Lecture 4: Direction

Bonds Point — Vectors in Chemistry

DIRECTION

The Hook

Look at these three molecules:

Molecule	Shape	Bond Angle
CO₂	Linear	180°
H₂O	Bent	104.5°
CH₄	Tetrahedral	109.5°

What do you notice?

Bonds point. They have direction. The angle between them matters enormously for a molecule’s properties.

Water is bent → it’s polar → it dissolves salt, forms hydrogen bonds, and makes life possible.

CO₂ is linear → it’s nonpolar → it’s a gas at room temperature.

Same atoms can give completely different molecules depending on direction.

When something points, when angles matter, when orientation determines behavior — you’re seeing DIRECTION.

The mathematical tool that captures this: vectors.

The Tool: Vectors

A vector is a quantity with both magnitude (size) and direction.

Notation

\[\vec{v} = \begin{pmatrix} v_x \\ v_y \\ v_z \end{pmatrix}\]

In chemistry, we use vectors for:

Bond directions
Dipole moments
Electric fields
Orbital orientations

Operations

Magnitude (length): \[|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}\]

Dot product (angle relationship): \[\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta\]

Key insight: The dot product tells you how aligned two vectors are:

Parallel → dot product is maximum (positive)
Perpendicular → dot product is zero
Opposite → dot product is negative

Interactive: Bond Angles

Move the slider to see how bond angle affects molecular polarity:

Code

viewof angle = Inputs.range([90, 180], {value: 104.5, step: 0.5, label: "H-O-H Angle (°)"})

{
  const svg = d3.create("svg")
    .attr("width", 400)
    .attr("height", 300)
    .attr("viewBox", "-200 -150 400 300");

  // Central atom (oxygen)
  svg.append("circle")
    .attr("cx", 0)
    .attr("cy", 0)
    .attr("r", 20)
    .attr("fill", "#ef4444");

  svg.append("text")
    .attr("x", 0)
    .attr("y", 5)
    .attr("text-anchor", "middle")
    .attr("fill", "white")
    .attr("font-weight", "bold")
    .text("O");

  // Calculate hydrogen positions
  const radians = angle * Math.PI / 180;
  const bondLength = 80;

  const h1x = bondLength * Math.cos(radians/2);
  const h1y = -bondLength * Math.sin(radians/2);
  const h2x = bondLength * Math.cos(radians/2);
  const h2y = bondLength * Math.sin(radians/2);

  // Bonds
  svg.append("line")
    .attr("x1", 0).attr("y1", 0)
    .attr("x2", h1x).attr("y2", h1y)
    .attr("stroke", "#666")
    .attr("stroke-width", 4);

  svg.append("line")
    .attr("x1", 0).attr("y1", 0)
    .attr("x2", h2x).attr("y2", h2y)
    .attr("stroke", "#666")
    .attr("stroke-width", 4);

  // Hydrogens
  svg.append("circle")
    .attr("cx", h1x).attr("cy", h1y)
    .attr("r", 15)
    .attr("fill", "#3b82f6");

  svg.append("circle")
    .attr("cx", h2x).attr("cy", h2y)
    .attr("r", 15)
    .attr("fill", "#3b82f6");

  svg.append("text")
    .attr("x", h1x).attr("y", h1y + 4)
    .attr("text-anchor", "middle")
    .attr("fill", "white")
    .attr("font-size", "12")
    .text("H");

  svg.append("text")
    .attr("x", h2x).attr("y", h2y + 4)
    .attr("text-anchor", "middle")
    .attr("fill", "white")
    .attr("font-size", "12")
    .text("H");

  // Dipole arrow
  const polarity = (180 - angle) / 90; // 0 at 180°, 1 at 90°
  const arrowLength = polarity * 60;

  svg.append("line")
    .attr("x1", 0).attr("y1", 0)
    .attr("x2", -arrowLength).attr("y2", 0)
    .attr("stroke", "#10b981")
    .attr("stroke-width", 3)
    .attr("marker-end", "url(#arrowhead)");

  // Arrowhead definition
  svg.append("defs").append("marker")
    .attr("id", "arrowhead")
    .attr("markerWidth", 10)
    .attr("markerHeight", 7)
    .attr("refX", 9)
    .attr("refY", 3.5)
    .attr("orient", "auto")
    .append("polygon")
    .attr("points", "0 0, 10 3.5, 0 7")
    .attr("fill", "#10b981");

  // Label
  svg.append("text")
    .attr("x", 0)
    .attr("y", 120)
    .attr("text-anchor", "middle")
    .attr("font-size", "14")
    .text(`Polarity: ${(polarity * 100).toFixed(0)}%`);

  return svg.node();
}

Notice: As the angle approaches 180° (linear), polarity drops to zero. The bond dipoles cancel!

Why bond angles matter:

H₂O at 104.5°: Bent → polar → hydrogen bonding → liquid at room temperature, dissolves ionic compounds, high heat capacity (regulates climate)
CO₂ at 180°: Linear → nonpolar → no hydrogen bonding → gas at room temperature, different solubility behavior

Same atoms. Different angles. Completely different chemistry.

Practice: DIRECTION Mastery

Summary

Concept	Formula	Chemistry Use
Vector	$\vec{v} = (v_x, v_y, v_z)$	Bond direction
Magnitude	$\\|\vec{v}\\| = \sqrt{v_x^2 + v_y^2 + v_z^2}$	Bond length
Dot product	$\vec{a} \cdot \vec{b} = \\|\vec{a}\\|\\|\vec{b}\\|\cos\theta$	Bond angle
Vector sum	$\vec{a} + \vec{b}$	Net dipole moment

Exercises

Water geometry: If the H-O bond length is 0.96 Å and the H-O-H angle is 104.5°, calculate the H-H distance.
Dipole cancellation: Explain why CCl₄ has no net dipole moment despite having four polar C-Cl bonds.
Vector addition: Two bond dipoles each have magnitude 1.5 D and point at 120° to each other. What is the magnitude of the net dipole?

Next: Lecture 5: Angles and Projections →

--- title: "Lecture 4: Direction" subtitle: "Bonds Point — Vectors in Chemistry" format: html: include-after-body: - text: | <script src="../js/practice-widget.js"></script> --- <span class="primitive-badge primitive-DIRECTION">DIRECTION</span> ## The Hook {.hook-section} Look at these three molecules: | Molecule | Shape | Bond Angle | |----------|-------|------------| | CO₂ | Linear | 180° | | H₂O | Bent | 104.5° | | CH₄ | Tetrahedral | 109.5° | **What do you notice?** Bonds *point*. They have direction. The angle between them matters enormously for a molecule's properties. Water is bent → it's polar → it dissolves salt, forms hydrogen bonds, and makes life possible. CO₂ is linear → it's nonpolar → it's a gas at room temperature. Same atoms can give completely different molecules depending on **direction**. --- ::: {.recognition-callout} When something **points**, when **angles matter**, when **orientation** determines behavior — you're seeing **DIRECTION**. The mathematical tool that captures this: **vectors**. ::: --- ## The Tool: Vectors {.tool-section} A **vector** is a quantity with both magnitude (size) and direction. ### Notation $$\vec{v} = \begin{pmatrix} v_x \\ v_y \\ v_z \end{pmatrix}$$ In chemistry, we use vectors for: - Bond directions - Dipole moments - Electric fields - Orbital orientations ### Operations **Magnitude (length):** $$|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$$ **Dot product (angle relationship):** $$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$$ **Key insight:** The dot product tells you how aligned two vectors are: - Parallel → dot product is maximum (positive) - Perpendicular → dot product is zero - Opposite → dot product is negative --- ## Interactive: Bond Angles Move the slider to see how bond angle affects molecular polarity: ```{ojs} //| echo: false viewof angle = Inputs.range([90, 180], {value: 104.5, step: 0.5, label: "H-O-H Angle (°)"}) { const svg = d3.create("svg") .attr("width", 400) .attr("height", 300) .attr("viewBox", "-200 -150 400 300"); // Central atom (oxygen) svg.append("circle") .attr("cx", 0) .attr("cy", 0) .attr("r", 20) .attr("fill", "#ef4444"); svg.append("text") .attr("x", 0) .attr("y", 5) .attr("text-anchor", "middle") .attr("fill", "white") .attr("font-weight", "bold") .text("O"); // Calculate hydrogen positions const radians = angle * Math.PI / 180; const bondLength = 80; const h1x = bondLength * Math.cos(radians/2); const h1y = -bondLength * Math.sin(radians/2); const h2x = bondLength * Math.cos(radians/2); const h2y = bondLength * Math.sin(radians/2); // Bonds svg.append("line") .attr("x1", 0).attr("y1", 0) .attr("x2", h1x).attr("y2", h1y) .attr("stroke", "#666") .attr("stroke-width", 4); svg.append("line") .attr("x1", 0).attr("y1", 0) .attr("x2", h2x).attr("y2", h2y) .attr("stroke", "#666") .attr("stroke-width", 4); // Hydrogens svg.append("circle") .attr("cx", h1x).attr("cy", h1y) .attr("r", 15) .attr("fill", "#3b82f6"); svg.append("circle") .attr("cx", h2x).attr("cy", h2y) .attr("r", 15) .attr("fill", "#3b82f6"); svg.append("text") .attr("x", h1x).attr("y", h1y + 4) .attr("text-anchor", "middle") .attr("fill", "white") .attr("font-size", "12") .text("H"); svg.append("text") .attr("x", h2x).attr("y", h2y + 4) .attr("text-anchor", "middle") .attr("fill", "white") .attr("font-size", "12") .text("H"); // Dipole arrow const polarity = (180 - angle) / 90; // 0 at 180°, 1 at 90° const arrowLength = polarity * 60; svg.append("line") .attr("x1", 0).attr("y1", 0) .attr("x2", -arrowLength).attr("y2", 0) .attr("stroke", "#10b981") .attr("stroke-width", 3) .attr("marker-end", "url(#arrowhead)"); // Arrowhead definition svg.append("defs").append("marker") .attr("id", "arrowhead") .attr("markerWidth", 10) .attr("markerHeight", 7) .attr("refX", 9) .attr("refY", 3.5) .attr("orient", "auto") .append("polygon") .attr("points", "0 0, 10 3.5, 0 7") .attr("fill", "#10b981"); // Label svg.append("text") .attr("x", 0) .attr("y", 120) .attr("text-anchor", "middle") .attr("font-size", "14") .text(`Polarity: ${(polarity * 100).toFixed(0)}%`); return svg.node(); } ``` **Notice:** As the angle approaches 180° (linear), polarity drops to zero. The bond dipoles cancel! --- ::: {.chemistry-connection} **Why bond angles matter:** - **H₂O at 104.5°**: Bent → polar → hydrogen bonding → liquid at room temperature, dissolves ionic compounds, high heat capacity (regulates climate) - **CO₂ at 180°**: Linear → nonpolar → no hydrogen bonding → gas at room temperature, different solubility behavior Same atoms. Different angles. Completely different chemistry. ::: --- ## Practice: DIRECTION Mastery <div id="practice-direction" class="practice-container"></div> <script> document.addEventListener('DOMContentLoaded', function() { if (typeof ChemPractice !== 'undefined') { new ChemPractice('practice-direction', { primitive: 'DIRECTION', topic: 'bond_angles', apiUrl: 'https://api.thebeakers.com' }); } }); </script> --- ## Summary | Concept | Formula | Chemistry Use | |---------|---------|---------------| | Vector | $\vec{v} = (v_x, v_y, v_z)$ | Bond direction | | Magnitude | $\|\vec{v}\| = \sqrt{v_x^2 + v_y^2 + v_z^2}$ | Bond length | | Dot product | $\vec{a} \cdot \vec{b} = \|\vec{a}\|\|\vec{b}\|\cos\theta$ | Bond angle | | Vector sum | $\vec{a} + \vec{b}$ | Net dipole moment | --- ## Exercises 1. **Water geometry**: If the H-O bond length is 0.96 Å and the H-O-H angle is 104.5°, calculate the H-H distance. 2. **Dipole cancellation**: Explain why CCl₄ has no net dipole moment despite having four polar C-Cl bonds. 3. **Vector addition**: Two bond dipoles each have magnitude 1.5 D and point at 120° to each other. What is the magnitude of the net dipole? --- **Next:** [Lecture 5: Angles and Projections →](05-projections.qmd)