Volume Reduction in Ethanol-Water Mixtures

Quote of the Day

The greater the artist, the greater the doubt. Perfect confidence is granted to the less talented as a consolation prize.

— Robert Hughes, art critic. This is a restatement of the Dunning-Kruger effect. I see this effect all the time with engineers and physicists – the best of them are full of doubt and the worst are cocksure.

Introduction

Figure 1: Excess Volume Vs Mole Fraction of
Ethanol-Water Mixture (Source).

I recently toured a winery and, during the tour, I saw a number of charts related to various properties of ethanol and water mixtures. I was a bit puzzled when I saw Figure 1, which is a chart that I had never seen before. Figure 1 shows the amount of volume reduction experienced by an ethanol-water mixture compared to the separate volumes of the water and ethanol. I became curious about the chart and decided to see if I could derive these results on my own.

This was a good exercise in basic mixtures and provided me some insight into what happens when you mix liquids. I was able to derive almost exactly the same chart using some data from the Wikipedia.

My Mathcad source and a PDF version are included here. One interesting aspect of this solution is that I demonstrate how to pass a function to a Mathcad program.

Background

Why a Volume Reduction?

You can view the net volume reduction in a number of ways. The best explanations I have seen for describing this effect were in this Quora discussion. The following quote from that discussion provides a simple, intuitive way of viewing what is happening (Source).

Water has a relatively loose structure because the multiple hydrogen bonds keep the molecules apart. This creates a lot of empty space between the molecules (see BIOdotEDU).

When mixed with other substances, this loose structure of liquid water is disrupted, and the empty space becomes less.

For more information on this topic, see the Wikipedia discussion on ideal solutions.

Modeling Overview

The calculation is actually quite simple and is performed in three phases:

• compute the volume of ethanol and water as separate liquids
• compute the volume of the mixture of ethanol and water
• compute the difference between the two volumes per mole

Analysis

Gather Ethanol-Water Mixture Density vs Mass Fraction Data

All the data needed to recreate Figure 1 is present in a table of specific gravity versus mass ratios for ethanol-water mixtures. There are numerous places to find this data – I used the Wikipedia's ethanol data page. You need to make sure that you grab the data for the correct mixture temperature – in my case, 25 °C ethanol and 25 °C water.

Figure 2: Ethanol-Water Mixture Density Data from Wikipedia.

Convert Mass Fraction to Mole Fraction

Figure 3 shows how I converted the mass fraction to a mole fraction. As you might expect, the conversion factor is the molar mass.

Figure 3: Mass Fraction to Mole Fraction Conversion.

Grab Reference Data and Curve Interpolation

Figure 5 shows my reference data, which is also plotted in Figure 1. I was impressed with the amount of chemical data on the Dortmund Data Bank web site.

Figure 5: Obtain Reference Curve and Interpolate Discrete Data.

Grab Reference Data and Curve Interpolation

Figure 6 shows how I computed the excess volume per mole:

• the volume of ethanol and water as separate liquids
• the volume of the mixture of ethanol and water
• the difference between the two volumes per mole

I then graphed this calculation as a function of the mass mix ratio. I should mention that the excess volume per mole is negative because the volume of ethanol and water mixtures is less than the sum of the individual volumes.

Figure 6: Generate Excess Volume Metric and Graph.

Conclusion

My results are very close to the published result and the differences are probably due to differences in the original ethanol-water mixture data. When I started hunting for data online, I saw that there was some variability between the different ethanol-water mixture data sets, which you would expect since it is empirical data.