Molarity, Molality, Mass Fraction Conversion Formula

Quote of the Day

A leader is best when people barely know that he exists. ... When his work is done, his aims fulfilled, they will all say, 'We did this ourselves.'

Lao Tzu, ancient philosopher. My best managers have taken Lao's approach. The first management advice I received was similar in spirit: (1) keep a low profile, and (2) always take less than my share of the credit and more than my share of the blame. This advice has served me well.


Introduction

Figure 1: Graphical Model of Concentration Formula

Figure 1: Graphical Model of Concentration Formula

I am doing some work with lead-acid batteries and their state of charge versus their H2SO4 electrolyte concentrations. While we normally use specific gravity to assess a battery's state of charge, some folks use molality (m), molarity (M), or mass fraction (w). These concentration metrics are all related by relatively simple formulas (Figure 1).

In this post, I will review the formulas used to convert between the different electrolyte metrics. As part of my review, I will illustrate how to use the formula by recreating a table of combined H2SO4 electrolyte metrics.

My Mathcad source and its PDF here. These calculations depend on specific gravity, which I always treat as equivalent to density with units of mass/volume. Technically, specific gravity is unitless.

Background

Definitions

I refer you to this post where I had previously defined molarity, molality, and mass fraction.

Reference Table

Figure 2 shows a screen capture of a table of H2SO4 concentrations that I used to test my formulas. For molarity and molality, I only used the 25 °C values. I grabbed this table from Google Books, used ABBY FineReader to OCR it, and tossed the data into Mathcad.

Figure 2: Table of H2SO4 Concentration Metrics.

Figure 2: Table of H2SO4 Concentration Metrics (Source).

Analysis

Formula Setup

Figure 3 shows how I setup the calculations. The key conversion formulas are highlighted in green.

Figure 3: Conversion Formulas.

Figure 3: Conversion Formulas.

Cross-Check

Figure 4 shows where I used the formulas to calculate the various concentration metrics and compared my results with the values from Figure 2. I used the formulas of Figure 3 to show that I can compute

  • Molality ⇨ Mass Fraction
  • Mass Fraction and Density ⇨ Molarity
  • Molarity and Density ⇨ Molality

The agreement between the calculations and the empirical results (Figure 2) is excellent.

Figure 4: Comparison with Empirical Results.

Figure 4: Comparison with Empirical Results.

Conclusion

This was just a quick calculation to verify that I had my conversion formulas were coded correctly.

Appendix A: Random Worked Examples.

Here is an example I found in many textbooks (here is an example).

Figure M: Text Book Example 1.

Figure 5: Text Book Example 1.

Here is are two examples I found in this test solution.

Figure 6: Textbook Example 2.

Figure 6: Textbook Exam Solution 1.

Figure 7: Exam Solution Example 2.

Figure 7: Exam Solution Example 2.

 
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