Lifestyle, history: how was acetaminophen discovered?

The tutor shares a few facts from his recent research of pain medications.

In 1886, Doctors Arnold Cahn and Paul Hepp were trying naphthalene as a treatment for intestinal worms. They ordered more, but a pharmacist mistakenly sent acetanilide.

One of the patients was not only suffering from worms, but also from fever. After they took the acetanilide, the fever subsided, although the intestinal worms remained. The doctors examined the medication and discovered it wasn’t naphthalene, but rather acetanilide.

Having noticed the fever-reducing performance of the acetanilide, the doctors performed more trials with it. They discovered it to be effective as a pain reliever (analgesic) as well. A derivative, phenacetin, became established by Bayer for relief of pain and fever.

Phenacetin, even in moderate doses, was found to be toxic. In 1899, Karl Morner of Germany discovered that the body metabolizes acetanilide into acetaminophen. (The same effect occurs with phenacetin.) In the UK, Sterling realized that acetaminophen is effective against pain and fever, without the toxicity of phenacetin.

Acetaminophen went to market in the US and the UK in the 1950s. It is also known as paracetamol, and sold under many brand names worldwide.


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

English: terror vs horror

Tutoring English, the horror genre can be prominent. The tutor examines the difference between terror and horror.

Apparently, terror refers to the dread of anticipating an attack. Just as likely, the nature of the attack is unknown. When someone is terrified, it’s because they are afraid a destructive event is likely. The event may not threaten the terrified person; they might be terrified on behalf of someone else.

Horror, on the other hand, is felt as the destructive event unfolds. At the point of horror, the mystery of the attack ceases; its nature manifests. The horrified person is experiencing or witnessing the damage, or else reliving it.


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

Chemistry: Graham’s Law of Effusion

Tutoring chemistry, you may mention kinetic energy of particles, diffusion, and effusion. The tutor gives a brief explanation.

In my post from Jan 18, 2017, I define effuse: it means to escape from a container through a porous boundary.

Graham’s Law of Effusion compares the rate at which two different gases will effuse, based on their comparative molecular masses. If eA is the effusion rate of gas A, and eB that of gas B, then

eA/eB = (MMB/MMA)1/2

where MMA is molecular mass of gas A, and MMB, that of gas B.

The reasoning behind the formula is that effusion depends on molecular motion, which is quantified by kinetic energy, KE:

KE = 0.5MMv2

where, once again, MM means molecular mass, while v means velocity.

Two gases of the same temperature have the same kinetic energy:


Dividing both sides by 0.5MMAvB2 gives

vA2/vB2 = MMB/MMA

square rooting both sides gives

vA/vB = (MMB/MMA)1/2

Since effusion is motion through pores, the ratio of velocities is the ratio of effusion:

eA/eB=vA/vB = (MMB/MMA)1/2

Example: Compare the effusion rate of methane, CH4, with that of propane, C3H8.

Solution: the ratio of effusion should be


Note that methane, CH4, has MM=16, while propane, C3H8, has MM=44. Therefore,


Methane should escape from a porous container 1.66 times the rate that propane escapes.

Note that, in a more general sense, this is a law of diffusion. Therefore, if propane and methane are released at one end of a room, methane should reach the other end 1.66 times as quickly as propane.



Mortimer, Charles E. Chemistry, sixth ed. Belmont: Wadsworth, 1986.

White, J. Edmund. Physical Chemistry, College Outline Series. New York: Harcourt Brace Jovanovich, 1987.

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

Perl: int() function: isolating a digit

Tutoring computer science, extracting a digit is a suggested exercise. The tutor shows a use of Perl’s int() function.

Let’s imagine you want to find the hundred-thousands digit of an input number. You can do so with the following code:


In the code above, the $input_number is first divided by 100000, which will likely give a decimal. Then, int() simply removes the decimal places, but doesn’t round. Next, %10 gives the remainder when the result is divided by 10. This is needed if the input number is a place past the hundred thousands; eg., the millions, ten millions, etc.


McGrath, Mike. Perl in easy steps. Southam: Computer Step, 2004.

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

Geography: what is the Dahomey Gap (aka Togo Gap)?

Tutoring geography, you notice features on maps.

Apparently, the Dahomey Gap, possibly aka the Togo Gap, is a strip of dry woodland that runs north from the coast of Benin, Togo, and eastern Ghana.

The Gap divides the equatorial rainforest of west Africa into two distinct regions. While no tall mountains surround it, uplands do, which may explain its local dryness.



O’Shea, Mark. Venomous Snakes of the World. Princeton: Princeton University Press, 2005.

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

Statistics, spreadsheets: confidence interval for the mean, population standard deviation unknown: CONFIDENCE.T() function on LibreOffice Calc

Tutoring statistics, the tutor is happy to share the CONFIDENCE.T() function from LibreOffice Calc.

My last couple of posts (here and here) I’ve talked about confidence intervals for the mean. Yesterday I mentioned finding one using Excel or LibreOffice Calc’s CONFIDENCE() function.

While the CONFIDENCE() function assumes the population standard deviation is known, I pointed out that, with sample size n≥31, the t-distribution approximates the normal closely enough that the sample standard deviation can be used. Today, I’ll make a direct comparison.

Yesterday’s post considered a sample mean of 67.3, known population standard deviation of 12.4, and sample size 42. The input


gave the result 3.75, meaning a confidence interval of 67.3±3.75, or 63.55 to 71.05.

LibreOffice Calc’s CONFIDENCE.T() function has the following format:

=confidence(1-confidence_level, sample_standard_deviation, sample_size)

Since it uses the sample standard deviation, CONFIDENCE.T() calculates the confidence interval from the t-distribution. By constrast, CONFIDENCE() takes the population standard deviation, so uses the normal distribution to calculate the confidence interval.

The following input

=confidence.t(0.05, 12.4, 42)

gives the result 3.864, implying a confidence interval of 67.3±3.864 or 63.44 to 71.16. Obviously this is not much different from the confidence interval 63.55 to 71.05 gotten using =confidence(0.05,12.4,42).

So, the CONFIDENCE.T() function seems to demonstrate that, for a sample size n≥31, the t-distribution approximates the normal distribution closely enough that the sample standard deviation can be used when the population standard deviation is unavailable.


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

Statistics, spreadsheets: confidence interval for population mean: CONFIDENCE() function on Excel and LibreOffice Calc

Tutoring statistics, you realize how convenient using a spreadsheet can be.

In yesterday’s post I mentioned some theoretical points about two-sided confidence intervals for the population mean.

On the practical side, if you simply need a confidence interval for the population mean, you can use Excel’s CONFIDENCE() function, which works the same on LibreOffice Calc. It has the following format:

=confidence(1-confidence_level, pop_standard_deviation, sample_size)

The formula assumes the population standard deviation is known. If not, you can just use a sample_size ≥31, calculate the sample standard deviation, and use it. This gives a pretty good approximation (see yesterday’s post).

The CONFIDENCE() formula gives the margin of error for the confidence interval. To get the actual lower and upper bounds, you both subtract and add its output to the sample mean.


Imagine an exam written by 706 students. A sample of 42 papers reveals a mean grade of 67.3 and standard deviation 12.4. Give a 95% confidence interval for the mean exam mark.


The confidence level is 95% = 0.95, so the first parameter is 1-0.95=0.05.

In a cell, key

=confidence(0.05, 12.4, 42)

Hopefully, you obtain the output 3.75, which means the confidence interval for the mean is given by



63.55 to 71.05

Apparently the mean, with 95% confidence, is between 63.55 and 71.05.


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

Statistics: confidence interval for the mean (two sided)

Tutoring statistics, confidence intervals are important.

A two-sided confidence interval for the population mean is given by

sample_mean – (standard_dev/n1/2)*sig_factor, sample_mean + (standard_dev/n1/2)*sig_factor

The sig_factor (significance factor) depends on the certainty (confidence level) with which we want the confidence interval to include the population mean; typically it’s around 2 (aka, 1.96) for 95% confidence.

The standard deviation might be known or might be calculated from the sample itself. If it’s known, the normal distribution is used; if calculated, then technically the t-distribution should be used (see point 3 below).

There are a few points that make the two-sided confidence interval for the population mean an elegant construct:

  1. Its lower and upper boundaries depend on the sample size, but not the population size.
  2. For sample size n≥31, the parent population needn’t be normal for the sample mean to be normally distriubted. This validates the confidence interval even for a non-normal population for n≥31. It’s a consequence of the Central Limit Theorem. (Actually, the rule of thumb is n≥30, but for the purpose of the next point, I like 31.)
  3. For n≥31, the t-distribution approximates the normal to around 4%, so the normal approximation can probably be used even for unknown population standard deviation.


Harnett, Donald L. and James L. Murphy. Statistical Analysis for Business and Economics, first Can. ed. Don Mills: Addison-Wesley, 1993.

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

Battery output vs discharge rate: Peukert’s Law

Tutoring physics or chemistry, you might encounter Peukert’s Law, although it’s probably used more by industry.

In my March 2 post I mentioned reserve capacity and Amp*hours as two ways to measure a battery’s potential output. Numerically they are convertible backwards and forwards, but the reality is not necessarily so simple, because the speed of discharge affects the total output a battery can manage. Specifically, a higher discharge rate lessens the battery’s efficiency, so that the total output will decrease the faster the battery is discharged.

Peukert’s Law gives the equation

t=T(C/(I*T))k, where

t is the actual time the battery will deliver arbitrary current I,

T is the discharge time corresponding to the given amp*hour rating,

C is the given amp*hour rating,

k is a physical constant that depends on the type of battery (around 1.4 for lead-acid).

Because of Peukert’s Law, an amp*hour rating must be given with a specific time for which it’s valid. (My reading suggests that 20 hours might be a typical time.) Therefore, an amp*hour rating might read “120Ah over 20 hours”. Such a rating implies a discharge rate of 6A for 20 hours. How long the battery can deliver a different amperage can be calculated by Peukert’s Law.

Example: Imagine a battery rated 120Ah over 20 hours. How long can it deliver 120A?

Solution: Theoretically, a 120Ah battery can deliver 120A for 1 hour, although we already know not to expect it. Peukert’s Law gives



t=0.30 hours, or about 18 minutes.

I’ll be talking more about lead acid batteries in future posts:)


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

English: what is a lido?

Tutoring English, finding the precise word is helpful.


  1. a beach resort that is accepted as a good place to be seen. (Merriam-Webster)
  2. An outdoor recreation area centered around aquatic activities. (Collins)

Short words, packed with meaning but seldom used, improve writing.


Merriam-Webster Dictionary, 2004.

Collins Essential Canadian English Dictionary & Thesaurus, 2006.

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