Measurements

Chemistry Word Problems

When solving any algabraic equation in chemistry there are generally 5 steps to do:

1. Identify what you need to solve for

2. Make a list of what you're given to solve for that (including units)

3. Figure out what equation to use for this (or make one, depending on the problem)

4. Plug in given #s - if you have more than 1 unknown to solve for you'll need to start with another formula (or use a different formula entirely to get one of the unknowns)

5. Solve 

Scientific Notation

Measurements and calculations in chemistry often require the use of very large numbers or very small numbers. Instead of writing these out fully, we use something called Scientific Notation. Scientific Notation Has 2 Components:

• • A number at least as big as 1 but smaller than 10 (1 to 9.99999999......)

  • A multiplier of 10 raised to some exponent. This exponent is known as the number's order of magnitude.

The number of places the decimal point has moved determines the power of 10

• • If a decimal point moved to the left then the power is positive

  • If a decimal point moved to the right then the power is negative

Systems of Units

Instead of Scientific Notation we can also use Metric Prefixes to denote very large and very small numbers when used in measurement. While the Metric System isn't used in the U.S. for some units often we still use its terminology all the time, such as when using milliliters or gigabytes.  The biggest advantage of the metric system is every prefix is a power of 10:

• Tera (T) is 10^12

• Giga (G) is 10^9

• Mega (M) is 10^6

• Kilo (k) is 10^3

• Deci (d) is 10^-1

• Centi (c) is 10^-2

• Milli (m) is 10^-3

• Micro (µ) is 10^-6

• Nano (n) is 10^-9

• Pico (p) is 10^-12

For example, 1 kilogram (km) is 10^3 grams (or 1000 grams if written out fully without scientific notation) and 15 centimeters is the same as 0.15 meters.

Dimensional Analysis

One of the main ways to use math in medicine (and science in general) is via conversions. To convert from one set of units to another a conversion factor is used, a fraction that allows converting by making both the top and bottom of the fraction equal and allows units to cancel out so the answer is in the units we want. For example, suppose we wanted to convert 100 minutes into hours:

Multiple conversion factors can be used one after another as dimensional analysis, ending with the units you're looking for and cancelling out any units that are no longer being used:

Conversion factors with metric prefixes tend to be fairly easy since they should always use their specific powers of 10. An example of this is converting 15 cm to m:

15 cm * (10^-2 m /1 cm) = 0.15 m

If multiple numbers in scientific notation are being used in dimensional analysis we can use our power rules to combine exponents to make things easier. An example of this is converting 15 cm to mm:

15  cm * (10^-2 m /1 cm) * (1 mm /10 ^-3 m) = 15 * 10^-2 / 10^-3 = 15 * 10^1 = 150 mm

Significant Figures

In math, you are often told to round to two or three decimal places. This is not the case for science. Instead, you need to round and estimate based on what's being measured or calculated. To do this we look at the uncertainty and math, keeping the Significant Figures. To determine the number of significant figures present in a number we look at the following:

• Any non-zero integer is always counted as a significant figure

• Leading zeros are those that precede all of the non-zero digits and are never counted as significant figures

•  Captive zeros are those that fall between non-zero digits and are always counted as significant figures

• Trailing zeros are those at the end of a number and are only counted if the number is written with a decimal point

It's also important to note that exact numbers (like ones generally used in math class) have an unlimited number of significant figures and that the 10^ part of the number (aka the order of magnitude) is never considered significant when using Scientific Notation.

Doing math when using significant figures can be tricky but it needs to be considered to make sure the error from tolerances doesn't get too out of hand. When we do this we need to consider what type of math we're doing. When adding or subtracting, we need to line up the math before rounding. The answer should be based on the original data whose last number is in the leftmost spot. This happens because anything past that point could get caught up in that specific number's rounding, making it impossible to correctly calculate. When multiplying or dividing, we do the math first then round to a number of places equal to the smallest number of significant figures in the original data. This happens because of how multiplication and division quickly amplify error.

Medical Math (PSA)

Additional Resources

• Crash Course: Unit Conversion and Significant Figures

• Dimensional Analysis Tutorial (aka Unit Conversion)

• How to Write Proper Scientific Notation

• Khan Academy Scientific Notation

• Metric Prefix Conversion Tutorial 

• Crash Course Conversions & Sig Figs

• Sig Fig Rules & Hints

Interesting Links

• Uncertainty and Probability for Devices

• Conversion Factor Google Sheet (Make a copy of it to use it yourself)

• Medical Math

• Scale of the Universe Video

• Do People Understand the Scale of the Universe?

• Mars Probe Lost due to Math Error

• Human Estimation & Number Sense

• Cursed Units

• Do we need all that Pi?