In chemistry, we use different tools for the purpose of quantifying physical and chemical properties. For example, a chemist might use a beaker to **roughly** measure out 50mL of solution. (Note that beakers should never be used for precise measuring) In this unit, we explore the way which scientists show appreciation and knowledge for how ‘unsure’ they are about their data.)

As we can see from the above meme, Andy Bernard from the office got marks taken off for being too precise. This is a classic rookie mistake! Errors and uncertainties, like all the other great physical properties, energy and momentum, is conserved at **ALL TIMES!**

One way which this is achieved is through the use of **significant figures**. Significant figures allow the communication of the **degree of precision** of a specific data point. For example:$$0.90$$

The number 0.90, has 2 significant figures as you will see later in this chapter. What does the 2 significant figures tell us? It tells us that the person/equipment that recorded this number is accurate to the hundredth place of a decimal point. Like the name suggests, significant figures (or sig figs, sf for short) is exactly the number of digits that are ‘significant’. Although some less so than other as we will see.

To find the number of significant figures, keep in mind the following rules:

Nonzero integers are always significant. i.e. the 9 in 0.90

Zeros are more complicated, but keep in mind these three rules:

Leading zeros are zeros that precede all the nonzero digits. These are not significant. i.e. the 000 in 0.00025 is non significant and thus the number has only 2 sig figs.

Captive zeros, or as I like to call them, ‘trapped zeros’ are zeros that are encapsulated between nonzero integers are always significant. i.e. 0.2005 the zeros between 2 and 5 are captive and therefore significant and yielding 4 sig figs for the number.

Trailing zeros are zeros at the right end of the numbers, but are only significant when there is a decimal point. i.e., 100 has only 1 sig fig where as 100. has 3 sig figs. and 1.00 has 3 sig figs.

Exact numbers. many calculations involve the use of exact numbers, this means that the number is exact and therefore exempt from the sig fig rules governing the arithmetic. i.e. 1 m= 100 cm. This we know, as a fact and therefore, need not to apply the rules of sig fig.

### The Rules of Sig Fig in Operations

Now that we are familiar with the basic rules of the sig figs, lets now put it into good use! To do this, we must have an appreciation that uncertainties **propagate** as they are operated upon. For example:

A student is looking for the volume of a right rectangular prism. The student measures the base length and width with a ruler and the height with a vernier caliber, and obtains the following data:

Length: 4.5 cm Width: 7.8 cm and Height: 3.47 cm

So now, we would use the formula for volume of the rectangular prism: area of base times height.

\[4.5 \times{} 7.8 \times{} 3.47 = 121.797 \text{cm}^3\]

Normally, you would think that we are done. But not now that you have learned sig figs! Note that the answer, has 6 sig figs, and our values have 2, 2, and 3 sig figs respectively. Now if that is the case, our answer is more precise than our measurement! That is in direct violation of the sig fig rules. The product must have a number of sig figs that equate to the least number of sig figs in the measurements. This is to say that our answer should only have 2 sig figs because our least precise data was 2 sig figs. So we have to round our answer to 2 sig figs, yielding: 120 cm^{3}

\(1.1+1.89=2.99 \approx{} 3.0\) (\(\approx{}\) represents approximates to)

The rules of operating with sig figs are as follow:

Multiplication and division, the answer will have the same number of sig figs as the input value that has the least number of sig figs.

Addition and subtraction, the answer will have the same decimal place as the input value that has the least number of decimal places. i.e. $1.1+1.89=2.99 \approx{} 3.0$ ($\approx{}$ represents approximates to).

### The Rules for Rounding

If your calculations require multiple steps, round to the appropriate sf/decimal places after all the calculations have been carried out.

If the digit to be removed

is less than 5, the preceding digit stays the same. For example, 1.33 round to 1.3.

is equal to or greater than 5, the preceding digit is to increase by 1. For example, 1.36 rounds to 1.4.

DO NOT round sequentially. i.e.: reducing the sig figs for 4.348 to 2 sig figs, look at the second 4 ( the digit right after the last digit of the final answer) and do not look at the 8, as in do not round the 4 to a 5 then to 4.4. But instead, round to 4.3 as 4 is less than 5.

### Uncertainties

### Absolute Error (ΔL)

**Hypothetical Situation**

If you are using an scale for weighing out some reagents, say 10 grams. According to literature data, per 1 gram of reagent produces 1 mL of gas product when heated. Assume that all temperature and pressure are consistent with the experiment carried out in the literature and that there is 100% confidence in the measurement of the volume of the gas, the gas produced was measured out to be 12.00 mL. Now can you explain why that is? There are several possibilities:

1. There were impurities in the reagents that produced gas aside from the desired product gas.

2. Flaw in methodology.

3. Error in the scale.

Now, as you move away from the world of theoretical chemistry and explore the fascinating realm of experimental chemistry, you will need to develop a sense of ‘problem solving’ as depicted above. We will go into this in later chapters.

Now, let us focus on the error in the scale. Most common laboratory equipments that are used to produce quantative data have uncertainties associated with them. They are usually found on the label of the equipment. Now, the scale we used earlier has a manufacturer rated uncertainty of ±3 grams. (It is a rather lousy scale, normal lab scales’ uncertainties are much less). So we can see that the **margin of error** of said scale is 3 g (up/down), so when the screen read 10 grams, in fact, it could have been anywhere between 7-13 grams, due to its margin of error. So that would explain a higher than anticipated amount of product.

We have now identified the source of the error. We are now ready to represent it in our calculations.

Because we are using the stoichiometric ratio 1:1 (gram of reactants: mL of gas produced), it is a known factor that has no uncertainties associated with it.

*To propagate the Absolute Uncertainties when adding/subtracting numbers, simply add all of the uncertainties associated with them.*

(10 g ±3 g ) x 1 mLg^{-1} = 10 mL ±3 mL

We have now propagated the uncertainties from weight of reactants to the volume of the products. Now compare the theoretical volume for the gas product, we will find that 12 mL is well within the margin of error.

### Fractional and Percentage Uncertainties (\(\frac{\Delta{}L}{L}\)and ΔL*100/L)

Fractional and percentage uncertainties are just another way of representing the uncertainty associated with a value. They come into use when you are trying to plot a graph with error bars or when you calculations involve operations aside from addition and subtraction.

Exactly as the symbol suggest, to calculate fractional uncertainty, just take the uncertainty of the value and divide it by the value itself. For percentage, take the result from the division, and time it by 100. It is that simple.

To propagate the uncertainties when doing multiplication and division, you add the percentage/fractional errors of the two/more values that were involved in the operation.