Chemistry: Ionic Compounds with the Transition Metals

Tutoring chemistry, ionic compounds with the transition metals often need extra attention because of the Roman Numerals.

You can read my previous posts (Oct 20 and Oct 25) about writing ionic formulas.  Assuming you’re up to speed, we’ll discuss how to do it with transition metals.

The transition metals are in the low area in the middle of the periodic table.  You’ll recognize iron (Fe), copper (Cu) and others among them.  Most transition metals have more than one possible combining capacity – in that way, they are different from the other elements.  Of course, we know that to write an ionic formula, we need to know the combining capacity of the metal and of the nonmetal.  How do we proceed?

The tell is the Roman Numeral in the written formula.  For instance, you might encounter iron(III)sulphate.  Note the III in the middle of the compound.  Only formulas with transition metals have a Roman Numeral.  The Roman Numeral tells you the combining capacity of the metal.  In the case of iron(III)sulphate, the combining capacity of iron is 3.  From the point of view of positive and negative, metals are always positive.  Therefore, you can imagine the charge on the iron to be +3.

Knowing (from the Roman Numeral) the combining capacity (i.e., the charge) of the metal, we proceed as I’ve described in previous posts:  positives have to equal negatives.

iron:  Fe3+
sulphate: SO42-

Iron brings 3+; sulphate brings 2-. The way to make the positives equal the negatives is to go to six each:  2×3+=6+, and 3×2-=6-.  Therefore, we need two irons and three sulphates:

Fe2(SO4)3.

So, the formula of iron(III)sulphate is Fe2(SO4)3.

Note that, when writing the word formula from the symbolic formula (the opposite way from what we’ve just done), you need to include the Roman Numeral. For example, suppose you have CuNO3. Consider the nitrate (you can look up its charge on a table of ions):
 
nitrate: NO3

So we know that the charge on nitrate is 1-. Since there is one nitrate and one copper, the charge on the copper must be 1+ so that the positives and negatives are equal. Therefore, we would write this formula as copper(I)nitrate: the Roman Numeral shows the numeric charge on the copper.

Remember, you only use Roman Numerals with transition metals – not the others – and only in written formulas.

Hope this helps.

Jack of Oracle Tutoring By Jack and Diane, Campbell River, BC.

Biology: How a Kidney Works

When you tutor biology or any science, often the basic concept is better without all the extra details thrown in.

The renal artery brings blood to the kidneys to be filtered.  Upon reaching a kidney, the renal artery branches into thousands of arterioles, each of which leads to a nephron.  A nephron is the basic working unit of the kidney; each kidney may contain around a million of them, according to wikipedia.

Upon entering the nephron, the blood is spun at high speed in the glomerulus.  The glomerulus is a “merry-go-round” of capillaries.   Water, glucose, amino acids, ions, urea, and uric acid can pass through it;  the high speed flings them out.  The solution thus expelled from the blood is called the filtrate.  It enters the convoluted tubule.  This phase of kidney function is called glomerular filtration.

The main point of the kidneys is excretion, which means ridding the blood of nitrogenous wastes – chiefly urea and uric acid.  Therefore, most of the filtrate now needs to be reclaimed.  Virtually all the glucose and amino acids, most of the water, and some of the ions are reabsorbed by the peritubular capillary network that surrounds the convoluted tubule.  This phase of kidney function is called tubular reabsorption.

At the same time, certain non-filtrable species (too big to exit the blood by glomerular filtration) are actively removed from the capillary into the convoluted tubule.  These non-filtrable species include drug remnants.  Filtrable ions that escaped glomerular filtration can also be removed from the blood in this way.  In either case, this phase of kidney function is called tubular secretion.

What remains of the filtrate after tubular reabsorption – with the targets of tubular secretion added in – flows into the collecting duct, the ureter, and finally into the urinary bladder.

There we have it:  basic kidney function.  It consists of three phases:  filtration, reabsorption, and secretion.  Of course, you can read further.  I use Sylvia S. Mader’s Inquiry into Life, 11th edition, McGraw-Hill.

I hope this gets you started, anyway:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC

Ionic Compounds with Radicals

Tutoring high school chemistry, ionic compounds are fundamental.  We’ll discuss how to write their formulas when radicals and involved.

In my Oct 20 post about the periodic table and ionic compounds, I mentioned the issue of metal vs nonmetal.  As I described in that article, it’s easy to tell for individual atoms.  However, what about something like NO3?

The answer is that if the species has a negative charge, it is a nonmetal; if its charge is positive, it’s a metal.  Therefore, Cr2O72- is a nonmetal, even though Cr itself is a metal.  By that same reasoning, NH4+ is a metal, even though N itself is a nonmetal.

Species like nitrate (NO3) and ammonium (NH4+) are called radicals or polyatomic ions. There are many of them; type “table of polyatomic ions” into your browser for lists.

To use a radical in a formula, you remember that the negative charges have to add up to the same as the positives.  If you need a multiple of the radical, you use brackets.  Here are some examples:

calcium hydroxide:  Look up “ion table” in your browser to find a list of common ions.  You’ll find calcium is Ca2+. Then, find (on the same table, or from a table of polyatomic ions) hydroxide. You’ll find it’s OH. The negative sign with no number means a charge of negative 1.

So, we have Ca2+ and OH.  Realizing positives must equal the negatives, we need two hydroxides (2x-1=-2).  So we have

calcium hydroxide:  Ca(OH)2

By a similar process, you can assemble the following:

sodium sulphate:  Na2SO4

magnesium nitrate:  Mg(NO3)2

One important point to remember is that with a polyatomic ion such as NO3, the 3 doesn’t mean there are three nitrates; it means there are three oxygens in one nitrate.  Therefore, if you need three nitrates – such as in the case of aluminum nitrate – you would put a three outside the brackets, as follows:

aluminum nitrate: Al(NO3)3

We have more to discuss about formulas of ionic compounds. Good luck with this installment.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Prime Factorization and Reducing Fractions

Continuing our discussion about the prime factorization, we focus on using it to reduce fractions containing large numbers.  In high school math tutoring, this application – happily – has resurfaced.

You can read about prime factorization in my Oct 14 post.  Let’s assume you’re old hat at it now.

We are familiar with reducing fractions.  For instance,

(1)   \begin{equation*}\frac{40}{35}=\frac{8}{7}\end{equation*}

We know that the reason for the reduction is that 5 divides into both 40 and 35, so can be cancelled out top and bottom:

(2)   \begin{equation*}\frac{40}{35}=\frac{8*5}{7*5}=\frac{8}{7}\end{equation*}

Of course, it’s easy to tell that 40/35 reduces to 8/7.  However, what about reducing a fraction like 98/154?  Well, you can cut both down by 2, then realize 7 goes into both, and so on.  However, let’s try the prime factorization method.  First, we’ll rewrite the fraction with each number’s prime factorization:

(3)   \begin{equation*}\frac{98}{154}=\frac{2*7*7}{2*7*11}\end{equation*}

Note that 2*7 occurs in both the top and the bottom.  It’s called the greatest common factor (the GCF), since it’s the largest combination that occurs in both prime factorizations.  Cancel the GCF top and bottom and you get the reduced fraction:

(4)   \begin{equation*}\frac{98}{154}=\frac{2*7*7}{2*7*11}=\frac{7}{11}\end{equation*}

Like with finding the LCM (see my Oct 17 post), the prime factorization method for reducing fractions becomes more useful as the numbers get bigger and less familiar.  Try it and see :)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Chemistry: the Periodic Table and Ionic Compounds

When you tutor high school chemistry, these basics come up every year.  Let’s sort them out.

From the high school chemistry point of view, there are two types compounds:  ionic and covalent.  Ionic names start with a metal; covalent names contain only nonmetals.

What is a metal and what is a nonmetal?  There are two ways to answer that question.  Talking about elements on the periodic table, the metals are on the left.  They stretch all the way to a boundary that starts at the 13th column and zigzags down to the right.  Usually it’s a red line, but if it’s not on yours, here’s the boundary:

boron is a nonmetal, but aluminum is a metal;

silicon is a nonmetal, but germanium is a metal;

arsenic is a nonmetal, but antimony is a metal;

tellurium is a nonmetal, but polonium is a metal.

Anything to the right of that boundary is a nonmetal.  Once again, anything to its left is a metal.

The important concept about forming an ionic compound – such as calcium chloride – is that the nonmetal “charges” (or combining capacities) must equal the metal charges (or combining capacities, depending on how you see it).

Some periodic tables have these charges right on them:  they’re either positive or negative, and most are between 1 and 5.  However, old ones often don’t.  If yours doesn’t, here’s how to tell what the combining capacity is:

Column H, Li, Na, etc Be B C N O F
Combining Capacity 1 2 3 4 3 2 1

To our example:  calcium chloride.  Calcium is under Be, so its combining capacity is 2.  On the other hand, chlorine is under F, so it has a combining capacity of 1.  Since the nonmetal combining capacity has to equal the metal, we need two chlorines.  2×1=2, which equals the calcium’s 2 (calcium being the metal, of course).  The formula for calcium chloride is, therefore, CaCl2.

By that same reasoning, the formula of sodium sulfide is Na2S. Finally, the formula of magnesium oxide is MgO (since both Mg and O have the same combining capacity, you only need one each).

We’ll make some observations:

1)  When written in words, the names are in all lower case.

2)  When written as formulas, only the first letter of each atomic symbol is capitalized.  For example, hydrogen is written H, but sodium is written Na.

3)  When naming a simple ionic, the second atom (the nonmetal) gets an -ide ending.

4)  The sunken middle portion of the table (which starts on the left with Sc) we didn’t discuss this time.  Its members get a slightly different treatment, which we’ll address soon.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Uses of Prime Factorization: Finding Least Common Multiple

We continue our exploration of prime factorization.  Any high school math tutor deals with the subject a few times a year.

Continuing from our previous post, let’s investigate one use of the prime factorization:  finding the least common multiple (LCM) of two (or more) numbers.

You can find the LCM just by listing the multiples of the numbers.  For instance, imagine we need the LCM of 12 and 14:

12, 24, 36, 48, 60, 72, 84, 96, 108….

14, 28, 42, 56, 70, 84, ….

We see 84 is the first number to show up in both lists.  Therefore, the LCM of 12 and 14 is 84.

Using the prime factorization method (once again, refer to the previous post for the quick details),

12=2x2x3

14=2×7

The LCM of 12 and 14 is the simplest combination that includes the prime factorizations of both:

2x2x3x7

Note that 2x2x3x7 includes the prime factorization of 12 (which is 2x2x3, from above), and also includes the prime factorization of 14 (which is 2×7, also from above).  2×7 does not occur written exactly that way, but since this is multiplication, we could change the order:

2x3x2x7

Now 2×7 does occur just as written.  Note that 2x2x3x7=2x3x2x7=84, which was the LCM we got from the list method.

This simple trick to find the LCM is very handy as the numbers get bigger.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC

 

Prime Numbers and Prime Factorization

In math tutoring, prime numbers and prime factorization are receiving fresh attention.  We’ll briefly look at the concepts.

Before we talk about prime numbers, let’s define factor.  In math, a factor is a number that divides into another number with no remainder.  Therefore, 4 is a factor of 12; 7 is a factor of 14.

We also need to define product.  A product is the answer to a multiplication.  Therefore, the product of 2 and 5 is 10.  We could also say that 36 is the product of 9 and 4.

A prime number is a number with only two factors:  itself and one.  The number 1 is not prime, since it only has one factor:  1.  However, 2 is prime, as are 3, 5, 7,11,13, 19, and so on.  Note that 9 is not prime, since 9 has 3 as a factor besides itself and 1.

The prime factorization of a number is the number stated as a product of primes.  Each number has a unique prime factorization, if you discount the different orders.  The prime factorization of 10 is 2×5.  The prime factorization of 48 is 2x2x2x2x3, or 24x3.  Note, once again, that the prime factorization only includes prime numbers as defined above.

To illustrate how to get the prime factorization of a number, you might use the ancient “factor tree” method.  For example, imagine you need the prime factorization of 396:

Notice each path ends with a prime number.  Collecting them all gives 396=2x2x3x3x11, or 396=2²x3²x11.

We’ll explore some uses of the prime factorization in future posts.

Jack of  Oracle Tutoring by Jack and Diane, Campbell River, BC

Study Strategy

One question a math tutor – or any tutor – might be asked: “How do I prepare for an exam?”   We’ll examine a basic strategy.

Exam preparation is an important process.  Few people like doing it, so a lot less of it gets done than is really needed.  Naturally, people wonder if there’s a “right” way to do it.  Well, there is!

The secret to exam preparation – and the harder the material, the more advantage it affords – is to start early.  For instance, starting the day the course starts is not absurd.  The other point is that you should do it in small daily portions – maybe a half hour a day or less in most cases.  Usually, if you start long before the exam, you can afford to study only 15 – 30 minutes a day for it.

How should you structure these daily study rituals?

The answer is, be casual.  You can afford to be, since you’re starting so early.  Look over your notes.  Spend a few extra readings on the more difficult parts.  Perhaps you want to highlight some points you still don’t understand, so you can bring them up quickly at the next lecture.  (Instructors usually like being asked about earlier lectures:  it shows you’re paying attention.)  You might not bring all the questions up at once; rather, maybe ask one question today, then one next time.

You might check out the course schedule to discover the future content and leaf through the textbook to get a little familiar with it before it’s taught.  Maybe you only spend five minutes doing this each time, but the pre-emptive strikes on course content usually pay off big.  People learn much easier when they’ve already heard of something than when it’s totally new.

Enjoy these half-hour sessions.  Have a cup of tea while you leaf through your notes.  You don’t have to go through your notes in order; using a random “here, then there” selection often works just as well.  Don’t feel you need to cover all your notes each time; spend an honest 30 min, then stop.

If you’re studying for math, you’ll have to attempt questions from the sections you’re looking back on.  Spread it out:  do a couple from this section, then a couple from another one.  Do a few from long ago, then a couple of recent ones.  After an honest half an hour, stop.  You can afford to – as long as you start this habit long before exam time.  Of course, you can adjust your durations to your personal taste; this is only a rough guide.

I don’t recommend listening to music or watching TV while studying.  Real studying can be a lonely business – no doubt about it.  However, if you start early, and do it daily, it should be a lot easier.

I should offer some reasoning for why this method works (or, at least, why it’s better than cramming).  Think of any biological process – muscle development, for example.  Working out for eight hours today won’t help you develop stronger muscles; we all know that.  However, working out for tweny minutes a day, three days a week, for eight weeks (which totals eight hours), will definitely produce substantial results in most cases.

Biology takes time; you can’t rush it.  However, when you give it time, you often get better results than you would have imagined.  You know that your lawn won’t grow measurably today.  However, you know that if you let it grow for two weeks, you might barely be able to pass the mower through it.

Your daily attention to your studies – even if it is only a few minutes (as long as real effort is achieved during that time) reminds your brain that it needs to continue learning that material.  Your brain – similar to your body – only believes something is worth adapting to if it keeps coming up over and over again.  You have to convince it that what you’re studying is important and here to stay.  That’s what the daily study sessions do – even if they’re only 15 minutes to half an hour.

For those who are reading this a couple of days before the big exam:  well, I guess you’ll have to cram for now.  But you’ll know for next time.

Good luck with your midterms.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Horizontal Asymptotes and Holes: some comments about rational function graphs

When tutoring math 12 or calculus, you encounter graphs of rational functions.  Let’s look at a couple of features:

Rational function graphs are defined by (and you get marks for)  the locations of the asymptotes (if any), as well as the x and y intercepts (once again, provided they exist) and holes (if any).  Today, we’ll look at two of these features:  horizontal asymptotes and holes.

First, to holes:  consider the following rational function:

(1)   \begin{equation*}f(x)=\frac{(x-1)(x+2)}{(x+2)(x-3)}\end{equation*}

You can see that, since (x+2) is both in the top and the bottom, it simplifies to

(2)   \begin{equation*}f(x)=\frac{(x-1)}{(x-3)}\end{equation*}

However, when you cancel, you are really dividing.  Since you can’t divide by zero, you can’t cancel x+2 when x=-2.  At x=-2, the equation remains undefined.  Therefore, you will get a hole there.  The graph of (1), above, will follow the graph of (2) identically, except for a hole at x=-2.

Now, to horizontal asymptotes:  you get them when the degree on top matches the degree on the bottom or is less than the degree on the bottom.

Case 1:  the degrees on top and bottom are the same.  Consider

(3)   \begin{equation*}f(x)=\frac{2x^2 - 2x -3}{x^2 +17x+11}\end{equation*}

To get the horizontal asymptote, divide the coefficient of the highest exponent term on top by the coefficient of the highest exponent term on the bottom.  The horizontal asymptote will be y=2/1, or just y=2.

Case 2:  the degree on the bottom is greater than that on top.

Simple:  the horizontal asymptote is y=0.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC