The content that follows is the substance of lecture 3. In this lecture we cover Density, Uncertainty, Accuracy and Precision, Significant Figures, Dimensional Analysis and Conversions and the Temperature Scales (oF, oC and K).
Density
Let's start with Density. Density is the property of matter that tells you how its atoms or molecules are packed together. Very dense substances have closely packed particles and less dense substances have greater space between their particles.
Look at the picture below:
In both circles the same number of particles are present, but the red circle is more dense than the green because its particles are closer packed. Looking at the pictures what has really changed to make the red circle more dense? What is immediately obvious is that the size of the red circle is much smaller than the green. It's area or Volume (since we are using it as a container) is much less. So to increase the density of a substance without changing its content in terms of its number of particles or MASS, you need to reduce the volume. This means that the density of a substance is inversely proportional (densityIncrease ~ 1/VolumeIncrease).
In the image above the Volume has been kept the same but mass has been added to the circle on the right. Because this packs more mass into the same volume the density increases. This means that increases in Density are directly proportional to increases in Mass (DensityIncrease ~ MassIncrease).
So if we combine these two relationships:
Density = Mass * 1/Volume = Mass/Volume.
Now that we have this relationship understood and have created a mathematical equation to use, we can determine the density of any object just by weighing it to get its mass and measuring it to get its volume. We can also use the density of an object to determine how much a given amount will weigh or how much room a certain mass might need for storage etc.
For example: What if you wanted to ship 10.0 liters of wine to a friend for their birthday. How much weight would this be? Well, the density of red wine is 0.983 g/mL. This means that if you multiply the volume of the wine in mL by the density you can calculate the mass:
Just so you know, this mass would be about 21 lbs (453g/lb) so it looks a lot heavier than it actually was because of the unit it was expressed in, right? You should also note that the answer was expressed in scientific notation so that the correct number of significant figures could be given. The least precise measurement given was the 10.0 liters of wine so even though the value 9830. would be correct numerically, the last 0 in the number is uncertain and therefore should not be expressed in the answer.
Now I just used some terms above that you might not be completely familiar with: Precision and Uncertainty. There is a third term Accuracy that we need to discuss as well.
Uncertainty, Accuracy and Precision
Let's start with the proper definitions of each of these and then continue on with some examples:
Relative uncertainty or relative erroris a measure of the uncertainty of measurement compared to the size of the measurement. It is calculated as:
Accuracy: The accuracy of a measurement is how close a result comes to the true value.
Precision:Precision refers to the closeness of two or more measurements to each other.
So what in layman's terms do these mean? Uncertainty, while it can be calculate using the equation above to report with a measured quantity can also be broken down into a simpler rule when applied in chemistry. Uncertainty means that you cannot report a value to a greater number of significant digits than the number of digits of the most uncertain value (smallest number of digits) in the calculation. So you can't multiply numbers with two or three digits together and report the answer to 5 digits because you cannot be certain of the accuracy of the numbers beyond two (the smallest number of digits in the calculation).
Example: 2.3 x 6.75 = 15.525 (raw answer) but you can only be sure of the value to two digits so you would have to report 15 (2 digits) as the answer.
Accuracy is probably the easiest of the terms to understand because it is just the comparison of a value observed or calculated to the known or true value. The closer the measured value is to the known/true value the more Accurate the measurement.
Precision refers to the ability to repeat a measurement and achieve the same result over and over. For example if you measure a distance between the door and the foot of your bed and get 2.7 meters over and over, the measurement is said to be Precise. BUT if the actual distance is 1.9 meters then while precise your measurement is not terribly accurate. In science it is very important to be both accurate and precise and to not overstate the certainty of either.
Significant Figures
In the paragraph above we mentioned significant figures and this is a term that many of you will grow to hate. In an effort not to overstate the certainty of a value, you must keep track of the number of digits that are significant (or certain) throughout the operations performed on them. AND there are different rules for keeping track of these digits based on what operation is being performed: Addition/Subtraction, Multiplication/Division and Logarithms/Antilogs.
What in a number is significant?
1) All zeros before a number are not significant. If they disappear when placing the value into scientific notation then they were just place holders and not significant. The number shown would be 3.400e-5 so note that the zeros before three disappear.
2) All non-zero numbers are significant.
3) All zeros after a non-zero number with a decimal are significant too.
4) Zeros after a number that lacks a decimal may or may not be significant. So a number like 500 would generally be considered as having 1 significant figure. If I add the decimal 500. then it has 3.
Exact numbers, such as the number of people in a room, have an infinite number of significant figures. Exact numbers are counting up how many of something are present, they are not measurements made with instruments. Another example of this are defined numbers, such as
1 foot=12 inches
There are exactly 12 inches in one foot. Therefore, if a number is exact, it DOES NOT affect the accuracy of a calculation nor the precision of the expression. Some more examples:
There are 100 years in a century.
Interestingly, the speed of light is now a defined quantity. By definition, the value is 299,792,458 meters per second.
Let's Practice:
Rules for Addition and Subtraction:
In addition and subtraction the number of significant figures that can be reported are based on the number of digits in the least precise number given. Specifically this means the number of digits after the decimal determine the number of digits that can be expressed in the answer.
Rules for Multiplication and Division:
In multiplication and division the number of significant figures is simply determined by the value of lowest digits. This means that if you multiplied or divided three numbers: 2.1, 4.005 and 4.5654, the value 2.1 which has the fewest number of digits would mandate that the answer be given only to two significant figures.
Rules for Logarithms:
Before we start discussing the significant digit rules for logarithms we need to define a term you may or may not be familiar with: The Mantissa. While this sounds like the name of a monster that would fight Godzilla it is actually what all the digits found to the right of the decimal are called in a number:
25.637
What is shown in red is the mantissa of this number.
So now we can use this term when discussing the rules for logarithms:
1) When you take a logarithm, assess the number of digits in the entire number:
Log10(4.5e-4) = -3.34679 so 2 sf in Green
So by logarithm rules you should report the answer so that the mantissa has two significant figures:
Log10(4.5e-4) = -3.34679 = -3.35 so 2 sf in Green
2) When you take an antilog of a number, assess the number of digits in the mantissa of the number:
Antilog(5.15) = 105.15 = 141253.7 so 2 sf in Green
So by logarithm rules you should report the answer so that the whole number in the answer has two significant figures:
Antilog(5.15) = 105.15 = 1.4e5 so 2 sf in Green
Below are some Practice Problems you can work on as needed:
Dimensional Analysis and Conversions:
Dimensional analysis is a fancy term for the rules that we must apply when performing any mathematical operation on a number that also has units. The simple rule is anything you do to the number also has to be done to the units. For example if you want to calculate the volume of a cube that measure 2cm on each side, you would simply cube the number 2 correct? Length x width x height = 23 = 8. BUT we also have a unit of cm in the mix so we have to cube that value as well to abide by the rule of dimensional analysis so our final answer must be 8 cm3.
The RULE of Dimensional Analysis:
Whatever mathematical operation is applied to a number MUST also be applied to its unit(s).
As you can see by the red, large size and bold this is important. Now let's look at the application of this rule in Conversions.
Conversions in Chemistry
Conversion is the process we conduct when changing a value from one unit to another. We do this in real life all the time without really thinking about it. For instance, if you go the laundramat and it takes $2 to wash a large load, but the machines only take quarters, how many quaters do you need to wash a load? Simple answer 8. HOW did you know this??? Simple answer you converted $2 to 8 quarters by knowing the conversion factor that says there are 4 quarters in a dollar or 4 quarters = $1. Mathematically we would set this up as:
2 Dollars x (4 quarters/1 Dollar) = 8 quarters
Notice that we place the conversion factor into the equation as a fraction. We do this because since the two values (4 quaters and $1) are the same in value the equation is essentially a multiplication by 1.
4 quarters/1 Dollar = 1
All we are doing in the equation is converting from one unit to another. The value remains the same since $2 = 8 quarters.
Conversions of Temperature Units
There is a special case of conversion for the units of temperature used around the world and in science. There are 3 common units for temperature:
Fahrenheit: Symbol oF Celsius: : Symbol oC Kelvin: Symbol K
The mathematical relationships between the scales is as follows:
1) All zeros before a number are not significant. If they disappear when placing the value into scientific notation then they were just place holders and not significant. The number shown would be 3.400e-5 so note that the zeros before three disappear.
