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Chemistry 101: General Chemistry14 chapters | 132 lessons | 11 flashcard sets

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Lesson Transcript

Instructor:
*Chelsea Schuyler*

Are 7.5 grams and 7.50 grams the same? How do scientists represent very large and very small quantities? Find out the answers to these questions in this video.

When scientists do experiments, they're always recording data and making measurements. Sometimes the information they record is based on observation. This is called *qualitative*, meaning that it is based on an observation, but it's not directly measured and recorded numerically. For example, 'The water in the beaker is warm' would be a qualitative observation. I didn't actually go and measure the temperature of the water in the beaker. The other type of information that scientists record is *quantitative*, meaning that it is based on a measurement, and it's reported numerically. An example would be 'The water in the beaker is 87 degrees.' Notice how there's a number in the quantitative observation and not in the qualitative observation.

If you have ever measured something more than once, you may have noticed that each time you may get a slightly different result. Any time you make a measurement there is some degree of uncertainty related to that measurement. This is because no measuring device is perfect. Usually the more high-quality the measuring instrument is the more precise your measurement will be. The **precision** of an instrument refers to *the smallest repeatable digit* that the instrument can measure to. For example, if you are measuring the mass of a pen and one balance reads 7.5 grams while another - more precise - balance measures 7.50 grams, the second balance will give you a more precise measurement.

When reporting these measurements, it's extremely important to report all the digits that are given. In math class, you may have learned that 7.50 is equivalent to 7.5, but when it comes to making and recording a measurement, the zero at the end is just as important as the seven and the five. This is because the zero tells the person reading the number that the balance measured out to the nearest hundredth place, which just happened to be a zero. It pretty much tells us that the second balance we used is a little more 'high-tech' than the first one because it measures out farther. This zero is so important that it is called a **significant figure**. A significant figure is *a number that plays a role in the precision of a measurement*. Don't confuse the word 'significant' with 'important' or 'certain.' If a number is significant, it's just kept track of when reporting measured results and making calculations. It's very important to be able work with significant figures correctly so both the measurement and the precision of the instrument used are communicated.

So, if the last zero in 7.50 is significant, what numbers are not significant? First of all, all non-zero numbers are considered significant, as in the number 524, which has three significant figures. Also, zeros between two non-zero numbers - I like to call them 'sandwiched zeros' - are significant, as in the number 9,201, which has four significant figures. Leading zeros are not significant, as in the number 0.003, which has one significant figure. These zeros just serve as placeholders. The same number could have easily been written as 3 x 10-3 without those leading zeros. Trailing zeros are only significant in numbers with a decimal point, as in the number 7.50, which has three significant figures. Trailing zeros in numbers that do not contain a decimal point are not significant, as in the number 25,000, which only has two significant figures. These zeros also just serve as placeholders. For example, when finding the population of a town, the number 25,000 implies that the actual value is around 25,000 rounded to the nearest thousand, whereas the number 25,000. - with a decimal at the end - implies that the actual value is 25,000. When you see a number, keep an eye out for significant figures and decimal points.

You will also need to pay close attention to numbers when you're doing calculations. Calculators do not identify significant figures, so you'll need to make sure you report the results of calculations with the correct number of significant figures. Any time you're adding or subtracting, you will always report your answer with the same number of decimal places as the number with the least number of decimal places. For example, if you are adding 5.113 and 2.0, your answer should be rounded to the nearest tenth place, or 7.1. You can't assume that the second measurement was 2.000, so your answer should never be more precise than the measurements used in the calculations.

Have you ever wanted to divide two numbers (like 54 divided by 7), and you end up getting a number with LOTS of numbers past the decimal? When you take 54 divided by 7 on a calculator, you'll get an answer of 7.715284714â€¦ and so on. Hopefully you know that it would be a little absurd to report the entire answer given on the calculator, but how do you know where to round? When multiplying and dividing, your answer must have the same number of significant figures as the number with the least number of significant figures. For example, if you are multiplying 40 (with one significant figure) and 9.2 (with two significant figures), your answer should contain only one significant figure, so it should be reported as 400 instead of the 368 your calculator would give you.

So back to dividing 54 and 7; 54 has two significant figures, but 7 only has one. If both of these numbers were measurements, our answer can only have one significant figure, so our answer would need to be 8! Exact numbers (as opposed to measured numbers) contain an infinite number of significant figures. Examples of exact numbers are counting numbers, like 'there are four beakers' and 'I have two hands'. Those are exact. There're no 2.1 hands. There're no 5.3 beakers. Those are exact, whole numbers. This category also contains many conversion factors. There are 12 inches in one foot; there are EXACTLY 12 inches in one foot. Examples of measured numbers are 'The length of the table is 5.3 feet' or 'The salt has a mass of 5.63 grams.' When you're doing calculations, ignore exact numbers when counting up significant figures.

One way to avoid having to worry about whether a zero is significant or not, is to use scientific notation. If you use scientific notation correctly, all digits are significant. **Scientific notation** is often used in chemistry as *a way of representing either very large or very small numbers*, and chemists often use very large or very small numbers. When converting standard notation to scientific notation, move the decimal to the left or right until there is one non-zero integer on its left. If you moved the decimal to the left, you'll multiply by ten times the number you moved it to the left, so 56,000,000 would be 5.6 x 107. When doing this, be sure to preserve the significant digits, so 2,400. (with a decimal point at the end) would be 2.400 x 103. If you moved the decimal to the right, you'll multiply by ten times the negative number of times you moved it to the right, so 0.0045 would be 4.5 x 10-3, and 0.0005340 would be 5.340 x 10-4.

When converting from scientific notation to standard notation, first remove the *10x at the end, then move the decimal x times to the right if x is positive and x times to the left if x is negative. Add zeros as placeholders where they are needed. The number 2.304 x 10-7 would become 0.0000002304, and the number 9.87 x 104 would become 98,700. You'll notice that if you have a negative exponent your number's going to be very small, and if you have a positive exponent on the 10 your number's going to be very large. A negative exponent on the 10 doesn't mean you have a negative number, it just means that your number is very, very small.

As you can see, when making measurements, reporting data and performing calculations, it's important to be very clear about not only what the measurement is, but how precise your instrument was. This can all be accomplished using significant figures.

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Chemistry 101: General Chemistry14 chapters | 132 lessons | 11 flashcard sets

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