Measurements can be done directly using appropriate instruments, e.g. length, temperature, pressure, weight, etc.…
Other measurements can be gotten indirectly from other measurements by calculation; e.g. surface area, density (weight/volume), etc.… Those use derived units, such g/ml for the density.
When any measurement is derived from other measurements through calculation, you must be careful and use appropriate units in the measurements used for calculation of the derived measurement. In a calculated measurement, the units used in the primary measurements must be in the same units or converted into the same units. Either way, we depend on units to help set up and solve the calculation.
Unit analysis and Dimensional analysis: although those two concepts are different, both aim at analyzing units of measurements used in a given operation to insure that the units and the relationships used are appropriate. That is why they often confused and used alternatively.
Unit analysis: means using rules of multiplying and reducing fractions to solve problems involving different units: Kg = 1000g; 1000L = 1m3
Normally, conversion factors are quantities that are equal to one another such as 1m = 100 cm, in which both values describe a length.
Dimensional analysis: it’s the analysis of the relationships between different physical quantities by identifying their base quantities and units of measure and tracking these dimensions.
Relationships are between two values that are not necessarily a measure of the same quantity, such as the density of water 1.00 g/mL. Grams are a measure of mass while milliliters measure the volume, so this is considered a relationship rather than a conversion factor.
Therefore, in operations related to calculations of quantities, a dimensional and unit analysis must be done.
Example: Volume = Length x Width x Height
In this case, the units used expressing the measurements in the calculation must be the same for the three terms at the right side of the equation: e.g. (i) if the three measurements are expressed in cm unit, then the volume will be expressed in cm3, (ii) if the three measurements are expressed in m unit, then the volume will be expressed in m3. If the 3 measurements are expressed in different units, they must be converted in the same unit before calculation, otherwise the calculated results will be wrong and meaningless.
Density = Weight/Volume (ex: g/mL)
In this calculation, the units used for the mass and for the volume must be well indicated; for instance (i) if the weight is expressed in grams and the volume in cm3, the density will be in g/cm3, (ii) if the weight is expressed in kilogram and the volume in dm3, the density will be in kg/dm3, (iii) if the weight is expressed in g and the volume in mL, the density will be in g/mL.
From the formula of the density, we can calculate the volume if the weight is known:
Volume = Weight/Density= for ex: g/(g/cm3) = cm3
What’s the weight in g of 50 ml of a liquid whose density is 2Kg/L? Weight = density x volume = 2Kg/L x 50 ml = (2000g/1000ml) x 50 ml = 100g
Molar concentration = Number of moles of the solute/L of solution = M
The molar concentration, [M], is expressed as number of moles per liter of solution; this must be taken into consideration when calculating the molar concentration or any calculation involving molar concentrations.