#### Learning Objective

- Apply knowledge of dimensional analysis to convert between units in chemistry problems

#### Key Points

- Chemistry, along with other sciences and engineering, makes use of many different units.
- In mathematics and chemistry, a conversion factor is used to convert a measured quantity to a different unit of measure without changing the relative amount.
- Units behave just like numbers in products and quotients—they can be multiplied and divided.

#### Term

- conversion factorA conversion factor changes one unit to a new unit.

## 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!

## Example Problem 1

Here is an example problem: How many hours are in 3 days?

1. Identify the given: 3 days

2. Identify conversion factors that will help you get from your original units to your desired unit: [latex]\frac {24\:hours}{1\:day}[/latex]

3. Set up your equation so that your undesired units cancel out to give you your desired units: [latex]3\:days \cdot \frac{24\:hours}{1\:day} [/latex]

4. Multiply through to get your final answer: 72 hours

## 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.

## Example Problem 2

You can also use these quotients to convert from inches to feet or from feet to inches. For example, how many inches are in 5 feet?

1. The given is 5 feet.

2. The conversion factor is [latex]\frac {12\:inches}{1\:foot}[/latex]

3. Set up the equation: [latex]5\:feet\cdot\frac{12\:inches}{1\:foot}[/latex]

4. Multiply through: 60 inches

Another example is: how many feet are in 30 inches?

[latex]30\:inches\cdot\frac{1\:foot}{12\:inches} = 2.5\:feet[/latex]

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.

## Converting Between Moles and Grams

You can also use dimensional analysis to convert between moles and grams. For example:

[latex]22.34\:g\:H_2O \cdot \frac {1\:mol\:H_2O}{18\:g\:H_2O} = 1.24\:moles\:H_2O[/latex]

Knowing that one molecule of H_{2}O contains two hydrogens (2 g/mol) and one oxygen (16g/mol), you can calculate the molecular weight of water equal to 18 g/mol. Then use this number in the equation to get 1.24 mol H_{2}O in 22.34g. The equation is set up correctly because all the units cancel out to give moles.