Changing units without losing your place

Unit conversions go wrong in one of two ways: a factor of a thousand in the wrong direction, or a conversion that needs a molar mass done without one. Both are caught by the same habit: carry the units through the arithmetic as if they were numbers.

Multiply by one

A ratio of two equal quantities is equal to one, and multiplying by one changes nothing but the unit. 2.5 mg × (1000 µg ÷ 1 mg) = 2500 µg. The mg cancels on paper, so the direction of the factor is never in doubt. Set every conversion up this way and the thousand-fold slips stop.

The prefixes

Mass to moles needs a molar mass

Grams and moles are different kinds of quantity, so no prefix table converts between them; the molar mass is the bridge. 50 µg of glucose:

50 µg × (1 g ÷ 10⁶ µg) = 5.0 × 10⁻⁵ g
5.0 × 10⁻⁵ g ÷ 180.156 g/mol = 2.775e-7 mol
= 277.5 nmol

Concentrations

• mg/mL → mmol/L. 1 mg/mL is 1 g/L; divide by the molar mass for mol/L; multiply by 1000 for mmol/L. In one step: mmol/L = mg/mL × 1000 ÷ M. For glucose, 1 mg/mL = 1000 ÷ 180.156 = 5.551 mmol/L.
• mmol/L → mg/mL. The reverse: mg/mL = mmol/L × M ÷ 1000.
• Percent w/v → mg/mL. Multiply by 10: a 2% w/v solution is 20 mg/mL.
• ppm. For dilute aqueous solutions, 1 ppm ≈ 1 mg/L ≈ 1 µg/mL.

Two slips worth naming

• µ and m look alike in handwriting, and “u” is often typed for µ. A factor of a thousand hides there; read a handwritten unit twice.
• Dividing by a thousand when you meant to multiply. 250 µL is 0.250 mL, not 250 000 mL. If the result is absurd, the factor went the wrong way.