Greetings,
Measurements
A measurement consists of 1) a number which indicates the magnitude of the measurement, 2) a unit which provides the scale used to take the measurement, and 3) a description of what's being measured. For example, using a top-loading balance, I could make the following measurement:
19.34 g reagent-grade sodium chloride.
No measurement is ever perfect, which means there will always be some error involved. There are two main types of #errors in chemical measurements: 1) random and 2) systematic. Random errors generally result from limitations of laboratory equipment and instrumentation. Estimating between measuring-device graduations/increments and instrument uncertainty due to electrical background noise contribute to random errors. Reducing random errors will tend to increase precision; precision being how well we can reproduce a measurement. Systematic errors tend to result from human error or errors from poorly calibrated instruments. Results from 3 or more laboratories testing the same batch of a chemical product using the same testing method would, very likely, display some systematic errors. Reducing the amount of systematic error will tend to increase accuracy; accuracy being the 'closeness' of a measurement to a true or known value. Generally speaking, the precision of a testing method (i.e. degree of random error) must be determined before its accuracy can be correctly evaluated.
Data but Not Measurements
There are two types of data that are not measurements because there is no error associated with them. These types of data result from 1) counting and 2) unit definitions. Here are examples: If I go and buy 5 apples, there can be no doubt that there are 5 apples in my shopping bag. Also, there are 2.54 cm = 1 in by definition, and so no uncertainty there either.
Significant Figures
Significant figures are those digits in a number (resulting from a measurement) which are considered to significantly contribute to the number's "acceptable-level" of precision. So, just what is an acceptable-level of precision with regard to a number obtained from a measurement, hum? Short answer: All certain digits plus the first uncertain digit (That's right!, the one we estimate).
Here's an example, using a ruler image. Let's measure out to the blue line, first using the 'cm' top and then the 'mm' bottom.
Using the top of the ruler first, we see our line is past '3' but not to '4', so we have 3 cm for sure (our certain digit) plus some fraction of a cm, which we'll need to estimate (our first uncertain digit). We have to imagine 10 increments between the '3' and the '4' and with that in mind let's estimate another 0.6 cm. So, we have 3 cm + 0.6 cm = 3.6 cm for a valid measurement using the cm 'side' of our ruler. The number of significant figures in a measurement is all of the digits that we can record, so we have 2 significant figures (aka sig figs).
Now for the bottom of the ruler, we have 35 mm, but not quite 36 mm, so 35 mm for sure (our certain digits). The blue line is just 'shy' of 36 mm, so let's "call it" another 0.9 mm (our first uncertain digit). We have 35 mm + 0.9 mm = 35.9 mm and this is 3 sig figs, which is more precise than our cm measurement because that was only 2 sig figs.
Another situation we need to consider is being presented with a number which we presume is produced from a measurement. How do we determine the number of sig figs in this situation? There are two main things here: 1) nonzero digits and 2) zero digits.
Nonzero digits are always significant while the significance of 0's depends on where they are in a number. There are three 'types' of zeros: a) leading zeros (on the left side of a decimal fraction and never significant), b) captive zeros (between nonzero digits and always significant), and c) trailing zeros (on the right side of a number and only significant when there is a decimal point in the number).
Here are a few examples (all sig figs are red text):
0.00300450 6 sig figs
300450 5 sig figs
300450.0 7 sig figs
300450. 6 sig figs
There are rules for rounding calculation results to the proper number of sig figs: those will be covered in the next post.
Derived Units
Derived units result from a combination of fundamental S.I. units of measurement. A good example is the derived unit for density; a combination of mass and volume S.I. units. In chemistry, the density of a liquid is commonly expressed using the derived unit, g/mL, which stated is "grams per milliliter". Units of volume and heat energy are also derived, as explained below.
The Volume Unit
The S.I. Unit of volume is derived from the S.I. unit of length, the meter (m). Imagine a large cube with dimensions of 1 m on every side. We determine the volume by multiplying together the dimensions of width, height, and depth, like this; Vcube = 1m x 1m x 1m = 1m^3 (a cubic meter). The cubic meter is OK for buying a 'fridge in Germany but is way too big for the chemistry lab, so we use the cubic centimeter (cm^3) instead. The cool thing about the cubic centimeter is that 1 cubic centimeter equals 1 milliliter; the volume unit commonly used in the chemistry lab!
Energy Units
In chemistry, there are two heat units commonly used; the Joule (J) and the calorie (cal). The Joule is the S.I. unit for heat, derived from the fundamental units of mass (kg), distance (m), and time (s). The amount of heat energy (q) is equal to force x distance. Using the fundamental units above, q can be expressed using (kg x m/s^2) x m = N x m; where N = Newtons (the S.I. unit of force). So, 1 Joule is the heat energy equal to the force of 1 newton acting over a distance of 1 meter.
The calorie is the amount of heat energy needed to increase the temperature of one gram of water by one degree Celsius. 4.184 J = 1 cal
The "diet calorie" is actually the heat content of 1 kilocalorie, that is 1000 cal.
Dimensional Analysis
The real usefulness of dimensional analysis is for more complex unit conversions. In those cases, the more complex problem can be broken down into a 'string' of simpler conversion steps. Here is an example:
How many pounds does one 'brick' of gold (Au) weigh given 1 brick Au = 1.0 L, 453.6 g = 1 lb, and
Density (Au) = 19.3 g/cm^3?
That's all for now. My next post will cover rules for rounding to correct sig figs in calculations.
Have a good day!
Now, we'll get into more details on #measurements, #significant figures, #derived #units, #volume units, #energy units, and #dimensional analysis.
Measurements
A measurement consists of 1) a number which indicates the magnitude of the measurement, 2) a unit which provides the scale used to take the measurement, and 3) a description of what's being measured. For example, using a top-loading balance, I could make the following measurement:
19.34 g reagent-grade sodium chloride.
No measurement is ever perfect, which means there will always be some error involved. There are two main types of #errors in chemical measurements: 1) random and 2) systematic. Random errors generally result from limitations of laboratory equipment and instrumentation. Estimating between measuring-device graduations/increments and instrument uncertainty due to electrical background noise contribute to random errors. Reducing random errors will tend to increase precision; precision being how well we can reproduce a measurement. Systematic errors tend to result from human error or errors from poorly calibrated instruments. Results from 3 or more laboratories testing the same batch of a chemical product using the same testing method would, very likely, display some systematic errors. Reducing the amount of systematic error will tend to increase accuracy; accuracy being the 'closeness' of a measurement to a true or known value. Generally speaking, the precision of a testing method (i.e. degree of random error) must be determined before its accuracy can be correctly evaluated.
Data but Not Measurements
There are two types of data that are not measurements because there is no error associated with them. These types of data result from 1) counting and 2) unit definitions. Here are examples: If I go and buy 5 apples, there can be no doubt that there are 5 apples in my shopping bag. Also, there are 2.54 cm = 1 in by definition, and so no uncertainty there either.
Significant Figures
Significant figures are those digits in a number (resulting from a measurement) which are considered to significantly contribute to the number's "acceptable-level" of precision. So, just what is an acceptable-level of precision with regard to a number obtained from a measurement, hum? Short answer: All certain digits plus the first uncertain digit (That's right!, the one we estimate).
Here's an example, using a ruler image. Let's measure out to the blue line, first using the 'cm' top and then the 'mm' bottom.
Using the top of the ruler first, we see our line is past '3' but not to '4', so we have 3 cm for sure (our certain digit) plus some fraction of a cm, which we'll need to estimate (our first uncertain digit). We have to imagine 10 increments between the '3' and the '4' and with that in mind let's estimate another 0.6 cm. So, we have 3 cm + 0.6 cm = 3.6 cm for a valid measurement using the cm 'side' of our ruler. The number of significant figures in a measurement is all of the digits that we can record, so we have 2 significant figures (aka sig figs).
Now for the bottom of the ruler, we have 35 mm, but not quite 36 mm, so 35 mm for sure (our certain digits). The blue line is just 'shy' of 36 mm, so let's "call it" another 0.9 mm (our first uncertain digit). We have 35 mm + 0.9 mm = 35.9 mm and this is 3 sig figs, which is more precise than our cm measurement because that was only 2 sig figs.
Another situation we need to consider is being presented with a number which we presume is produced from a measurement. How do we determine the number of sig figs in this situation? There are two main things here: 1) nonzero digits and 2) zero digits.
Nonzero digits are always significant while the significance of 0's depends on where they are in a number. There are three 'types' of zeros: a) leading zeros (on the left side of a decimal fraction and never significant), b) captive zeros (between nonzero digits and always significant), and c) trailing zeros (on the right side of a number and only significant when there is a decimal point in the number).
Here are a few examples (all sig figs are red text):
0.00300450 6 sig figs
300450 5 sig figs
300450.0 7 sig figs
300450. 6 sig figs
There are rules for rounding calculation results to the proper number of sig figs: those will be covered in the next post.
Derived Units
Derived units result from a combination of fundamental S.I. units of measurement. A good example is the derived unit for density; a combination of mass and volume S.I. units. In chemistry, the density of a liquid is commonly expressed using the derived unit, g/mL, which stated is "grams per milliliter". Units of volume and heat energy are also derived, as explained below.
The Volume Unit
The S.I. Unit of volume is derived from the S.I. unit of length, the meter (m). Imagine a large cube with dimensions of 1 m on every side. We determine the volume by multiplying together the dimensions of width, height, and depth, like this; Vcube = 1m x 1m x 1m = 1m^3 (a cubic meter). The cubic meter is OK for buying a 'fridge in Germany but is way too big for the chemistry lab, so we use the cubic centimeter (cm^3) instead. The cool thing about the cubic centimeter is that 1 cubic centimeter equals 1 milliliter; the volume unit commonly used in the chemistry lab!
Energy Units
In chemistry, there are two heat units commonly used; the Joule (J) and the calorie (cal). The Joule is the S.I. unit for heat, derived from the fundamental units of mass (kg), distance (m), and time (s). The amount of heat energy (q) is equal to force x distance. Using the fundamental units above, q can be expressed using (kg x m/s^2) x m = N x m; where N = Newtons (the S.I. unit of force). So, 1 Joule is the heat energy equal to the force of 1 newton acting over a distance of 1 meter.
The calorie is the amount of heat energy needed to increase the temperature of one gram of water by one degree Celsius. 4.184 J = 1 cal
The "diet calorie" is actually the heat content of 1 kilocalorie, that is 1000 cal.
Dimensional Analysis
The real usefulness of dimensional analysis is for more complex unit conversions. In those cases, the more complex problem can be broken down into a 'string' of simpler conversion steps. Here is an example:
How many pounds does one 'brick' of gold (Au) weigh given 1 brick Au = 1.0 L, 453.6 g = 1 lb, and
Density (Au) = 19.3 g/cm^3?
That's all for now. My next post will cover rules for rounding to correct sig figs in calculations.
Have a good day!
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