Error Analysis of Hydrometer Calibration Method

This analysis examines sources of error involved in calibration of hydrometers using a 1:1 dilution of a saturated salt (NaCl) solution in water.

The first source of variability to be examined is the effect of temperature on the concentration of salt in the saturated solution. The following data were taken from Solubilities of Inorganic and Metal Organic Compounds, Vol. 2, by William Flinke.

Temperature ºC	Grams NaCl per 100 grams H2O in saturated solution	Grams NaCl per 100 grams saturated solution	Density g/ml
10	35.72	26.31	1.2048 *
15	35.76	26.34	1.2024
20	35.82	26.38	1.2001
25	35.92	26.43	1.1979
30	36.04	26.50	1.1958 *

(* density data points marked with asterisk are extrapolated from a quadratic curve fit on the three given points.)

Now consider a 1:1 dilution of the saturated solution with plain water at the same temperature. The density of water vs temperature is

Temperature ºC	Density g/ml
10	0.9997
15	0.9991
20	0.9982
25	0.9970
30	0.9957

The resulting mass fraction, molality, density and specific gravity of the dilution are

Temperature ºC	Mass Fraction (mass NaCl)/(mass solution)	Molality (moles NaCl)/(kg water)	Density g/ml	Specific Gravity	Specific Gravity of Solution Brought to 20C After Dilution
10	0.1438	2.8737	1.1078	1.1082	1.1058
15	0.1439	2.8757	1.1059	1.1069	1.1059
20	0.1440	2.8789	1.1040	1.1060	1.1060
25	0.1442	2.8843	1.1020	1.1053	1.1062
30	0.1446	2.8926	1.1001	1.1048	1.1064
35	0.1450	2.9025	1.0980	1.1045	1.1068
40	0.1454	2.9124	1.0959	1.1048	1.1071

The density data were calculated using two-dimensional cubic spline interpolation of tabular data "Volumetric Properties of Aqueous Sodium Chloride Solutions", specifically the table of specific volume versus temperature and molality, in the CRC Handbook of Chemistry and Physics, 79^th Edition.

The 40C data is questionable since the saturated solution density extrapolation is a bit far removed from the actual data range.

So we can conclude that the 1:1 diluted solution has a specific gravity of 1.106 +/- .001 over the range of 15-35C. When the diluted solution is brought to 20C, the specific gravity is 1.106 -0/+.001 for saturated solutions made up and diluted at temperatures over 15-35C.

The next source of variability is dilution of the saturated solution with water at a different temperature. We will examine just one point to see if there is a significant effect.

Assume that the saturated solution is made up at 35C, and is diluted with an equal volume of water at 15C, and the diluted solution is then brought to 20C.

The specific gravity of the resulting 20C solution is 1.1065.

Therefore the calibration process is not significantly sensitive to the dilution water temperature not being the same as the saturated solution temperature.

The third source of error is lack of repeatability in measuring volumes. If the volume of dilutant is not exactly equal to the volume of the saturated solution, then there will be an error in the specific gravity of the diluted solution.

Let's consider just one example. Suppose that the volume of dilutant is one percent less than the volume of saturated solution, and consider this at 20C only.

The resulting specific gravity will be 1.1063 instead of 1.1060. This implies that three percent error can be tolerated before the error introduced is one gravity point. One percent repeatability should be readily achievable with a reasonably tall form factor container. This author has demonstrated repeatability of better than 1% measuring about 325 ml in a 12 oz. beer "pilsner" glass, using a mark on a piece of Scotch tape on the glass to mark the level of the liquid.

It is concluded that with reasonable care in measuring liquid volumes, no significant error due to volume measurement will be introduced.

© Copyright 1999, 2002 Stanley E. Prevost All Rights Reserved. This document may be copied and distributed for noncommercial use provided that this copyright notice is maintained with the document.