Chemistry 12 First Assignment Readings

Math in Science

The language of science is mathematics. In order to study sciences seriously, one needs to learn mathematics that took generations of brilliant people centuries to work out. Algebra, for example, was cutting-edge mathematics when it was being developed in Baghdad in the 9th century. But today it's just the first step along the journey.

Measurement https://www.thoughtco.com/definition-of-measurement-605880In science, a measurement is a collection of quantitative or numerical data that describes a property of an object or event. A measurement is made by comparing a quantity with a standard unit. Since this comparison cannot be perfect, measurements inherently include error, which is how much a measured value deviates from the true value. The study of measurement is called metrology.There are many measurement systems that have been used throughout history and across the world, but progress has been made since the 18th century in setting an international standard. The modern International System of Units (SI) bases all types of physical measurements on seven base units.

Methods of Measurement

The length of a piece of string can be measured by comparing the string against a meter stick.

The volume of a drop of water may be measured using a graduated cylinder. The mass of a sample may be measured using a scale or balance. The temperature of a fire may be measured using a thermocouple.

Comparing Measurements

Measuring the volume of a cup of water with an Erlenmeyer flask will give you a better measurement than trying to gauge its volume by putting it into a bucket, even if both measurements are reported using the same unit (e.g., milliliters). Accuracy matters, so there are criteria that scientists use to compare measurements: type, magnitude, unit, and uncertainty.

The level or type is the methodology used for taking the measurement. Magnitude is the actual numerical value of a measurement (e.g., 45 or 0.237). Unit is the ratio of the number against the

standard for the quantity (e.g., gram, candela, micrometer). Uncertainty reflects the systematic and random errors in the measurement. Uncertainty is a description of confidence in the accuracy and precision of a measurement that is typically expressed as an error.

Measurement Systems

Measurements are calibrated, which is to say they are compared against a set of standards in a system so that the measuring device can deliver a value that matches what another person would obtain if the measurement were repeated. There are a few common standard systems you may encounter:

International System of Units (SI): SI comes from the French name Système International d'Unités. It is the most commonly used metric system.

Metric System: SI is a specific metric system, which is a decimal system of measurement. Examples of two common forms of the metric system are the MKS system (meter, kilogram, second as base units) and CGS system (centimeter, gram, and second as base units). There are many units in SI and other forms of the metric system that are built upon combinations of base units. These are called derived units.

English System: The British or Imperial system of measurements was common before SI units were adopted internationally. Although Britain has largely adopted the SI system, the United States and some Caribbean countries still use the English system for non-scientific purposes. This system is based on the foot-pound-second units, for units of length, mass, and time.

SI Unit Prefixes

Metric system  or SI units are based on factors of ten. However, most units prefixes with names are 1000 times apart. The exception are near the base unit (centi-, deci-, deca-, hecto-). Usually, a measurement is reported using a unit with one of these prefixes. It's a good idea to become comfortable converting between factors as they are used in all scientific disciplines.

FactorsPrefixSymbol1024 yotta Y1021 zetta Z1018 exa E1015 peta P1012 tera T199 giga G106 mega M103 kilo k102 hecto h101 deca da

FactorsPrefixSymbol10-1 deci d10-2 centi c10-3 milli m10-6 micro µ10-9 nano n10-12 pico p10-15 femto f10-18 atto a

The ascending prefixes (e.g., tera, peta, exa) are derived from Greek prefixes. Within 1000 factors of a base unit, there are prefixes for each factor of 10. The exception is 1010, which is used in distance measurements for the angstom..Beyond this, factors of 1000 are used. Very large or very small measurements are usually expressed using scientific notation.

A unit prefix is applied with the word for a unit, while its symbol is applied together with a unit's symbol. For example, it is correct to cite a value in units of either kilograms or kg, but it is incorrect to give the value as kilog or kgrams.

Conversion Factors https://www.thoughtco.com/definition-of-conversion-factor-604954

A conversion factor is the number or formula you need to convert a measurement in one set of units to the same measurement in another set of units. The number is usually given as a numerical ratio or fraction that can be used as a multiplication factor. For example, say you have a length that is measured in feet and you wish to report on it in meters. If you know that there are 3.048 feet in a meter, then you can use that as a conversion factor to determine what the same distance is in meters. 

One foot is 12 inches long, and the conversion factor of 1 foot to inches is 12. In yards, 1 foot is equal to 1/3 yard (conversion factor of 1 foot to yards is 1/3) so forth. The same length is 0.3048 meters, and it is also 30.48 centimeters.

To convert 10 feet to inches, multiply 10 times 12 (the conversion factor) = 120 inches To convert 10 feet to yards, multiply 10 x 1/3 = 3.3333 yards (or 3 1/3 yards) To convert 10 feet to meters, multiply 10 x .3048 = 3.048 meters To convert 10 feet to centimeters, multiply 10 x 30.48 = 304.8 centimeters

Examples of Conversion FactorsThere many different types of measurements that sometimes require conversions: length (linear), area (two dimensional) and volume (three dimensional) are the most common, but you can also use conversion factors to convert mass, speed, density, and force. Conversion factors are used for conversions within the imperial system (feet, pounds, gallons), within

the International System of Units (SI, and the modern form of the metric system: meters, kilograms, liters) or across the two. 

Remember, the two values must represent the same quantity as each other. For example, it's possible to convert between two units of mass (e.g., grams to pounds), but you generally can't convert between units of mass and volume (e.g., grams to gallons).

Examples of conversion factors include:

1 gallon = 3.78541 liters (volume) 1 pound = 16 ounces (mass)  1 kilogram = 1,000 grams (mass)  1 pound = 453.592 grams (mass) 1 minute = 60000 milliseconds (time)  1 square mile = 2.58999 square kilometers (area) 

Using a Conversion Factor

For example, to change a time measurement from hours to days, use a conversion factor of 1 day = 24 hours.

time in days = time in hours x (1 day/24 hours)

The (1 day/24 hours) is the conversion factor.

Note that following the equal sign, the units for hours cancel out, leaving only the unit for days.

Scientific Notation in Chemistry https://www.thoughtco.com/scientific-notation-in-chemistry-606205Scientists and engineers often work with very large or very small numbers, which are more easily expressed in exponential form or scientific notation. A classic chemistry example of a number written in scientific notation is Avogadro's number (6.022 x 1023). Scientists commonly perform calculations using the speed of light (3.0 x 108 m/s). An example of a very small number is the electrical charge of an electron (1.602 x 10-19 Coulombs). You write a very large number in scientific notation by moving the decimal point to the left until only one digit remains to the left. The number of moves of the decimal point gives you the exponent, which is always positive for a big number. For example:3,454,000 = 3.454 x 106

For very small numbers, you move the decimal point to the right until only one digit remains to the left of the decimal point. The number of moves to the right gives you a negative exponent:

0.0000005234 = 5.234 x 10-7

Addition Example Using Scientific Notation

Addition and subtraction problems are handled the same way.

1. Write the numbers to be added or subtracted in scientific notation.2. Add or subtract the first part of the numbers, leaving the exponent portion unchanged.3. Make sure your final answer is written in scientific notation.

(1.1 x 103) + (2.1 x 103) = 3.2 x 103

Subtraction Example Using Scientific Notation(5.3 x 10-4) - (2.2 x 10-4) = (5.3 - 1.2) x 10-4 = 3.1 x 10-4

Multiplication Example Using Scientific Notation

You do not have to write numbers to be multiplied and divided so that they have the same exponents. You can multiply the first numbers in each expression and add the exponents of 10 for multiplication problems.

(2.3 x 105)(5.0 x 10-12) =When you multiply 2.3 and 5.3 you get 11.5. When you add the exponents you get 10-7. At this point, your answer is:11.5 x 10-7

You want to express your answer in scientific notation, which has only one digit to the left of the decimal point, so the answer should be rewritten as:

1.15 x 10-6

Division Example Using Scientific Notation

In division, you subtract the exponents of 10.

(2.1 x 10-2) / (7.0 x 10-3) = 0.3 x 101 = 3

Using Scientific Notation on Your CalculatorNot all calculators can handle scientific notation, but you can perform scientific notation calculations easily on a scientific calculator. To enter in the numbers, look for a ^ button, which means "raised to the power of" or else yx or xy, which means y raised to the power x or x raised to the y, respectively. Another common button is 10x, which makes scientific notation easy. The way these button function depends on the brand of calculator, so you'll need to either read the instructions or else test the function. You will either press 10x and then enter your value for x or else you enter the x value and then press the 10x button. Test this with a number you know, to get the hang of it.

Also remember not all calculators follow the order of operations, where multiplication and division are performed before addition and subtraction. If your calculator has parentheses, it's a good idea to use them to make certain the calculation is carried out correctly.

Significant Figures in Precise Scientific Measurementhttps://www.thoughtco.com/using-significant-figures-2698885When making a measurement, a scientist can only reach a certain level of precision, limited either by the tools being used or the physical nature of the situation. The most obvious example is measuring distance.

Consider what happens when measuring the distance an object moved using a tape measure (in metric units). The tape measure is likely broken down into the smallest units of millimeters. Therefore, there's no way that you can measure with a precision greater than a millimeter. If the object moves 57.215493 millimeters, therefore, we can only tell for sure that it moved 57 millimeters (or 5.7 centimeters or 0.057 meters, depending on the preference in that situation).

In general, this level of rounding is fine. Getting the precise movement of a normal-sized object down to a millimeter would be a pretty impressive achievement, actually. Imagine trying to measure the motion of a car to the millimeter, and you'll see that, in general, this isn't necessary. In the cases where such precision is necessary, you'll be using tools that are much more sophisticated than a tape measure.The number of meaningful numbers in a measurement is called the number of significant figures of the number. In the earlier example, the 57-millimeter answer would provide us with 2 significant figures in our measurement.Zeroes and Significant Figures

Consider the number 5,200.

Unless told otherwise, it is generally the common practice to assume that only the two non-zero digits are significant. In other words, it is assumed that this number was rounded to the nearest hundred.However, if the number is written as 5,200.0, then it would have five significant figures. The decimal point and following zero is only added if the measurement is precise to that level.

Similarly, the number 2.30 would have three significant figures, because the zero at the end is an indication that the scientist doing the measurement did so at that level of precision.

Some textbooks have also introduced the convention that a decimal point at the end of a whole number indicates significant figures as well. So 800. would have three significant figures while 800 has only one significant figure. Again, this is somewhat variable depending on the textbook.

Following are some examples of different numbers of significant figures, to help solidify the concept:

One significant figure49000.00002

Two significant figures3.70.005968,0005.0Three significant figures9.640.0036099,9008.00900. (in some textbooks)

To Summarize the Rules of SigFigs:

All non-zero numbers ARE significant. ... Zeros between two non-zero digits ARE significant. ... Leading zeros are NOT significant. ... Trailing zeros to the right of the decimal ARE significant. ... Trailing zeros in a whole number with the decimal shown ARE significant.

Mathematics With Significant Figures

Scientific figures provide some different rules for mathematics than what you are introduced to in your mathematics class. The key in using significant figures is to be sure that you are maintaining the same level of precision throughout the calculation. In mathematics, you keep all of the numbers from your result, while in scientific work you frequently round based on the significant figures involved.

When adding or subtracting scientific data, it is only last digit (the digit the furthest to the right) which matters. For example, let's assume that we're adding three different distances:

5.324 + 6.8459834 + 3.1

The first term in the addition problem has four significant figures, the second has eight, and the third has only two. The precision, in this case, is determined by the shortest decimal point. So you will perform your calculation, but instead of 15.2699834 the result will be 15.3, because you will round to the tenths place (the first place after the decimal point), because while two of your measurements are more precise the third can't tell you anything more than the tenths place, so the result of this addition problem can only be that precise as well.

Note that your final answer, in this case, has three significant figures, while none of your starting numbers did. This can be very confusing to beginners, and it's important to pay attention to that property of addition and subtraction.

When multiplying or dividing scientific data, on the other hand, the number of significant figures do matter. Multiplying significant figures will always result in a solution that has the same significant figures as the smallest significant figures you started with. So, on to the example:

5.638 x 3.1

The first factor has four significant figures and the second factor has two significant figures. Your solution will, therefore, end up with two significant figures. In this case, it will be 17 instead of 17.4778. You perform the calculation then round your solution to the correct number of significant figures. The extra precision in the multiplication won't hurt, you just don't want to give a false level of precision in your final solution.Using Scientific Notation

Physics deals with realms of space from the size of less than a proton to the size of the universe. As such, you end up dealing with some very large and very small numbers. Generally, only the first few of these numbers are significant. No one is going to (or able to) measure the width of the universe to the nearest millimeter.

In order to manipulate these numbers easily, scientists use scientific notation. The significant figures are listed, then multiplied by ten to the necessary power. The speed of light is written as: 2.997925 x 108 m/s

There are 7 significant figures and this is much better than writing 299,792,500 m/s.

Note

The speed of light is frequently written as 3.00 x 108 m/s, in which case there are only three significant figures. Again, this is a matter of what level of precision is necessary.

This notation is very handy for multiplication. You follow the rules described earlier for multiplying the significant numbers, keeping the smallest number of significant figures, and then you multiply the magnitudes, which follows the additive rule of exponents. The following example should help you visualize it:

2.3 x 103 x 3.19 x 104 = 7.3 x 107

The product has only two significant figures and the order of magnitude is 107 because 103 x 104 = 107

Adding scientific notation can be very easy or very tricky, depending on the situation. If the terms are of the same order of magnitude (i.e. 4.3005 x 105 and 13.5 x 105), then you follow the addition rules discussed earlier, keeping the highest place value as your rounding location and keeping the magnitude the same, as in the following example:

4.3005 x 105 + 13.5 x 105 = 17.8 x 105

If the order of magnitude is different, however, you have to work a bit to get the magnitudes the same, as in the following example, where one term is on the magnitude of 105 and the other term is on the magnitude of 106:

4.8 x 105 + 9.2 x 106 = 4.8 x 105 + 92 x 105 = 97 x 105or4.8 x 105 + 9.2 x 106 = 0.48 x 106 + 9.2 x 106 = 9.7 x 106

Both of these solutions are the same, resulting in 9,700,000 as the answer.

Similarly, very small numbers are frequently written in scientific notation as well, though with a negative exponent on the magnitude instead of the positive exponent. The mass of an electron is:

9.10939 x 10-31 kg

This would be a zero, followed by a decimal point, followed by 30 zeroes, then the series of 6 significant figures. No one wants to write that out, so scientific notation is our friend. All the rules outlined above are the same, regardless of whether the exponent is positive or negative.

The Limits of Significant Figures

Significant figures are a basic means that scientists use to provide a measure of precision to the numbers they are using. The rounding process involved still introduces a measure of error into the numbers, however, and in very high-level computations there are other statistical methods that get used. For virtually all of the physics that will be done in the high school and college-level classrooms, however, correct use of significant figures will be sufficient to maintain the required level of precision.