6 Dimensional Analysis

Converting from One Unit to Another

Converting units using dimensional analysis makes working with large and small measurements more convenient.

Base and Derived Units

For most quantities, a unit is absolutely necessary to communicate values of that physical quantity. Imagine you need to buy some rope to tie something onto the roof of a car. How would you tell the salesperson how much rope you need without using some unit of measurement?

However, not all quantities require a unit of their own. Using physical laws, units of quantities can be expressed as combinations of units of other quantities. Therefore, only a small set of units is required. These units are called base units, and other units are derived units. Derived units are a matter of convenience, as they can be expressed in terms of basic units.

Different systems of units are based on different choices of base units. The most widely used system of units is the International System of Units, or SI. There are seven SI base units, and all other SI units can be derived from these base units.

The seven base SI units are: [Physical Quantity: unit symbol (unit name)]

• Length: m (meter)
• Mass: kg (kilogram)
• Time: s (second)
• Electric Current: A (Ampere)
• Thermodynamic Temperature: K (degrees Kelvin)
• Amount of Substance: mol (mole)
• Luminous Intensity: cd (candela)

The base units of SI are actually not the smallest set possible; smaller sets have been defined. For example, there are unit sets in which the electric and magnetic field have the same unit. This is based on physical laws that show that electric and magnetic fields are actually different manifestations of the same phenomenon.

Derived units are based on units from the SI system of units. For example, volume is a derived unit because volume is based on length. To calculate the volume of something, you multiply the width x length x height, all in meters. Therefore, the derived unit for volume is m³. Here is a list of some commonly derived units:

• Area: m²
• Volume: m³
• Velocity: m/s
• Acceleration: m/s²
• Density: g/mL or g/cm³
• Force: [latex]kg \cdot m/s^2[/latex], or the Newton (N)
• Energy: [latex]N \cdot m[/latex], or the Joule (J)

Dimensional Analysis

Sometimes, it is necessary to deal with measurements that are very small (as in the size of an atom) or very large (as in numbers of atoms). In these cases, it is often necessary to convert between units of metric measurement. For example, a mass measured in grams may be more convenient to work with if it was expressed in mg (10^–3 g). Converting between metric units is called unit analysis or dimensional analysis.

Unit analysis is a form of proportional reasoning where a given measurement can be multiplied by a known proportion or ratio to give a result having a different unit or dimension. Algebraically, we know that any number multiplied by one will be unchanged. If, however, the number has units, and we multiply it by a ratio containing units, the units in the number will multiply and divide by the units of the ratio, giving the original number (remember you are multiplying by one) but with different units.

This method can be generalized as multiply or divide a given number by a known ratio to find your answer. The given number is a numerical quantity (with its units). The ratios used are based upon the units and are set up so that the units in the denominator of the ratio match the numerator units of the given and the units in the numerator of the ratio match those in either the next ratio or the final answer. When these are multiplied, the given number will now have the correct units for your answer.

“Converting Units with Conversion Factors” – YouTube: How to convert units using conversion factors and canceling units.

Strategy for General Problem Solving

To convert a measured quantity to a different unit of measure without changing the relative amount, use a conversion factor.

Dimensional Analysis

Chemistry, along with other sciences and engineering, makes use of many different units. Some of the common ones include mass (ton, pounds, ounces, grains, grams), length (yard, feet, inches, meters), and energy (Joule, erg, kcal, eV). Since there are so many different units that can be used, it is necessary to be able to convert between the various units. To do this, one uses a conversion factor.

In mathematics, specifically algebra, a conversion factor is used to convert a measured quantity to a different unit of measure without changing the relative amount. To accomplish this, a ratio (fraction) is established that equals one (1). In the ratio, the conversion factor is a multiplier that, when applied to the original unit, converts the original unit into a new unit, by multiplication with the ratio.

When doing dimensional analysis problems, follow this list of steps:

1. Identify the given (see previous concept for additional information).
2. Identify conversion factors that will help you get from your original units to your desired unit.
3. Set up your equation so that your undesired units cancel out to give you your desired units. A unit will cancel out if it appears in both the numerator and the denominator during the equation.
4. Multiply through to get your final answer. Don’t forget the units and sig figs (significant figures)!

Flipping the Conversion Factor

Don’t forget that if need be, you can flip a conversion factor. After all, if a = b, then a/b = 1 and b/a = 1. For example, days are converted to hours by multiplying the days by the conversion factor of 24. The conversion can be reversed by dividing the hours by 24 to get days. The reciprocal 1/24 could be considered the reverse conversion factor for an hours-to-days conversion. The term “conversion factor” is the multiplier, not divisor, which yields the result.

Consider the relationship between feet and inches.

1 foot = 12 inches

1 foot/12 inches = 1 = 12 inches/1 foot.

Both fractions are equal to 1. If the units are ignored, the quotients do not numerically equal 1, but 1/12 or 12. However, with the inclusions of the units, both the numerators and denominators describe the exact same length, so the quotients are equal to 1. Since the two quotients are equal to 1, multiplying or dividing by the quotients is the same as multiplying or dividing by 1. It does not change the equation, only the relative numerical values within the various units.

If there is confusion regarding which quotient to use in the conversion, just make sure the units cancel out correctly. In the first equation, the unit (feet) is in both the numerator and denominator of the expression, so they cancel. The units behave just like numbers in products and quotients—they can be multiplied and divided.