Significant Figures
Exercise
Dr. MJ Patterson
The driving force behind significant figures is the accurate reporting of experimentally measured data. Any time you use an instrument to measure something in the lab, there is some uncertainty associated with that measurement. The golden rule for reporting measurements is:
Report all of the digits that you know with certainty, plus the first digit that you must estimate.
Digital Instruments:
For measuring tools with a digital display, it is understood that the last
digit shown on the display is the first uncertain digit. The balances in
our lab have this type of readout, and you may notice that the last digit frequently oscillates between two values. This
occurs because the actual value lies in between the two values.
Consequently, the last digit has some uncertainty associated with it.
Analog Instruments:
Measuring tools which have a scale over which some type of indicator moves are
referred to as analog instruments. In our lab, you will encounter many
analog instruments such as beakers, graduated cylinders, burets,
thermometers and volt meters. The general rule for these instruments
is to estimate one decimal place beyond the markings or graduations on the
instrument.
For example, if you use a graduated cylinder that is marked to the nearest mL, you might observe
that the liquid inside is between the 42 mL and 43 mL markings. You know with certainty that the
measurement is 42 mL.
However, you should also estimate the next digit. If it is closer to 42
than
43, you might estimate 42.3 mL. If it is closer
to 43, you might estimate 42.8 mL. Recording
the answer to the tenths place reports all of the certain digits plus the first
uncertain one.
In our introductory and general chemistry labs, we commonly use burets. These burets are
marked
to the tenth mL, or one decimal place. How many
decimal places should your reported value have
any time you use a buret? Well, since you know
with certainty to the nearest tenth, you need to
estimate to the nearest hundredth, or 2 decimal places.
Conventions for reading and writing significant figures:
Calculations with Significant Figures:
The rule for multiplication and division is to look at the total number of
significant digits in all of the
numbers involved in the calculation. Then, decide what
is the smallest number of sig figs you are
working with. Report your answer to this smallest number of sig
figs.
For addition and subtraction, focus on just the decimal places. Decide which number has the fewest significant decimal places, and then report your answer to this smallest number of decimal places.
Activity:
We will use the ruler provided to measure your quarter. The reported
measurements will depend on which side of the ruler is being used. Then,
we will practice the rules for carrying significant figures through
calculations by solving some problems incorporating our measurements.
1. Use the side of the ruler that is only marked to the nearest cm.
What is the diameter of the coin in cm? Consider how many significant
digits should be reported in your answer.
2. Use the other side of the ruler – the one marked to the nearest mm.
What is the diameter of the coin in cm?
3. Did the diameter of the coin change between the two measurements?
Which one is right? Which one is better?
4. Using your answer to #2, calculate the area of
the quarter.
5. Let's say you counted the number of quarters in a bag, and came up with
exactly 17. What is the area of the quarters in the bag?
6. Instead of counting the quarters in the bag, your lab partner
guesstimated that there were 15 quarters in the bag. What is the area of
the quarters in the bag?