Rearranging equations to change the subject

Quantitative chemistry • Volumes of gases

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How to isolate V from c = n / V?

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Multiply both sides by V then divide by c to get V = n / c.

Key concepts

What you'll likely be quizzed about

Subject of an equation

The subject of an equation is the symbol that stands alone on one side of the equals sign. Changing the subject means isolating that symbol so that it appears by itself. Isolation requires removal of other terms that multiply, divide, add or subtract from the chosen symbol using inverse operations on both sides of the equality. The subject becomes the dependent quantity that can be calculated directly from the other symbols.

Inverse operations and order of reversal

Inverse operations reverse the effect of an operation: subtraction reverses addition, division reverses multiplication, roots reverse powers, and reciprocals reverse reciprocals. Rearrangement follows the reverse order of operations: remove additions/subtractions first, then remove multiplications/divisions, and resolve powers or roots last. The same inverse operation applies to both sides of the equation to preserve equality.

Fractions, multiple terms and collecting like terms

Terms that do not multiply the subject must move across the equals sign using addition or subtraction. Terms that multiply the subject require division by the whole multiplying factor. When the subject appears inside a fraction, multiply both sides by the denominator to remove the fraction. When the subject appears in more than one term, algebraic rearrangement requires collecting like terms or factorising to express the subject as a single factor.

Application to gas equations and unit limitations

Application examples include pV = nRT and concentration relations c = n/V. Rearrangement yields V = nRT/p for the ideal gas law and V = n/c for concentration. Units determine the correct algebraic result: R requires SI units (R = 8.31 J mol^-1 K^-1) so pressure must be in pascals and volume in cubic metres. Failure to convert units causes incorrect numerical answers even when algebraic rearrangement is correct.

Key notes

Important points to keep in mind

Perform inverse operations on both sides to keep the equation balanced.

Reverse the order of operations when isolating the subject: undo addition/subtraction first, then multiplication/division, then powers or roots.

Convert units to SI when using R = 8.31 in pV = nRT: p in Pa, V in m3, T in K.

Remove denominators by multiplying both sides by the denominator before isolating the variable.

Collect like terms or factorise when the subject appears in multiple terms.

Check rearrangements by substituting numerical values into the original and the rearranged equation.

Apply physical limits after algebraic steps (for example, volume and temperature must be positive).

Use consistent units across all terms to avoid numerical errors even when algebra is correct.

Reciprocals and roots introduce sign or domain considerations that require context checks.

When using molar gas volumes given in dm3, include explicit conversion to m3 if R in SI units is used.