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If a reaction takes 20 seconds at high concentration but 80 seconds at low concentration, did the rate get four times faster, or four times slower?

The speed of a chemical reaction is called its **reaction rate**, which measures how quickly reactants turn into products over time.

The **mean rate of reaction** is defined as the quantity of **reactant used** or **product formed** divided by the **time taken**:

$$\text{Mean Rate of Reaction} = \frac{\text{Quantity of Reactant Used or Product Formed}}{\text{Time Taken}}$$

Depending on the experiment, this quantity can be measured in grams of mass lost, cubic centimetres ($\text{cm}^3$) of gas produced, or moles transformed.

The **mean rate of reaction** is defined as the amount of **reactant used** or **product formed** divided by the **time taken**:
$$\text{Mean Rate of Reaction} = \frac{\text{Quantity of Reactant Used or Product Formed}}{\text{Time Taken}}$$

In 1850, German chemist Ludwig Wilhelmy conducted the first quantitative rate experiment, discovering that raw time alone doesn't directly equal speed without a mathematical conversion.

Because a faster reaction finishes in less time, rate is **inversely proportional** to time ($t$), which we write mathematically as $\text{rate} \propto \frac{1}{t}$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Rate vs Time</title>
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    }

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    .grid-line { stroke: #f1f5f9; stroke-width: 1; stroke-dasharray: 3 3; }
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      font-weight: 600;
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<body>

  <div class="container">
    
    <div class="btn-row">
      <button class="preset-btn" id="btn-fast">⚡ Fast</button>
      <button class="preset-btn active" id="btn-med">⏱ Medium</button>
      <button class="preset-btn" id="btn-slow">🐢 Slow</button>
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    <div class="graph-card">
      <svg viewBox="0 0 320 180" id="graph">
        
        <line x1="45" y1="40" x2="295" y2="40" class="grid-line"/>
        <line x1="45" y1="85" x2="295" y2="85" class="grid-line"/>
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        <line id="proj-x" x1="45" y1="145" x2="170" y2="145" class="guide-line"/>
        <line id="proj-y" x1="170" y1="145" x2="170" y2="145" class="guide-line"/>

        
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        <polygon points="300,142 306,145 300,148" fill="#94a3b8"/>
        <polygon points="42,20 45,14 48,20" fill="#94a3b8"/>

        
        <text x="300" y="162" text-anchor="end" class="axis-label">Time (t)</text>
        <text x="40" y="16" text-anchor="start" class="axis-label">Rate (1/t)</text>

        
        <path id="curve" class="curve-line" d=""/>

        
        <circle id="dot" cx="170" cy="85" r="6" fill="#22b7ff" stroke="#ffffff" stroke-width="2"/>
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    <div class="slider-card">
      <div class="bar-row">
        <span class="bar-label">Time (t)</span>
        <div class="bar-track">
          <div id="bar-time" class="bar-fill fill-time" style="width: 50%;"></div>
        </div>
        <span id="txt-time" class="bar-val">25 s</span>
      </div>

      <div class="bar-row">
        <span class="bar-label">Rate (1/t)</span>
        <div class="bar-track">
          <div id="bar-rate" class="bar-fill fill-rate" style="width: 50%;"></div>
        </div>
        <span id="txt-rate" class="bar-val">0.040 s⁻¹</span>
      </div>

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    const barTime = document.getElementById('bar-time');
    const barRate = document.getElementById('bar-rate');
    const txtTime = document.getElementById('txt-time');
    const txtRate = document.getElementById('txt-rate');

    const btnFast = document.getElementById('btn-fast');
    const btnMed = document.getElementById('btn-med');
    const btnSlow = document.getElementById('btn-slow');

    // Graph Mapping: t in [10, 50] -> x in [60, 285]
    // Rate = 1/t in [0.02, 0.10] -> y in [135, 30]
    function tToX(t) {
      return 60 + ((t - 10) / 40) * 225;
    }

    function rToY(r) {
      return 135 - ((r - 0.02) / 0.08) * 105;
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    // Build Curve Path mathematically
    const pts = [];
    for (let t = 10; t <= 50; t += 0.5) {
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      txtRate.textContent = r.toFixed(3) + ' s⁻¹';

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    btnFast.addEventListener('click', () => { slider.value = 10; update(10); });
    btnMed.addEventListener('click', () => { slider.value = 25; update(25); });
    btnSlow.addEventListener('click', () => { slider.value = 50; update(50); });

    update(25);
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</body>
</html>

How do we apply this inverse relationship to a real laboratory investigation like the classic disappearing cross experiment?

When sodium thiosulfate reacts with hydrochloric acid, yellow solid sulfur precipitate forms and gradually turns the clear mixture cloudy.

Scientists place a conical flask over a printed black cross and record the time ($t$) it takes for the cross to become completely invisible.

<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta name="aspect-ratio" content="4:3">
<title>Interpreting Rate Data</title>
<style>
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    color: #0284c7;
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    position: relative;
    width: 110px;
    height: 110px;
    border-radius: 50%;
    background: #ffffff;
    border: 4px solid #cbd5e1;
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    align-items: center;
    justify-content: center;
    overflow: hidden;
    margin: 6px 0;
    box-shadow: inset 0 2px 6px rgba(0,0,0,0.08);
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  .cross {
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    align-items: center;
    justify-content: center;
    font-size: 46px;
    font-weight: 900;
    color: #0f172a;
    line-height: 1;
    z-index: 1;
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  .cloud-layer {
    position: absolute;
    inset: 0;
    background: #fef08a;
    opacity: 0;
    z-index: 2;
    transition: opacity 0.1s linear;
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    gap: 4px;
    margin-top: 8px;
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    width: 100%;
    max-width: 580px;
    background: #ffffff;
    border-radius: 12px;
    padding: 10px 14px;
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    align-items: center;
    gap: 12px;
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</head>
<body>

  <div class="main-grid">
    
    <div class="flask-card">
      <div class="badge">10 g/dm³</div>
      <div class="view-container">
        <div class="cross">✕</div>
        <div class="cloud-layer" id="cloud1"></div>
      </div>
      <div class="hidden-tag" id="status1"></div>
      <div class="stats">
        <div class="stat-row">
          <span>Time (t)</span>
          <span class="stat-val" id="t1">100 s</span>
        </div>
        <div class="stat-row">
          <span>Rate (1/t)</span>
          <span class="stat-val" id="r1">0.010 s⁻¹</span>
        </div>
      </div>
    </div>

    
    <div class="flask-card">
      <div class="badge highlight">20 g/dm³ (2×)</div>
      <div class="view-container">
        <div class="cross">✕</div>
        <div class="cloud-layer" id="cloud2"></div>
      </div>
      <div class="hidden-tag" id="status2"></div>
      <div class="stats">
        <div class="stat-row">
          <span>Time (t)</span>
          <span class="stat-val" id="t2">50 s</span>
        </div>
        <div class="stat-row">
          <span>Rate (1/t)</span>
          <span class="stat-val" id="r2">0.020 s⁻¹</span>
        </div>
      </div>
    </div>
  </div>

  <div class="controls">
    <button class="btn" id="playBtn">▶ Play</button>
    <input type="range" id="timeSlider" class="slider" min="0" max="100" value="0" step="1">
    <div class="timer-display" id="timeText">0 s</div>
  </div>

<script>
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  let playing = false;
  let interval = null;

  const playBtn = document.getElementById('playBtn');
  const slider = document.getElementById('timeSlider');
  const timeText = document.getElementById('timeText');
  const cloud1 = document.getElementById('cloud1');
  const cloud2 = document.getElementById('cloud2');
  const status1 = document.getElementById('status1');
  const status2 = document.getElementById('status2');

  function update() {
    timeText.textContent = Math.round(time) + ' s';
    slider.value = time;

    // 10 g/dm3 flask: completely obscured at t = 100s
    const op1 = Math.min(1, time / 100);
    cloud1.style.opacity = op1 * 0.98;
    status1.textContent = op1 >= 0.98 ? '✕ Invisible' : '';

    // 20 g/dm3 flask: completely obscured at t = 50s (2x rate)
    const op2 = Math.min(1, time / 50);
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    status2.textContent = op2 >= 0.98 ? '✕ Invisible' : '';
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    playing = true;
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    else play();
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    time = parseFloat(e.target.value);
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  };

  update();
</script>
</body>
</html>

Doubling the sodium thiosulfate concentration from $10\text{ g/dm}^3$ to $20\text{ g/dm}^3$ cuts the time in half from $100\text{ s}$ to $50\text{ s}$, which doubles the rate from $0.010\text{ s}^{-1}$ to $0.020\text{ s}^{-1}$.

This proves the reaction rate is **directly proportional** to reactant concentration: multiplying the concentration by any factor multiplies the rate by that exact same factor.

A common mistake in exams is plotting raw time against concentration and expecting a straight line—raw time creates a downward curve, whereas plotting $\frac{1}{t}$ produces the true straight-line rate graph through the origin.

What happens when an experiment produces an expanding gas instead of a cloudy precipitate?

When calcium carbonate chips react with hydrochloric acid, carbon dioxide gas forms and steadily pushes out the plunger of a **gas syringe**.

We calculate the **mean rate** across any time window using the equation $\text{mean rate} = \frac{\Delta \text{volume}}{\Delta \text{time}}$, where $\Delta$ represents the difference between two measurements.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Mean Rate of Reaction Graph</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
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      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      user-select: none;
    }

    .btn-group {
      display: flex;
      gap: 8px;
      margin-bottom: 8px;
      width: 100%;
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      font-size: 13px;
      font-weight: 600;
      border-radius: 8px;
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      background: #ffffff;
      color: #4b5563;
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      text-align: center;
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      max-width: 420px;
      background: #ffffff;
      border-radius: 12px;
      border: 1px solid #e5e7eb;
      padding: 8px 10px 4px 6px;
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    svg {
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    .calc-card {
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      background: #ffffff;
      border-radius: 12px;
      border: 1px solid #e5e7eb;
      padding: 10px 14px;
      margin-top: 8px;
      display: flex;
      align-items: center;
      justify-content: space-around;
      box-shadow: 0 1px 3px rgba(0,0,0,0.05);
      font-size: 15px;
      font-weight: 600;
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    .fraction {
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      vertical-align: middle;
      text-align: center;
      padding: 0 4px;
      font-weight: 700;
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    .fraction-top {
      border-bottom: 2px solid #1e2945;
      padding-bottom: 2px;
      color: #0284c7;
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    .fraction-bottom {
      padding-top: 2px;
      color: #f97316;
    }

    .rate-result {
      color: #1e2945;
      font-size: 16px;
      font-weight: 800;
      background: #f0fdf4;
      border: 1px solid #86efac;
      padding: 6px 10px;
      border-radius: 8px;
    }
  </style>
</head>
<body>

  <div class="btn-group">
    <button class="preset-btn active" id="btn0" onclick="setPreset(0)">0 – 20 s</button>
    <button class="preset-btn" id="btn1" onclick="setPreset(1)">20 – 60 s</button>
    <button class="preset-btn" id="btn2" onclick="setPreset(2)">0 – 60 s</button>
  </div>

  <div class="graph-container">
    <svg viewBox="0 0 360 210">
      
      <g stroke="#f1f5f9" stroke-width="1">
        <line x1="45" y1="165" x2="330" y2="165" />
        <line x1="45" y1="135" x2="330" y2="135" />
        <line x1="45" y1="105" x2="330" y2="105" />
        <line x1="45" y1="75" x2="330" y2="75" />
        <line x1="45" y1="45" x2="330" y2="45" />
        <line x1="45" y1="15" x2="330" y2="15" />

        <line x1="45" y1="15" x2="45" y2="165" />
        <line x1="92" y1="15" x2="92" y2="165" />
        <line x1="140" y1="15" x2="140" y2="165" />
        <line x1="187" y1="15" x2="187" y2="165" />
        <line x1="235" y1="15" x2="235" y2="165" />
        <line x1="282" y1="15" x2="282" y2="165" />
        <line x1="330" y1="15" x2="330" y2="165" />
      </g>

      
      <line x1="45" y1="165" x2="335" y2="165" stroke="#94a3b8" stroke-width="1.5" />
      <line x1="45" y1="10" x2="45" y2="165" stroke="#94a3b8" stroke-width="1.5" />

      
      <text x="38" y="169" font-size="10" fill="#64748b" text-anchor="end">0</text>
      <text x="38" y="139" font-size="10" fill="#64748b" text-anchor="end">10</text>
      <text x="38" y="109" font-size="10" fill="#64748b" text-anchor="end">20</text>
      <text x="38" y="79" font-size="10" fill="#64748b" text-anchor="end">30</text>
      <text x="38" y="49" font-size="10" fill="#64748b" text-anchor="end">40</text>
      <text x="38" y="19" font-size="10" fill="#64748b" text-anchor="end">50</text>
      <text x="14" y="90" font-size="10" fill="#1e2945" font-weight="600" text-anchor="middle" transform="rotate(-90 14 90)">Volume (cm³)</text>

      
      <text x="45" y="180" font-size="10" fill="#64748b" text-anchor="middle">0</text>
      <text x="140" y="180" font-size="10" fill="#64748b" text-anchor="middle">20</text>
      <text x="235" y="180" font-size="10" fill="#64748b" text-anchor="middle">40</text>
      <text x="330" y="180" font-size="10" fill="#64748b" text-anchor="middle">60</text>
      <text x="187" y="198" font-size="10" fill="#1e2945" font-weight="600" text-anchor="middle">Time (s)</text>

      
      <polygon id="slopePoly" points="45,165 140,165 140,75" fill="#e0f2fe" opacity="0.4" />

      
      <path id="curvePath" d="" fill="none" stroke="#0f172a" stroke-width="2.5" stroke-linecap="round" />

      
      <line id="deltaTLine" x1="45" y1="165" x2="140" y2="165" stroke="#f97316" stroke-width="2.5" stroke-dasharray="4 3" />
      <text id="deltaTLabel" x="92" y="157" font-size="11" font-weight="700" fill="#ea580c" text-anchor="middle">Δt = 20 s</text>

      
      <line id="deltaVLine" x1="140" y1="165" x2="140" y2="75" stroke="#0284c7" stroke-width="2.5" stroke-dasharray="4 3" />
      <text id="deltaVLabel" x="146" y="123" font-size="11" font-weight="700" fill="#0284c7" text-anchor="start">ΔV = 30 cm³</text>

      
      <line id="rateChord" x1="45" y1="165" x2="140" y2="75" stroke="#9333ea" stroke-width="2.5" stroke-linecap="round" />

      
      <circle id="ptA" cx="45" cy="165" r="5" fill="#9333ea" stroke="#ffffff" stroke-width="2" />
      <circle id="ptB" cx="140" cy="75" r="5" fill="#9333ea" stroke="#ffffff" stroke-width="2" />
    </svg>
  </div>

  <div class="calc-card">
    <div style="color: #6b7280; font-size: 13px;">Mean Rate =</div>
    <div>
      <div class="fraction">
        <span class="fraction-top" id="mathTop">ΔV (30 cm³)</span>
        <span class="fraction-bottom" id="mathBot">Δt (20 s)</span>
      </div>
    </div>
    <div style="font-size: 16px;">=</div>
    <div class="rate-result" id="rateValue">1.50 cm³/s</div>
  </div>

  <script>
    // Exact curve coordinates
    // Time 0 -> 60 s maps to X: 45 -> 330 (span: 285 px)
    // Volume 0 -> 50 cm³ maps to Y: 165 -> 15 (span: 150 px)
    function getV(t) {
      // Smooth gas curve passing through (0,0), (20,30), (60,50)
      return 50 * (1 - Math.exp(-0.04581 * t));
    }

    function toSvg(t, v) {
      const x = 45 + (t / 60) * 285;
      const y = 165 - (v / 50) * 150;
      return { x, y };
    }

    // Generate accurate path
    const pathElem = document.getElementById('curvePath');
    let d = '';
    for (let t = 0; t <= 60; t += 1) {
      const pt = toSvg(t, getV(t));
      d += (t === 0 ? 'M ' : ' L ') + pt.x.toFixed(2) + ' ' + pt.y.toFixed(2);
    }
    pathElem.setAttribute('d', d);

    const presets = [
      { t1: 0, t2: 20, v1: 0, v2: 30, rate: '1.50' },
      { t1: 20, t2: 60, v1: 30, v2: 50, rate: '0.50' },
      { t1: 0, t2: 60, v1: 0, v2: 50, rate: '0.83' }
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      const pt2 = toSvg(p.t2, p.v2);

      // Update Points
      const ptA = document.getElementById('ptA');
      const ptB = document.getElementById('ptB');
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      ptB.setAttribute('cy', pt2.y);

      // Update Secant Chord
      const chord = document.getElementById('rateChord');
      chord.setAttribute('x1', pt1.x);
      chord.setAttribute('y1', pt1.y);
      chord.setAttribute('x2', pt2.x);
      chord.setAttribute('y2', pt2.y);

      // Update Delta Lines
      const deltaT = document.getElementById('deltaTLine');
      deltaT.setAttribute('x1', pt1.x);
      deltaT.setAttribute('y1', pt1.y);
      deltaT.setAttribute('x2', pt2.x);
      deltaT.setAttribute('y2', pt1.y);

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      deltaV.setAttribute('x1', pt2.x);
      deltaV.setAttribute('y1', pt1.y);
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      deltaV.setAttribute('y2', pt2.y);

      // Update Slope Polygon
      const poly = document.getElementById('slopePoly');
      poly.setAttribute('points', `${pt1.x},${pt1.y} ${pt2.x},${pt1.y} ${pt2.x},${pt2.y}`);

      // Update Labels
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      dtLabel.setAttribute('x', (pt1.x + pt2.x) / 2);
      dtLabel.setAttribute('y', pt1.y - 6);
      dtLabel.textContent = `Δt = ${p.t2 - p.t1} s`;

      const dvLabel = document.getElementById('deltaVLabel');
      dvLabel.setAttribute('x', pt2.x + 6);
      dvLabel.setAttribute('y', (pt1.y + pt2.y) / 2 + 4);
      dvLabel.textContent = `ΔV = ${p.v2 - p.v1} cm³`;

      // Update Math Calculation Card
      document.getElementById('mathTop').textContent = `${p.v2 - p.v1} cm³`;
      document.getElementById('mathBot').textContent = `${p.t2 - p.t1} s`;
      document.getElementById('rateValue').textContent = `${p.rate} cm³/s`;
    }

    setPreset(0);
  </script>
</body>
</html>

In the first 20 seconds, gas volume jumps from $0\text{ cm}^3$ to $30\text{ cm}^3$, giving a rapid mean rate of $\frac{30 - 0}{20 - 0} = 1.5\text{ cm}^3/\text{s}$.

Between $60\text{ s}$ and $80\text{ s}$, volume only increases by $4\text{ cm}^3$, giving a much slower rate of $\frac{58 - 54}{80 - 60} = 0.2\text{ cm}^3/\text{s}$.

The rate steadily decreases over time because reactant particles get used up, leading to fewer successful collisions between particles per second.

How can you tell from a numerical data table the exact moment the chemical reaction has stopped?

When two consecutive time points show the exact same volume—such as $58\text{ cm}^3$ at both $80\text{ s}$ and $100\text{ s}$—the reaction has reached a **plateau**.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
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    }
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      display: block;
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</head>
<body>

  
  <div class="controls">
    <button class="btn active" id="btn-fast" onclick="setTime(15)">1. Fast Start</button>
    <button class="btn" id="btn-mid" onclick="setTime(45)">2. Slowing</button>
    <button class="btn" id="btn-end" onclick="setTime(90)">3. Plateau</button>
  </div>

  
  <div class="graph-wrap">
    <svg viewBox="0 0 500 300" id="rateGraph">
      <defs>
        <pattern id="grid" width="42" height="41.66" patternUnits="userSpaceOnUse">
          <path d="M 42 0 L 0 0 0 41.66" fill="none" stroke="#f1f5f9" stroke-width="1"/>
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      <rect x="55" y="25" width="420" height="230" fill="url(#grid)" />

      
      <rect x="391" y="25" width="84" height="230" fill="#f0fdf4" opacity="0.7"/>
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      <line x1="55" y1="53.3" x2="475" y2="53.3" stroke="#10b981" stroke-dasharray="4 3" stroke-width="1.5"/>
      <text x="433" y="44" fill="#059669" font-size="12" font-weight="700" text-anchor="middle">Plateau (58 cm³)</text>

      
      <line x1="55" y1="255" x2="475" y2="255" stroke="#94a3b8" stroke-width="2"/>
      <line x1="55" y1="25" x2="55" y2="255" stroke="#94a3b8" stroke-width="2"/>

      
      <text x="48" y="259" fill="#64748b" font-size="11" text-anchor="end">0</text>
      <text x="48" y="176" fill="#64748b" font-size="11" text-anchor="end">20</text>
      <text x="48" y="93" fill="#64748b" font-size="11" text-anchor="end">40</text>
      <text x="48" y="57" fill="#059669" font-size="11" font-weight="700" text-anchor="end">58</text>
      <text x="20" y="145" fill="#1e2945" font-size="12" font-weight="700" text-anchor="middle" transform="rotate(-90 20 145)">Volume (cm³)</text>

      
      <text x="55" y="275" fill="#64748b" font-size="11" text-anchor="middle">0</text>
      <text x="139" y="275" fill="#64748b" font-size="11" text-anchor="middle">20</text>
      <text x="223" y="275" fill="#64748b" font-size="11" text-anchor="middle">40</text>
      <text x="307" y="275" fill="#64748b" font-size="11" text-anchor="middle">60</text>
      <text x="391" y="275" fill="#059669" font-size="11" font-weight="700" text-anchor="middle">80</text>
      <text x="475" y="275" fill="#64748b" font-size="11" text-anchor="middle">100</text>
      <text x="265" y="294" fill="#1e2945" font-size="12" font-weight="700" text-anchor="middle">Time (s)</text>

      
      <path id="curvePath" fill="none" stroke="#22b7ff" stroke-width="3.5" stroke-linecap="round"/>

      
      <polygon id="tangentTriangle" fill="#ffedd5" opacity="0.75"/>
      <line id="runLine" stroke="#ea580c" stroke-width="1.5" stroke-dasharray="3 2"/>
      <line id="riseLine" stroke="#ea580c" stroke-width="1.5" stroke-dasharray="3 2"/>
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      <text id="riseLabel" fill="#ea580c" font-size="10" font-weight="700" text-anchor="start">Rise</text>

      
      <line id="tangentLine" stroke="#f97316" stroke-width="2.5" stroke-linecap="round"/>

      
      <circle id="curvePoint" r="6" fill="#f97316" stroke="#ffffff" stroke-width="2"/>
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  </div>

  
  <div class="metric-bar">
    <div class="slider-container">
      <label for="timeRange" style="font-size:12px; font-weight:700; color:#64748b;">Time:</label>
      <input type="range" id="timeRange" min="2" max="98" value="15" oninput="updateTime(this.value)">
    </div>
    <div class="stat-badge" id="rateDisplay">
      Gradient: <span>1.18 cm³/s</span>
    </div>
  </div>

  <script>
    const X0 = 55, Y0 = 255;
    const X_SCALE = 4.2;      // 420px for 100s
    const Y_SCALE = 3.477;    // 58 cm3 at y = 53.3 -> 201.7px / 58 = 3.477px/cm3

    // Mathematical Reaction Rate Curve (Quadratic ease to plateau at t=80, V=58)
    function getV(t) {
      if (t <= 0) return 0;
      if (t >= 80) return 58;
      return 58 * (1 - Math.pow(1 - t / 80, 2));
    }

    // Exact Derivative / Instantaneous Rate (dV/dt)
    function getRate(t) {
      if (t < 0) return 0;
      if (t >= 80) return 0;
      return (58 * 2 * (80 - t)) / 6400;
    }

    function toSvgX(t) { return X0 + t * X_SCALE; }
    function toSvgY(v) { return Y0 - v * Y_SCALE; }

    // Build the SVG reaction curve path
    function initCurve() {
      let d = `M ${toSvgX(0)} ${toSvgY(0)}`;
      for (let t = 1; t <= 100; t += 1) {
        d += ` L ${toSvgX(t)} ${toSvgY(getV(t))}`;
      }
      document.getElementById('curvePath').setAttribute('d', d);
    }

    function renderTangent(t) {
      const v = getV(t);
      const rate = getRate(t);

      const px = toSvgX(t);
      const py = toSvgY(v);

      // Tangent line span
      const dt = 16;
      const t1 = Math.max(0, t - dt);
      const t2 = Math.min(100, t + dt);
      const v1 = v + rate * (t1 - t);
      const v2 = v + rate * (t2 - t);

      const x1 = toSvgX(t1);
      const y1 = toSvgY(v1);
      const x2 = toSvgX(t2);
      const y2 = toSvgY(v2);

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      tLine.setAttribute('x1', x1);
      tLine.setAttribute('y1', y1);
      tLine.setAttribute('x2', x2);
      tLine.setAttribute('y2', y2);

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      pt.setAttribute('cy', py);

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      const rLine = document.getElementById('runLine');
      const riseLine = document.getElementById('riseLine');
      const runLbl = document.getElementById('runLabel');
      const riseLbl = document.getElementById('riseLabel');

      if (rate > 0.05) {
        tri.setAttribute('points', `${x1},${y1} ${x2},${y1} ${x2},${y2}`);
        tri.style.display = 'block';

        rLine.setAttribute('x1', x1);
        rLine.setAttribute('y1', y1);
        rLine.setAttribute('x2', x2);
        rLine.setAttribute('y2', y1);
        rLine.style.display = 'block';

        riseLine.setAttribute('x1', x2);
        riseLine.setAttribute('y1', y1);
        riseLine.setAttribute('x2', x2);
        riseLine.setAttribute('y2', y2);
        riseLine.style.display = 'block';

        runLbl.setAttribute('x', (x1 + x2) / 2);
        runLbl.setAttribute('y', y1 + 12);
        runLbl.style.display = 'block';

        riseLbl.setAttribute('x', x2 + 4);
        riseLbl.setAttribute('y', (y1 + y2) / 2 + 3);
        riseLbl.style.display = 'block';
      } else {
        tri.style.display = 'none';
        rLine.style.display = 'none';
        riseLine.style.display = 'none';
        runLbl.style.display = 'none';
        riseLbl.style.display = 'none';
      }

      // Metric update
      const display = document.getElementById('rateDisplay');
      if (rate === 0) {
        display.innerHTML = `Gradient: <span>0 cm³/s (Stopped)</span>`;
      } else {
        display.innerHTML = `Gradient: <span>${rate.toFixed(2)} cm³/s</span>`;
      }
    }

    function updateTime(val) {
      const t = parseFloat(val);
      renderTangent(t);

      // Sync buttons state
      document.getElementById('btn-fast').classList.toggle('active', Math.abs(t - 15) < 8);
      document.getElementById('btn-mid').classList.toggle('active', Math.abs(t - 45) < 8);
      document.getElementById('btn-end').classList.toggle('active', t >= 80);
    }

    function setTime(val) {
      document.getElementById('timeRange').value = val;
      updateTime(val);
    }

    initCurve();
    updateTime(15);
  </script>
</body>
</html>

To find the instantaneous rate at any specific second rather than an average interval, draw a **tangent line** to the curve at that point and calculate its gradient ($\frac{\text{rise}}{\text{run}}$).

Whenever you analyze rate questions in an exam, check whether you are looking at raw time ($t$), change in amount over time ($\frac{\Delta V}{\Delta t}$), or the relative rate factor ($\frac{1}{t}$) before drawing your conclusion.

science

Interpreting Rate Data from Experiments. Analyse numerical tables and trends from rate investigations (such as the disappearing cross or gas syringe method) to draw valid scientific conclusions.

GCSE Chemistry: Measuring and Calculating Rates of Reaction

Rates of Reaction: Practical Videos and Interpreting Rate Data

Rate of Reaction Graphs & Practical Data Interpretation (GCSE)

Pedagogical strategy focuses on bridging concrete experimental observations to mathematical models. Emphasize early data hygiene (identifying anomalies before taking averages) and mathematical fluency in transforming inverse raw time data into linear rate measures ($1/t$) to establish direct proportionality.

Work through this lesson with your tutor.

Interpreting Rate Data from Experiments

Converting Time to Rate

The Disappearing Cross Investigation