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Imagine trying to count individual grains of sand in a bucket — atoms are billions of times smaller, so counting them one by one is impossible. Instead, chemists count particles simply by weighing them.

The bridge between mass and particle count is the **mole**, a counting unit equal to $6.022 \times 10^{23}$ particles. The **relative formula mass** ($M_r$) tells you the mass in grams of one mole of that substance.

The core relationship connects all three variables in a single equation: $$\text{mass} = \text{moles} \times M_r$$

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If your balance measures grams in the lab, how do you find out exactly how many moles of a compound you have?

Since multiplication and division are inverse operations, dividing both sides of $\text{mass} = \text{moles} \times M_r$ by $M_r$ cancels it out on the right side and isolates **moles**.

$$\text{moles} = \frac{\text{mass}}{M_r}$$

For example, calculate the moles in $25\text{ g}$ of calcium carbonate ($\text{CaCO}_3$), which has an $M_r$ of $100\text{ g/mol}$.

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What if you synthesize a brand-new mystery compound and need to determine its identity using its formula mass?

## Rearranging to Solve for Formula Mass ($M_r$)

Using the same algebraic principle, divide both sides of the starting equation by **moles** to isolate $M_r$ on one side.

$$M_r = \frac{\text{mass}}{\text{moles}}$$

Suppose a $0.5\text{ mol}$ sample of an unknown gas has a mass of $22\text{ g}$. Substituting into the formula gives $M_r = \frac{22\text{ g}}{0.5\text{ mol}} = 44\text{ g/mol}$, identifying the gas as carbon dioxide ($\text{CO}_2$).

A common mistake is forgetting that mass must always be in **grams** — if a question provides mass in kilograms or milligrams, convert to grams first before rearranging.

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    padding: 14px 20px;
    display: flex;
    align-items: center;
    justify-content: center;
    gap: 12px;
    box-shadow: 0 2px 8px rgba(0,0,0,0.05);
    min-height: 64px;
  }

  .eq-target {
    font-size: 22px;
    font-weight: 700;
  }

  .eq-equals {
    font-size: 20px;
    color: #94a3b8;
    font-weight: 600;
  }

  .eq-body {
    display: inline-flex;
    align-items: center;
    font-size: 20px;
    font-weight: 700;
  }

  .fraction {
    display: inline-flex;
    flex-direction: column;
    align-items: center;
    vertical-align: middle;
    padding: 0 4px;
  }

  .frac-top {
    border-bottom: 2px solid #1e2945;
    padding: 0 6px 2px;
    line-height: 1.1;
  }

  .frac-bot {
    padding: 2px 6px 0;
    line-height: 1.1;
  }

  .target-m { color: #2563eb; }
  .target-n { color: #16a34a; }
  .target-mr { color: #ca8a04; }
  .var-m { color: #2563eb; }
  .var-n { color: #16a34a; }
  .var-mr { color: #ca8a04; }
  .op-mul { color: #64748b; margin: 0 6px; }
</style>
</head>
<body>

<div class="main-wrap">
  
  <div class="tab-row" role="tablist">
    <button class="tab-btn active" id="btn-m" onclick="setTarget('m')">Find m</button>
    <button class="tab-btn" id="btn-n" onclick="setTarget('n')">Find n</button>
    <button class="tab-btn" id="btn-mr" onclick="setTarget('mr')">Find Mᵣ</button>
  </div>

  
  <div class="triangle-stage" id="stage">
    <svg class="triangle-svg" viewBox="0 0 300 260">
      
      <defs>
        <clipPath id="tri-clip">
          <polygon points="150,15 285,245 15,245" />
        </clipPath>
      </defs>

      
      <g clip-path="url(#tri-clip)">
        
        <polygon class="tri-zone zone-top" points="0,0 300,0 300,130 0,130" onclick="setTarget('m')" />
        
        <polygon class="tri-zone zone-bl" points="0,130 150,130 150,260 0,260" onclick="setTarget('n')" />
        
        <polygon class="tri-zone zone-br" points="150,130 300,130 300,260 150,260" onclick="setTarget('mr')" />
      </g>

      
      <polygon points="150,15 285,245 15,245" fill="none" stroke="#cbd5e1" stroke-width="3" stroke-linejoin="round" />
      <line x1="82" y1="130" x2="218" y2="130" stroke="#cbd5e1" stroke-width="3" stroke-linecap="round" />
      <line x1="150" y1="130" x2="150" y2="245" stroke="#cbd5e1" stroke-width="3" stroke-linecap="round" />

      
      
      <text class="tri-text text-top" x="150" y="80">m</text>
      <text class="tri-unit unit-top" x="150" y="105">mass (g)</text>

      
      <text class="tri-text text-bl" x="88" y="180">n</text>
      <text class="tri-unit unit-bl" x="88" y="205">moles (mol)</text>

      
      <text class="tri-text text-br" x="212" y="180">Mᵣ</text>
      <text class="tri-unit unit-br" x="212" y="205">formula mass</text>

      
      <text class="op-label" x="150" y="188">×</text>
      <text class="op-label" x="65" y="125">÷</text>
      <text class="op-label" x="235" y="125">÷</text>
    </svg>
  </div>

  
  <div class="formula-card" id="formula-card">
    
  </div>
</div>

<script>
  const states = {
    m: {
      target: 'm',
      targetClass: 'target-m',
      body: '<span class="var-n">n</span><span class="op-mul">×</span><span class="var-mr">Mᵣ</span>'
    },
    n: {
      target: 'n',
      targetClass: 'target-n',
      body: '<div class="fraction"><span class="frac-top var-m">m</span><span class="frac-bot var-mr">Mᵣ</span></div>'
    },
    mr: {
      target: 'Mᵣ',
      targetClass: 'target-mr',
      body: '<div class="fraction"><span class="frac-top var-m">m</span><span class="frac-bot var-n">n</span></div>'
    }
  };

  let current = 'm';

  function setTarget(key) {
    current = key;

    // Update active tab
    ['m', 'n', 'mr'].forEach(k => {
      document.getElementById('btn-' + k).classList.toggle('active', k === key);
    });

    // Update triangle classes
    const stage = document.getElementById('stage');
    stage.className = 'triangle-stage active-' + key;

    // Update dynamic formula output
    const data = states[key];
    const card = document.getElementById('formula-card');
    card.innerHTML = `
      <span class="eq-target ${data.targetClass}">${data.target}</span>
      <span class="eq-equals">=</span>
      <div class="eq-body">${data.body}</div>
    `;
  }

  // Initial render
  setTarget('m');
</script>

</body>
</html>

Whenever you encounter a quantitative chemistry problem, check your units first, identify your unknown, and let inverse operations do the heavy lifting.

science

Rearranging the Mole, Mass, and Formula Mass Equation. Change the subject of mass = moles × Mr to solve for moles or relative formula mass Mr algebraically.

GCSE Chemistry: Moles and Mass Calculations

Moles and Masses KS4 Chemistry Lesson Resources

Linking Moles, Mass & Mr Revision Notes & Worked Solutions

Pedagogical best practice for teaching quantitative chemistry equation manipulation at Key Stage 4 centres on explicit dual-coding and gradual release ('I Do, We Do, You Do'). Teachers should first solidify conceptual understanding of the mole as a counting unit linked to mass via molar mass before introducing pure algebraic manipulation. While formula triangles provide useful initial scaffolding for lower-attaining students, lessons should transition pupils toward formal algebraic balancing (inverse operations) to prevent cognitive errors in multi-step chemical stoichiometry and A-Level transition. Emphasise dimensional analysis and unit checks (e.g., verifying that mass is in grams before substitution) in every worked example.

Rearranging the Mole, Mass, and Formula Mass Equation

The Fundamental Equation

Rearranging to Solve for Moles

Rearranging to Solve for Formula Mass ( $M_r$ )

Rearranging the Mole, Mass, and Formula Mass Equation

The Fundamental Equation

Rearranging to Solve for Moles

Rearranging to Solve for Formula Mass (MrM_rMr​)

Rearranging to Solve for Formula Mass ( $M_r$ )