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

I am doing some work with lead-acid batteries and their state of charge versus their H_{2}SO_{4} 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 H_{2}SO_{4} 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 H_{2}SO_{4} 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.

## Analysis

### Formula Setup

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

### 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.

## 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).

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

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