## When Does the Kinetic Theory of Gases Fail? Examining its Postulates with Assistance from Simple Linear Regression in R

#### Introduction

The Ideal Gas Law, $\text{PV} = \text{nRT}$, is a very simple yet useful relationship that describes the behaviours of many gases pretty well in many situations.  It is “Ideal” because it makes some assumptions about gas particles that make the math and the physics easy to work with; in fact, the simplicity that arises from these assumptions allows the Ideal Gas Law to be easily derived from the kinetic theory of gases.  However, there are situations in which those assumptions are not valid, and, hence, the Ideal Gas Law fails.

Boyle’s law is inherently a part of the Ideal Gas Law.  It states that, at a given temperature, the pressure of an ideal gas is inversely proportional to its volume.  Equivalently, it states the product of the pressure and the volume of an ideal gas is a constant at a given temperature.

$\text{P} \propto \text{V}^{-1}$

#### An Example of The Failure of the Ideal Gas Law

This law is valid for many gases in many situations, but consider the following data on the pressure and volume of 1.000 g of oxygen at 0 degrees Celsius.  I found this data set in Chapter 5.2 of ”General Chemistry” by Darrell Ebbing and Steven Gammon.

               Pressure (atm)      Volume (L)              Pressure X Volume (atm*L)
[1,]           0.25                2.8010                  0.700250
[2,]           0.50                1.4000                  0.700000
[3,]           0.75                0.9333                  0.699975
[4,]           1.00                0.6998                  0.699800
[5,]           2.00                0.3495                  0.699000
[6,]           3.00                0.2328                  0.698400
[7,]           4.00                0.1744                  0.697600
[8,]           5.00                0.1394                  0.697000

The right-most column is the product of pressure and temperature, and it is not constant.  However, are the differences between these values significant, or could it be due to some random variation (perhaps round-off error)?

Here is the scatter plot of the pressure-volume product with respect to pressure.

These points don’t look like they are on a horizontal line!  Let’s analyze these data using normal linear least-squares regression in R.

## How do Dew and Fog Form? Nature at Work with Temperature, Vapour Pressure, and Partial Pressure

In the early morning, especially here in Canada, I often see dew – water droplets formed by the condensation of water vapour on outside surfaces, like windows, car roofs, and leaves of trees.  I also sometimes see fog – water droplets or ice crystals that are suspended in air and often blocking visibility at great distances.  Have you ever wondered how they form?  It turns out that partial pressure, vapour pressure and temperature are the key phenomena at work.

Dew ( and Fog )

Source: Wikimedia

## Estimating the Decay Rate and the Half-Life of DDT in Trout – Applying Simple Linear Regression with Logarithmic Transformation

This blog post uses a function and a script written in R that were displayed in an earlier blog post.

#### Introduction

This is the second of a series of blog posts about simple linear regression; the first was written recently on some conceptual nuances and subtleties about this model.  In this blog post, I will use simple linear regression to analyze a data set with a logarithmic transformation and discuss how to make inferences on the regression coefficients and the means of the target on the original scale.  The data document the decay of dichlorodiphenyltrichloroethane (DDT) in trout in Lake Michigan; I found it on Page 49 in the book ”Elements of Environmental Chemistry” by Ronald A. Hites.  Future posts will also be written on the chemical aspects of this topic, including the environmental chemistry of DDT and exponential decay in chemistry and, in particular, radiochemistry.

Dichlorodiphenyltrichloroethane (DDT)

Source: Wikimedia Commons

A serious student of statistics or a statistician re-learning the fundamentals like myself should always try to understand the math and the statistics behind a software’s built-in function rather than treating it like a black box.  This is especially worthwhile for a basic yet powerful tool like simple linear regression.  Thus, instead of simply using the lm() function in R, I will reproduce the calculations done by lm() with my own function and script (posted earlier on my blog) to obtain inferential statistics on the regression coefficients.  However, I will not write or explain the math behind the calculations; they are shown in my own function with very self-evident variable names, in case you are interested.  The calculations are arguably the most straightforward aspects of linear regression, and you can easily find the derivations and formulas on the web, in introductory or applied statistics textbooks, and in regression textbooks.

## Why Does Diabetes Cause Excessive Urination and Thirst? A Lesson on Osmosis

#### A TABA Seminar on Diabetes

I have the pleasure of being an executive member of the Toronto Applied Biostatistics Association (TABA), a volunteer-run professional organization here in Toronto that organizes seminars on biostatistics.  During this past Tuesday, Dr. Loren Grossman from the LMC Diabetes and Endocrinology Centre generously donated his time to deliver an introductory seminar on diabetes for biostatisticians.  The Institute for Clinical and Evaluative Sciences (ICES) at Sunnybrook Hospital kindly hosted us and provided the venue for the seminar.  As a chemist and a former pre-medical student who studied physiology, I really enjoyed this intellectual treat, especially since Loren was clear, informative, and very knowledgeable about the subject.

The blue circle is a global symbol for diabetes.

Source: Wikimedia Commons

## The Gold Foil Experiment and The 250-Million-Ton Pea: The Composition of the Atom

#### This Atom Is Not To Scale

In a recent post about isotopic abundance, I used a prototypical image of a lithium atom to illustrate the basic structure of an atom.  However, the image was deliberately not drawn to scale to make the protons, neutrons, and electrons visible.  Let’s look at the basic composition of the atom to see why, and we owe this understanding to Ernest Rutherford.  First, let’s give some historical background about what motivated Rutherford to conduct this experiment; we first turn to the Plum Pudding Model by J.J. Thomson.

#### The Plum Pudding Model

Before 1911, the dominant theory of atomic composition was J.J. Thomson‘s “plum pudding” model.  Thomson hypothesized that an atom consisted of electrons as negatively charged particles (the “plums”) “floating” in a “pudding” of positive charge.

Plum Pudding Model of the Atom

Source: Wikimedia Commons