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

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 (Source).

Analysis

Formula Setup

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

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.

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 5: Text Book Example 1.

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

Figure 6: Textbook Exam Solution 1.

Figure 7: Exam Solution Example 2.