IINEQUALITIES

Solving Inequalities

Sometimes we need to solve Inequalities like these:

Symbol	Words	Example
>	greater than	x + 3 > 2
<	less than	7x < 28
≥	greater than or equal to	5 ≥ x - 1
≤	less than or equal to	2y + 1 ≤ 7

Solving

Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:

Something like:		x < 5
or:		y ≥ 11

We call that "solved".

How to Solve

Solving inequalities is very like solving equations ... you do most of the same things ...

... but you must also pay attention to the direction of the inequality.

Direction: Which way the arrow "points"

Some things you do will change the direction!

< would become >

> would become <

≤ would become ≥

≥ would become ≤

Safe Things To Do

These are things you can do without affecting the direction of the inequality:

Add (or subtract) a number from both sides
Multiply (or divide) both sides by a positive number
Simplify a side

Example: 3x < 7+3

You can simplify 7+3 without affecting the inequality:

3x < 10

But these things will change the direction of the inequality ("<" becomes ">" for example):

Multiply (or divide) both sides by a negative number
Swapping left and right hand sides

Example: 2y+7 < 12

When you swap the left and right hand sides, you must also change the direction of the inequality:

12 > 2y+7

Here are the details:

Adding or Subtracting a Value

We can often solve inequalities by adding (or subtracting) a number from both sides (just as in Introduction to Algebra), like this:

Solve: x + 3 < 7

If we subtract 3 from both sides, we get:

x + 3 - 3 < 7 - 3    

x < 4

And that is our solution: x < 4
In other words, x can be any value less than 4.

What did we do?

We went from this: To this:				x+3 < 7 x < 4

And that works well for adding and subtracting, because if you add (or subtract) the same amount from both sides, it does not affect the inequality

Example: Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.

What If I Solve It, But "x" Is On The Right?

No matter, just swap sides, but reverse the sign so it still "points at" the correct value!

Example: 12 < x + 5
If we subtract 5 from both sides, we get:

12 - 5 < x + 5 - 5    

7 < x

That is a solution!
But it is normal to put "x" on the left hand side ...

... so let us flip sides (and the inequality sign!):

x > 7

Do you see how the inequality sign still "points at" the smaller value (7) ?
And that is our solution: x > 7

Note: "x" can be on the right, but people usually like to see it on the left hand side.

Multiplying or Dividing by a Value

Another thing we do is multiply or divide both sides by a value (just as in Algebra - Multiplying).
But we need to be a bit more careful (as you will see).

Positive Values

Everything is fine if you want to multiply or divide by a positive number:

Solve: 3y < 15

If we divide both sides by 3 we get:

3y/3 < 15/3

y < 5

And that is our solution: y < 5

Negative Values

When you multiply or divide by a negative number you have to reverse the inequality.

Why?

Well, just look at the number line!

For example, from 3 to 7 is an increase,
but from -3 to -7 is a decrease.

-7 < -3	7 > 3

See how the inequality sign reverses (from < to >) ?

Let us try an example:

Solve: -2y < -8

Let us divide both sides by -2 ... and reverse the inequality!

-2y < -8

-2y/-2 > -8/-2

y > 4

And that is the correct solution: y > 4

(Note that I reversed the inequality on the same line I divided by the negative number.)
So, just remember:

When multiplying or dividing by a negative number, reverse the inequality

Multiplying or Dividing by Variables

Here is another (tricky!) example:

Solve: bx < 3b

It seems easy just to divide both sides by b, which would give us:

x < 3

... but wait ... if b is negative we need to reverse the inequality like this:

x > 3

But we don't know if b is positive or negative, so we can't answer this one!

To help you understand, imagine replacing b with 1 or -1 in that example:

• if b is 1, then the answer is simply x < 3
• but if b is -1, then you would be solving -x < -3, and the answer would be x > 3

So:

Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).

A Bigger Example

Solve: (x-3)/2 < -5

First, let us clear out the "/2" by multiplying both sides by 2.
Because you are multiplying by a positive number, the inequalities will not change.

(x-3)/2 ×2 < -5 ×2  

(x-3) < -10

Now add 3 to both sides:

x-3 + 3 < -10 + 3    

x < -7

And that is our solution: x < -7

Two Inequalities At Once!

How could you solve something where there are two inequalities at once?

Solve:

-2 < (6-2x)/3 < 4

First, let us clear out the "/3" by multiplying each part by 3:
Because you are multiplying by a positive number, the inequalities will not change.

-6 < 6-2x < 12

Now subtract 6 from each part:

-12 < -2x < 6

Now multiply each part by -(1/2).
Because you are multiplying by a negative number, the inequalities change direction.

6 > x > -3

And that is the solution!
But to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over (and make sure the inequalities point correctly):

-3 < x < 6

Summary

• Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.
• But these things will change direction of the inequality:
• Multiplying or dividing both sides by a negative number
• Swapping left and right hand sides
• Don't multiply or divide by a variable (unless you know it is always positive or always negative)

ALGEBRA (equation)

Equations and Formulas

What is an Equation?

An equation says that two things are equal. It will have an equals sign "=" like this:

x

+

2

=

6

That equations says: what is on the left (x + 2) is equal to what is on the right (6)
So an equation is like a statement "this equals that"

What is a Formula?

A formula is a special type of equation that shows the relationship between different variables.
(A variable is a symbol like x or V that stands in for a number we don't know yet).

Example: The formula for finding the volume of a box is:

V = lwh

V stands for volume, l for length, w for width, and h for height.

When l=10, w=5, and h=4, then V = 10 × 5 × 4 = 200

A formula will have more than one variable.
These are all equations, but only some are formulas:

x = 2y - 7	Formula (relating x and y)
a2 + b2 = c2	Formula (relating a, b and c)
x/2 + 7 = 0	Not a Formula (just an equation)

Without the Equals

Sometimes a formula is written without the "=":

Example: The formula for the volume of a box is:

lwh

But in a way the "=" is still there, because you could write V = lwh if you wanted to.

Subject of a Formula

The "subject" of a formula is the single variable (usually on the left of the "=") that everything else is equal to.

Example: in the formula

s = ut + ½ at²

"s" is the subject of the formula

Changing the Subject

One of the very powerful things that Algebra can do is to "rearrange" a formula so that another variable is the subject.

Rearrange the volume of a box formula (V = lwh) so that the width is the subject:

Start with:	V = lwh
divide both sides by h:	V / h = lw
divide both sides by l:	V / hl = w
swap sides:	w = V / hl

So now if you have a box with a length of 2m, a height of 2m and a volume of 12m³, you can calculate its width:

w = V / hl

w = 12m³ / (2m×2m) = 12/4 = 3m

Algebra - Substitution

"Substitute" means to put in the place of another.

Substitution

In Algebra "Substitution" means putting numbers where the letters are:

	If you have:	x − 2
	And you know that x=6 ...
	... then you can "substitute" 6 for x:	6 − 2 = 4

Example: If x=5 then what is 10/x + 4 ?

Put "5" where "x" is:

10/5 + 4 = 2 + 4 = 6

Example: If x=3 and y=4, then what is x² + xy ?

Put "3" where "x" is, and "4" where "y" is:

3² + 3×4 = 3×3 + 12 = 21

Example: If x=3 (but you don't know "y"), then what is x² + xy ?

Put "3" where "x" is:

3² + 3y = 9 + 3y

(that is as far as you can get)

As that last example showed, you may not always get a number for an answer, sometimes just a simpler formula.

Negative Numbers

When substituting negative numbers, put () around them so you get the calculations right.

Example: If x = −2, then what is 1 − x + x² ?

Put "(−2)" where "x" is:

1 − (−2) + (−2)² = 1 + 2 + 4 = 7

In that last example:

• the − (−2) became +2
• the (−2)² became +4

because of these special rules:

	Rule	Adding or Subtracting	Multiplying or Dividing
	Two like signs become a positive sign	3+(+2) = 3 + 2 = 5	3 × 2 = 6
		6−(−3) = 6 + 3 = 9	(−3) × (−2) = 6
	Two unlike signs become a negative sign	7+(−2) = 7 - 2 = 5	3 × (−2) = −6
		8−(+2) = 8 − 2 = 6	(−3) × 2 = −6

BASIC ORGANIC CHEMISTRY 3

BONDING IN ETHENE
	Important!  You will find this much easier to understand if you first read the article about the bonding in methane. You may also find it useful to read the article on orbitals if you aren't sure about simple orbital theory.
Ethene, C2H4 The simple view of the bonding in ethene At a simple level, you will have drawn ethene showing two bonds between the carbon atoms. Each line in this diagram represents one pair of shared electrons. Ethene is actually much more interesting than this. An orbital view of the bonding in ethene Ethene is built from hydrogen atoms (1s1) and carbon atoms (1s22s22px12py1). The carbon atom doesn't have enough unpaired electrons to form the required number of bonds, so it needs to promote one of the 2s2 pair into the empty 2pz orbital. This is exactly the same as happens whenever carbon forms bonds - whatever else it ends up joined to. So the first thing that happens is . . . Promotion of an electron There is only a small energy gap between the 2s and 2p orbitals, and an electron is promoted from the 2s to the empty 2p to give 4 unpaired electrons. The extra energy released when these electrons are used for bonding more than compensates for the initial input. The carbon atom is now said to be in an excited state.
	Note:  If you haven't read about bonding in methane, follow this link before you go any further. Use the BACK button on your browser to come back here when you have finished. It is important that you have first met the idea of hybridisation in the more simple methane case.
Hybridisation In the case of ethene, there is a difference from, say, methane or ethane, because each carbon is only joining to three other atoms rather than four. When the carbon atoms hybridise their outer orbitals before forming bonds, this time they only hybridise three of the orbitals rather than all four. They use the 2s electron and two of the 2p electrons, but leave the other 2p electron unchanged. The new orbitals formed are called sp2 hybrids, because they are made by an s orbital and two p orbitals reorganising themselves. sp2 orbitals look rather like sp3 orbitals that you have already come across in the bonding in methane, except that they are shorter and fatter. The three sp2 hybrid orbitals arrange themselves as far apart as possible - which is at 120° to each other in a plane. The remaining p orbital is at right angles to them. The two carbon atoms and four hydrogen atoms would look like this before they joined together: The various atomic orbitals which are pointing towards each other now merge to give molecular orbitals, each containing a bonding pair of electrons. These are sigma bonds - just like those formed by end-to-end overlap of atomic orbitals in, say, ethane. The p orbitals on each carbon aren't pointing towards each other, and so we'll leave those for a moment. In the diagram, the black dots represent the nuclei of the atoms. Notice that the p orbitals are so close that they are overlapping sideways. This sideways overlap also creates a molecular orbital, but of a different kind. In this one the electrons aren't held on the line between the two nuclei, but above and below the plane of the molecule. A bond formed in this way is called a pi bond. For clarity, the sigma bonds are shown using lines - each line representing one pair of shared electrons. The various sorts of line show the directions the bonds point in. An ordinary line represents a bond in the plane of the screen (or the paper if you've printed it), a broken line is a bond going back away from you, and a wedge shows a bond coming out towards you.
	Note:  The really interesting bond in ethene is the pi bond. In almost all cases where you will draw the structure of ethene, the sigma bonds will be shown as lines.
Be clear about what a pi bond is. It is a region of space in which you can find the two electrons which make up the bond. Those two electrons can live anywhere within that space. It would be quite misleading to think of one living in the top and the other in the bottom.
	Taking chemistry further:  This is another example of the curious behaviour of electrons. How do the electrons get from one half of the pi bond to the other if they are never found in between? It's an unanswerable question if you think of electrons as particles.
Even if your syllabus doesn't expect you to know how a pi bond is formed, it will expect you to know that it exists. The pi bond dominates the chemistry of ethene. It is very vulnerable to attack - a very negative region of space above and below the plane of the molecule. It is also somewhat distant from the control of the nuclei and so is a weaker bond than the sigma bond joining the two carbons.
	Important!  Check your syllabus! Find out whether you actually need to know how a pi bond is formed. Don't forget to look in the bonding section of your syllabus as well as under ethene. If you don't need to know it, there's no point in learning it! You will, however, need to know that a pi bond exists - that the two bonds between the carbon atoms in ethene aren't both the same. If you are working to a UK-based syllabus for 16 - 18 year olds, and haven't got a copy of your syllabus, find out how to download one
All double bonds (whatever atoms they might be joining) will consist of a sigma bond and a pi bond. The shape of ethene The shape of ethene is controlled by the arrangement of the sp2 orbitals. Notice two things about them: They all lie in the same plane, with the other p orbital at right angles to it. When the bonds are made, all of the sigma bonds in the molecule must also lie in the same plane. Any twist in the molecule would mean that the p orbitals wouldn't be parallel and touching any more, and you would be breaking the pi bond. There is no free rotation about a carbon-carbon double bond. Ethene is a planar molecule. The sp2 orbitals are at 120° to each other. When the molecule is constructed, the bond angles will also be 120°. (That's approximate! There will be a slight distortion because you are joining 2 hydrogens and a carbon atom to each carbon, rather than 3 identical groups.) Questions to test your understanding If this is the first set of questions you have done, please read the introductory page before you start. You will need to use the BACK BUTTON on your browser to come back here afterwards. questions on bonding in ethene answers

BASIC ORGANIC CHEMISTRY 2

BONDING IN METHANE AND ETHANE
	Warning!  If you aren't happy with describing electron arrangements in s and p notation, and with the shapes of s and p orbitals, you really should read about orbitals. Use the BACK button on your browser to return quickly to this point.
Methane, CH4 The simple view of the bonding in methane You will be familiar with drawing methane using dots and crosses diagrams, but it is worth looking at its structure a bit more closely. There is a serious mis-match between this structure and the modern electronic structure of carbon, 1s22s22px12py1. The modern structure shows that there are only 2 unpaired electrons to share with hydrogens, instead of the 4 which the simple view requires. You can see this more readily using the electrons-in-boxes notation. Only the 2-level electrons are shown. The 1s2 electrons are too deep inside the atom to be involved in bonding. The only electrons directly available for sharing are the 2p electrons. Why then isn't methane CH2? Promotion of an electron When bonds are formed, energy is released and the system becomes more stable. If carbon forms 4 bonds rather than 2, twice as much energy is released and so the resulting molecule becomes even more stable. There is only a small energy gap between the 2s and 2p orbitals, and so it pays the carbon to provide a small amount of energy to promote an electron from the 2s to the empty 2p to give 4 unpaired electrons. The extra energy released when the bonds form more than compensates for the initial input. The carbon atom is now said to be in an excited state.
	Note:  People sometimes worry that the promoted electron is drawn as an up-arrow, whereas it started as a down-arrow. The reason for this is actually fairly complicated - well beyond the level we are working at. Just get in the habit of writing it like this because it makes the diagrams look tidy!
Now that we've got 4 unpaired electrons ready for bonding, another problem arises. In methane all the carbon-hydrogen bonds are identical, but our electrons are in two different kinds of orbitals. You aren't going to get four identical bonds unless you start from four identical orbitals. Hybridisation The electrons rearrange themselves again in a process called hybridisation. This reorganises the electrons into four identical hybrid orbitals called sp3 hybrids (because they are made from one s orbital and three p orbitals). You should read "sp3" as "s p three" - not as "s p cubed". sp3 hybrid orbitals look a bit like half a p orbital, and they arrange themselves in space so that they are as far apart as possible. You can picture the nucleus as being at the centre of a tetrahedron (a triangularly based pyramid) with the orbitals pointing to the corners. For clarity, the nucleus is drawn far larger than it really is. What happens when the bonds are formed? Remember that hydrogen's electron is in a 1s orbital - a spherically symmetric region of space surrounding the nucleus where there is some fixed chance (say 95%) of finding the electron. When a covalent bond is formed, the atomic orbitals (the orbitals in the individual atoms) merge to produce a new molecular orbital which contains the electron pair which creates the bond. Four molecular orbitals are formed, looking rather like the original sp3 hybrids, but with a hydrogen nucleus embedded in each lobe. Each orbital holds the 2 electrons that we've previously drawn as a dot and a cross. The principles involved - promotion of electrons if necessary, then hybridisation, followed by the formation of molecular orbitals - can be applied to any covalently-bound molecule. The shape of methane When sp3 orbitals are formed, they arrange themselves so that they are as far apart as possible. That is a tetrahedral arrangement, with an angle of 109.5°. Nothing changes in terms of the shape when the hydrogen atoms combine with the carbon, and so the methane molecule is also tetrahedral with 109.5° bond angles. Ethane, C2H6 The formation of molecular orbitals in ethane Ethane isn't particularly important in its own right, but is included because it is a simple example of how a carbon-carbon single bond is formed. Each carbon atom in the ethane promotes an electron and then forms sp3 hybrids exactly as we've described in methane. So just before bonding, the atoms look like this: The hydrogens bond with the two carbons to produce molecular orbitals just as they did with methane. The two carbon atoms bond by merging their remaining sp3 hybrid orbitals end-to-end to make a new molecular orbital. The bond formed by this end-to-end overlap is called a sigma bond. The bonds between the carbons and hydrogens are also sigma bonds. In any sigma bond, the most likely place to find the pair of electrons is on a line between the two nuclei. The shape of ethane around each carbon atom The shape is again determined by the way the sp3 orbitals are arranged around each carbon atom. That is a tetrahedral arrangement, with an angle of 109.5°. When the ethane molecule is put together, the arrangement around each carbon atom is again tetrahedral with approximately 109.5° bond angles. Why only "approximately"? This time, each carbon atoms doesn't have four identical things attached. There will be a small amount of distortion because of the attachment of 3 hydrogens and 1 carbon, rather than 4 hydrogens. Free rotation about the carbon-carbon single bond The two ends of this molecule can spin quite freely about the sigma bond so that there are, in a sense, an infinite number of possibilities for the shape of an ethane molecule. Some possible shapes are: In each case, the left hand CH3 group has been kept in a constant position so that you can see the effect of spinning the right hand one. Other alkanes All other alkanes will be bonded in the same way: The carbon atoms will each promote an electron and then hybridise to give sp3 hybrid orbitals. The carbon atoms will join to each other by forming sigma bonds by the end-to-end overlap of their sp3 hybrid orbitals. Hydrogen atoms will join on wherever they are needed by overlapping their 1s1 orbitals with sp3 hybrid orbitals on the carbon atoms. Questions to test your understanding If this is the first set of questions you have done, please read the introductory page before you start. You will need to use the BACK BUTTON on your browser to come back here afterwards. questions on bonding in methane and ethane answers

BASIC ORGANIC CHEMISTRY

ELECTRONIC STRUCTURE AND ATOMIC ORBITALS A simple view In any introductory chemistry course you will have come across the electronic structures of hydrogen and carbon drawn as:
	Note:  There are many places where you could still make use of this model of the atom at A' level. It is, however, a simplification and can be misleading. It gives the impression that the electrons are circling the nucleus in orbits like planets around the sun. As you will see in a moment, it is impossible to know exactly how they are actually moving.
The circles show energy levels - representing increasing distances from the nucleus. You could straighten the circles out and draw the electronic structure as a simple energy diagram. Atomic orbitals Orbits and orbitals sound similar, but they have quite different meanings. It is essential that you understand the difference between them. The impossibility of drawing orbits for electrons To plot a path for something you need to know exactly where the object is and be able to work out exactly where it's going to be an instant later. You can't do this for electrons.
	Note:  In order to plot a plane's course, it is no use knowing its exact location in mid-Atlantic if you don't know its direction or speed. Equally it's no use knowing that it is travelling at 500 mph due west if you have no idea whether it is near Iceland or the Azores at that particular moment.
The Heisenberg Uncertainty Principle (not required at A'level) says - loosely - that you can't know with certainty both where an electron is and where it's going next. That makes it impossible to plot an orbit for an electron around a nucleus. Is this a big problem? No. If something is impossible, you have to accept it and find a way around it. Hydrogen's electron - the 1s orbital
	Note:  In this diagram (and the orbital diagrams that follow), the nucleus is shown very much larger than it really is. This is just for clarity.
Suppose you had a single hydrogen atom and at a particular instant plotted the position of the one electron. Soon afterwards, you do the same thing, and find that it is in a new position. You have no idea how it got from the first place to the second. You keep on doing this over and over again, and gradually build up a sort of 3D map of the places that the electron is likely to be found. In the hydrogen case, the electron can be found anywhere within a spherical space surrounding the nucleus. The diagram shows a cross-section through this spherical space. 95% of the time (or any other percentage you choose), the electron will be found within a fairly easily defined region of space quite close to the nucleus. Such a region of space is called an orbital. You can think of an orbital as being the region of space in which the electron lives.
	Note:  If you wanted to be absolutely 100% sure of where the electron is, you would have to draw an orbital the size of the Universe!
What is the electron doing in the orbital? We don't know, we can't know, and so we just ignore the problem! All you can say is that if an electron is in a particular orbital it will have a particular definable energy. Each orbital has a name. The orbital occupied by the hydrogen electron is called a 1s orbital. The "1" represents the fact that the orbital is in the energy level closest to the nucleus. The "s" tells you about the shape of the orbital. s orbitals are spherically symmetric around the nucleus - in each case, like a hollow ball made of rather chunky material with the nucleus at its centre. The orbital on the left is a 2s orbital. This is similar to a 1s orbital except that the region where there is the greatest chance of finding the electron is further from the nucleus - this is an orbital at the second energy level. If you look carefully, you will notice that there is another region of slightly higher electron density (where the dots are thicker) nearer the nucleus. ("Electron density" is another way of talking about how likely you are to find an electron at a particular place.) 2s (and 3s, 4s, etc) electrons spend some of their time closer to the nucleus than you might expect. The effect of this is to slightly reduce the energy of electrons in s orbitals. The nearer the nucleus the electrons get, the lower their energy. 3s, 4s (etc) orbitals get progressively further from the nucleus. p orbitals Not all electrons inhabit s orbitals (in fact, very few electrons live in s orbitals). At the first energy level, the only orbital available to electrons is the 1s orbital, but at the second level, as well as a 2s orbital, there are also orbitals called 2p orbitals. A p orbital is rather like 2 identical balloons tied together at the nucleus. The diagram on the right is a cross-section through that 3-dimensional region of space. Once again, the orbital shows where there is a 95% chance of finding a particular electron.
	Beyond A'level:   If you imagine a horizontal plane through the nucleus, with one lobe of the orbital above the plane and the other beneath it, there is a zero probability of finding the electron on that plane. So how does the electron get from one lobe to the other if it can never pass through the plane of the nucleus? For A'level chemistry you just have to accept that it does! If you want to find out more, read about the wave nature of electrons.
Unlike an s orbital, a p orbital points in a particular direction - the one drawn points up and down the page. At any one energy level it is possible to have three absolutely equivalent p orbitals pointing mutually at right angles to each other. These are arbitrarily given the symbols px, py and pz. This is simply for convenience - what you might think of as the x, y or z direction changes constantly as the atom tumbles in space. The p orbitals at the second energy level are called 2px, 2py and 2pz. There are similar orbitals at subsequent levels - 3px, 3py, 3pz, 4px, 4py, 4pz and so on. All levels except for the first level have p orbitals. At the higher levels the lobes get more elongated, with the most likely place to find the electron more distant from the nucleus. Fitting electrons into orbitals Because for the moment we are only interested in the electronic structures of hydrogen and carbon, we don't need to concern ourselves with what happens beyond the second energy level. Remember: At the first level there is only one orbital - the 1s orbital. At the second level there are four orbitals - the 2s, 2px, 2py and 2pz orbitals. Each orbital can hold either 1 or 2 electrons, but no more. "Electrons-in-boxes" Orbitals can be represented as boxes with the electrons in them shown as arrows. Often an up-arrow and a down-arrow are used to show that the electrons are in some way different.
	Beyond A'level:  The need to have all electrons in an atom different comes out of quantum theory. If they live in different orbitals, that's fine - but if they are both in the same orbital there has to be some subtle distinction between them. Quantum theory allocates them a property known as "spin" - which is what the arrows are intended to suggest.
A 1s orbital holding 2 electrons would be drawn as shown on the right, but it can be written even more quickly as 1s2. This is read as "one s two" - not as "one s squared". You mustn't confuse the two numbers in this notation: The order of filling orbitals Electrons fill low energy orbitals (closer to the nucleus) before they fill higher energy ones. Where there is a choice between orbitals of equal energy, they fill the orbitals singly as far as possible. The diagram (not to scale) summarises the energies of the various orbitals in the first and second levels. Notice that the 2s orbital has a slightly lower energy than the 2p orbitals. That means that the 2s orbital will fill with electrons before the 2p orbitals. All the 2p orbitals have exactly the same energy. The electronic structure of hydrogen Hydrogen only has one electron and that will go into the orbital with the lowest energy - the 1s orbital. Hydrogen has an electronic structure of 1s1. We have already described this orbital earlier. The electronic structure of carbon Carbon has six electrons. Two of them will be found in the 1s orbital close to the nucleus. The next two will go into the 2s orbital. The remaining ones will be in two separate 2p orbitals. This is because the p orbitals all have the same energy and the electrons prefer to be on their own if that's the case.
	Note:  People sometimes wonder why the electrons choose to go into the 2px and 2py orbitals rather than the 2pz. They don't! All of the 2p orbitals are exactly equivalent, and the names we give them are entirely arbitrary. It just looks tidier if we call the orbitals the electrons occupy the 2px and 2py.
The electronic structure of carbon is normally written 1s22s22px12py1. Questions to test your understanding If this is the first set of questions you have done, please read the introductory page before you start. You will need to use the BACK BUTTON on your browser to come back here afterwards. questions on orbitals answers

INTRODUCTION TO ALGEBRA 2

Introduction to Algebra - Multiplication

Please read Introduction to Algebra first

A Puzzle

What is the missing number?

×

4

=

8

The answer is 2, right? Because 2 × 4 = 8.
Well, in Algebra we don't use blank boxes, we use a letter. So we might write:

x

×

4

=

8

But the "x" looks like the "×"! ... that could be very confusing ... so in Algebra we don't use the multiply symbol (×) between numbers and letters, we simply put the number next to the letter to mean multiply:

4x

=

8

You would say in English "four x equals eight", meaning that 4 x's make 8. And the answer would be written:

x

=

2

How to Solve

On the previous page we showed this neat step-by-step approach:

• Work out what to remove to get "x = ..."
• Remove it by doing the opposite
• Do that to both sides

It still works, but you have to know that dividing is the opposite of multiplying. Have a look at this example:

We want to remove the "4"	To remove it, do the opposite, in this case divide by 4:	Do it to both sides:	Which is ...	Solved!

Why did we divide by 4 on both sides?

Because of the need for balance ...

	Divide Left by 4	Divide Right by 4 Also
In Balance	Out of Balance!	In Balance Again

Just remember ...

To keep the balance, what you do to one side of the "=" you should also do to the other side!

Another Puzzle

Solve this one:

x

/

3

=

5

Start with:	x/3 = 5
What you are aiming for is an answer like "x = ...", and the divide by 3 is in the way of that! If you multiply by 3 you can cancel out the divide by 3 (because 3/3=1)
So, let us have a go at multiplying by 3 on both sides:	x/3 ×3 = 5 ×3
A little arithmetic (3/3 = 1 and 5×3 = 15) becomes:	1x = 15
Which is just:	x = 15
	Solved!
(Quick Check: 15/3 = 5)

Have a Try Yourself

Now practice on this Algebra Multiplication Worksheet and then check your answers on the page after. Try to use the steps we have shown you here, rather than just guessing!

More Complicated Example

How would you solve this?

x

/

3

+

2

=

5

It might look hard, but not if you solve it in stages.

First let us get rid of the "+2":

Start with:	x/3 + 2 = 5
To remove the plus 2 use minus 2 (because 2-2=0)	x/3 + 2 -2 = 5 -2
A little arithmetic (2-2 = 0 and 5-2 = 3) becomes:	x/3 + 0 = 3
Which is just:	x/3 = 3

Now, get rid of the "/3":

Start with:	x/3 = 3
If you multiply by 3 you can cancel out the divide by 3:	x/3 ×3 = 3 ×3
A little arithmetic (3/3 = 1 and 3×3 = 9) becomes:	1x = 9
Which is just:	x = 9
	Solved!
(Quick Check: 9/3 + 2 = 3+2 = 5)

When you get more experienced:

When you get more experienced, you can solve it like this:

Start with:	x/3 + 2 = 5
Subtract 2 from both sides:	x/3 + 2 -2 = 5 -2
Simplify:	x/3 = 3
Multiply by 3 on both sides:	x/3 ×3 = 3 ×3
Simplify:	x = 9

Or even like this:

Start with:	x/3 + 2 = 5
Subtract 2:	x/3 = 3
Multiply by 3:	x = 9

Real World Example

Example: Sam bought 3 boxes of chocolates online.
Postage was $9 and the total cost was $45.
How much was each box?

Let's use x for the price of each box.

3 times x plus $9 is $45:

3x + 9 = 45

Let's solve!

Start with:	3x + 9 = 45
Subtract 9 from both sides:	3x + 9 − 9 = 45 − 9
Simplify:	3x = 36
Divide by 3:	3x /3 = 36 /3
Simplify:	x = 12

So each box was $12

Advanced: you can also do the "divide by 3" first (but you must do it to all terms):

Start with:	3x + 9 = 45
Divide by 3:	3x/3 + 9/3 = 45/3
Simplify:	x + 3 = 15
Subtract 3 from both sides:	x + 3 − 3 = 15 − 3
Simplify:	x = 12

Intoduction To Algebra

Introduction to Algebra

Algebra is great fun - you get to solve puzzles!

A Puzzle

What is the missing number?

-

2

=

4

OK, the answer is 6, right? Because 6 - 2 = 4. Easy stuff.
Well, in Algebra we don't use blank boxes, we use a letter (usually an x or y, but any letter is fine). So we would write:

x

-

2

=

4

It is really that simple. The letter (in this case an x) just means "we don't know this yet", and is often called the unknown or the variable.
And when you solve it you write:

x

=

6

Why Use a Letter?

	Because:
	it is easier to write "x" than drawing empty boxes (and easier to say "x" than "the empty box").
	if there were several empty boxes (several "unknowns") we can use a different letter for each one.

So x is simply better than having an empty box. We aren't trying to make words with it!
And it doesn't have to be x, it could be y or w ... or any letter or symbol you like.

How to Solve

Algebra is just like a puzzle where you start with something like "x-2 = 4" and you want to end up with something like "x = 6".
But instead of saying "obviously x=6", use this neat step-by-step approach:

• Work out what to remove to get "x = ..."
• Remove it by doing the opposite (adding is the opposite of subtracting)
• Do that to both sides

Here is an example:

We want to remove the "-2"	To remove it, do the opposite, in this case add 2:	Do it to both sides:	Which is ...	Solved!

Why did we add 2 to both sides?

To "keep the balance"...

	Add 2 to Left Side	Add 2 to Right Side Also
In Balance	Out of Balance!	In Balance Again

Just remember this:

To keep the balance, what you do to one side of the "=" you should also do to the other side!

Another Puzzle

Solve this one:

x

+

5

=

12

Start with:	x + 5 = 12
What you are aiming for is an answer like "x = ...", and the plus 5 is in the way of that! If you subtract 5 you can cancel out the plus 5 (because 5-5=0)
So, let us have a go at subtracting 5 from both sides:	x+5 -5 = 12 -5
A little arithmetic (5-5 = 0 and 12-5 = 7) becomes:	x+0 = 7
Which is just:	x = 7
	Solved!
(Quick Check: 7+5=12)