Statistics and Chemistry Lesson of the Day – Illustrating Basic Concepts in Experimental Design with the Synthesis of Ammonia

To summarize what we have learned about experimental design in the past few Applied Statistics Lessons of the Day, let’s use an example from physical chemistry to illustrate these basic principles.

Ammonia (NH3) is widely used as a fertilizer in industry.  It is commonly synthesized by the Haber process, which involves a reaction between hydrogen gas and nitrogen gas.

N2 + 3 H2 → 2 NH3   (ΔH = −92.4 kJ·mol−1)

Recall that ΔH is the change in enthalpy.  Under constant pressure (which is the case for most chemical reactions), ΔH is the heat absorbed or released by the system.

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Physical Chemistry Lesson of the Day: Pressure-Volume Work

In chemistry, a common type of work is the expansion or compression of a gas under constant pressure.  Recall from physics that pressure is defined as force applied per unit of area.

P = F \div A

P \times A = F

Consider a chemical reaction that releases a gas as its product inside a sealed cylinder with a movable piston.

 

Gax_expanding_doing_work_on_a_piston_in_a_cylinder

Image from Dpumroy via Wikimedia.

As the gas expands inside the cylinder, it pushes against the piston, and work is done by the system against the surroundings.  The atmospheric pressure on the cylinder remains constant while the cylinder expands, and the volume of the cylinder increases as a result.  The volume of the cylinder at any given point is the area of the piston times the length of the cylinder.  The change in volume is equal to the area of the piston times the distance along which the piston was pushed by the expanding gas.

w = -P \times \Delta V

w = -P \times A \times \Delta L

w = -F \times \Delta L

Note that this last line is just the definition of work under constant force in the same direction as the displacement, multiplied by the negative sign to follow the sign convention in chemistry.

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.

scatter plot pv vs 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.

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