Bromine > Iodine.
• Halogenation selectivity: tertiary > secondary > primary hydrogens, due to the stability of the formed free radicals.

Example: Chlorination of Methane

• Chlorination yields a mixture of chlorinated products:
  • $CH_4 + Cl_2 \xrightarrow{light} CH_3Cl + HCl$
  • $CH_3Cl + Cl_2 \xrightarrow{light} CH_2Cl_2 + HCl$
  • $CH_2Cl_2 + Cl_2 \xrightarrow{light} CHCl_3 + HCl$
  • $CHCl_3 + Cl_2 \xrightarrow{light} CCl_4 + HCl$
• $CH_3Cl$ is chloromethane.
• $CH_2Cl_2$ is dichloromethane.
• $CHCl_3$ is trichloromethane (chloroform).
• $CCl_4$ is tetrachloromethane (carbon tetrachloride).

Key Concepts

• Alcanes: Saturated hydrocarbons with single bonds only.
• IUPAC Nomenclature: Rules for naming organic compounds.
• Radical Halogenation: Reaction of alkanes with halogens under light or heat.
• Reactivity and Selectivity: Factors affecting halogenation reactions.

Table of Relative Reactivity of Halogens

Real Numbers

Introduction

• Advanced calculus requires a precise understanding of real numbers.
• Focus is on basic properties and completeness.

The Field of Real Numbers

• Real numbers form a field, satisfying specific axioms.

Axioms for Addition

• Closure: $a + b$ is a real number.
• Commutativity: $a + b = b + a$.
• Associativity: $(a + b) + c = a + (b + c)$.
• Identity: There exists 0 such that $a + 0 = a$.
• Inverse: There exists -a such that $a + (-a) = 0$.
• These apply for all real numbers $a$, $b$, and $c$.

Axioms for Multiplication

• Closure: $ab$ is a real number.
• Commutativity: $ab = ba$.
• Associativity: $(ab)c = a(bc)$.
• Identity: There exists $1 \neq 0$ such that $a \cdot 1 = a$.
• Inverse: There exists $a^{-1}$ such that $a \cdot a^{-1} = 1$ if $a \neq 0$.
• These apply for all real numbers $a$, $b$, and $c$.

Distributive Axiom

• Expressed as: $a(b + c) = ab + ac$ for all real numbers $a, b, c$.

Order Axioms

• Trichotomy: For $a, b \in \mathbb{R}$, exactly one holds: $a < b$, $a = b$, or $a > b$.
• Transitivity: If $a < b$ and $b < c$, then $a < c$.
• Addition: If $a < b$, then $a + c < b + c$.
• Multiplication: If $a < b$ and $c > 0$, then $ac < bc$.

Completeness Axiom

• Every nonempty set of real numbers bounded above has a least upper bound (supremum) in $\mathbb{R}$.
• Completeness distinguishes real numbers from rational numbers.

Principle of Mathematical Induction

• If a set $S \subseteq \mathbb{N}$ satisfies $1 \in S$ and if $k \in S$, then $k + 1 \in S$, then $S = \mathbb{N}$.

The Laplace Transform

Definition

• For a function $f(t)$ defined for $t \geq 0$, the Laplace transform, denoted $F(s)$ or $\mathcal{L}{f(t)}$, is:

$\qquad \mathcal{L}{f(t)} = F(s) = \int_{0}^{\infty} e^{-st} f(t) dt$

• Where $s$ is a complex number.

Example 1

• Laplace transform of $f(t) = 1$, $t \geq 0$: $\qquad \mathcal{L}{1} = \int_{0}^{\infty} e^{-st} dt = \frac{1}{s}$, $s>0$.

Example 2

• Laplace transform of $f(t) = e^{at}$, $t \geq 0$: $\qquad \mathcal{L}{e^{at}} = \int_{0}^{\infty} e^{-st} e^{at} dt = \frac{1}{s-a}$, $s>a$.

Example 3

• Laplace transform of $f(t) = t$, $t \geq 0$: $\qquad \mathcal{L}{t} = \int_{0}^{\infty} e^{-st} t dt = \frac{1}{s^2}$, $s>0$.

Example 4

• Laplace transform of $f(t) = \sin(at)$, $t \geq 0$: $\qquad \mathcal{L}{\sin(at)} = \int_{0}^{\infty} e^{-st} \sin(at) dt =\frac{a}{s^2 + a^2}$, $s>0$.

Linearity of the Laplace Transform

• For functions $f(t)$ and $g(t)$ with Laplace transforms existing for $s>a$, and constants $c_1$, $c_2$:

$\qquad \mathcal{L}{c_1 f(t) + c_2 g(t)} = c_1 \mathcal{L}{f(t)} + c_2 \mathcal{L}{g(t)}$ for $s>a$

Laplace Transforms of Derivatives

• For a function $f(t)$ with an existing Laplace transform and $f'(t)$ continuous for $t \geq 0$:

$\qquad \mathcal{L}{f'(t)} = s \mathcal{L}{f(t)} - f(0)$ $\qquad \mathcal{L}{f''(t)} = s^2 \mathcal{L}{f(t)} - sf(0) - f'(0)$

• In general: $\qquad \mathcal{L}{f^{(n)}(t)} = s^n \mathcal{L}{f(t)} - s^{n-1} f(0) - s^{n-2} f'(0) - \dots - f^{(n-1)}(0)$

Matplotlib

Definition

• Matplotlib is a Python library used for creating static, animated, and interactive visualizations.
• It simplifies both basic and complex plot creation.

Key Features

• Enables creation of publication-quality plots.
• Supports interactive figures with zoom, pan, and update capabilities.
• Offers visual style and layout customization.
• Allows exporting to various file formats.
• Can be embedded in JupyterLab and graphical user interfaces (GUIs).
• Features a wide range of third-party packages extending its functionalities.

Basic Plotting Example

Code Snippet

import matplotlib.pyplot as plt
import numpy as np

## Data for plotting
t = np.arange(0.0, 2.0, 0.01)
s = 1 + np.sin(2 * np.pi * t)

fig, ax = plt.subplots()
ax.plot(t, s)

ax.set(xlabel='time (s)', ylabel='voltage (mV)',
       title='About as simple as it gets, folks')
ax.grid()

fig.savefig("test.png")
plt.show()

Plot Description

• The plot displays a sine wave.
• X-axis: Labeled "time (s)", ranges from 0.0 to 2.0.
• Y-axis: Labeled "voltage (mV)", ranges approximately from 0 to 2.
• Title: "About as simple as it gets, folks".
• A grid is present to aid in data visualization.