Water: Density vs Temperature and Salinity

Like all substances, water changes density as a function of thermodynamic properties like temperature and pressure.  For most substances, as you decrease the temperature, the density will increase.  This makes some logical sense: temperature is just a measure of the typical kinetic energy of the water molecules, and if you decrease the temperature then they will have less energetic collisions and spread out less.  Water behaves this way for most temperatures (4 - 100) ℃, but when you decrease the temperature below 4 ℃ the density decreases rather than increases.  It continues to decrease slowly until you reach 0 ℃, at which point water freezes into ice and the density decreases by nearly 10%.  This is a well known and studied property of water, and the reason icebergs float with 10% above the surface of the water and 90% below.  However, this behavior is not true for all types of water.  In particular, when you start dissolving substances in water (e.g. - salt in sea water) the relationships change: the freezing point decreases, the temperature of maximum density decreases, etc.  Generally speaking, the density of water as a function of temperature and salinity is complicated.

Given how well studied this phenomenon is, we expected it to be easy to obtain high-accuracy values for water density as a function of temperature and salinity from the internet.  Unfortunately, this isn't the case.  Sites often provide incomplete tables, or values from curve-fits that are only valid in certain temperature ranges (and sometimes not specifying that range).  Here we summarize the information available on the internet, providing some sites/formulas that can be used for high accuracy.  The suggested methods are intended for Armchair Scientists: those of us without access to professional journals or specialized python libraries, but who still want to do calculations relating to the density of water.

Density of Water vs Temperature and Salinity
Image by USGS

Conclusions

We provide and compare several different sources:

• Canonical - the "true" answer, as decided by panels of scientists.  The links we provide here are not necessarily the most recent versions, as modifications are made as new definitions for things like Salinity are decided and/or new measurements are made, but should suffice for most purposes.
• Suggested - the suggested source for Armchair Scientists.  These might differ from the Canonical sources because they are more accessible without having Journal subscriptions and/or downloading python libraries.

Pure Water

• Canonical Source: https://webbook.nist.gov/chemistry/fluid/
  • Accurate for Temperatures (Melting Point, 1273 K), Pressures up to 1000 MPa
• Suggested Source: https://webbook.nist.gov/chemistry/fluid/
  • Accurate for Temperatures (Melting Point, 1273 K), Pressures up to 1000 MPa

Sea Water

• Canonical Source: http://www.iapws.org/relguide/seawater.pdf or https://www.teos-10.org/
  • Accurate for Temperatures (-6 - 80) ℃, Salinity (0 - 120) g/kg, Pressures up to 100 MPa
  • Values for (0 - 50) ℃ in 0.1 ℃ increments: https://ittc.info/media/4048/75-02-01-03.pdf
• Suggested Source: https://os.copernicus.org/articles/5/91/2009/os-5-91-2009.pdf
  • Accurate for Temperatures (T) (0 - 90) ℃ and Salinity (S) (0 - 70) g / kg
  • Provides a modification to Pure Water Density, that we provide here so readers don't have to dive into the article:
    • 𝞺(T,S) = 𝞺_o(T) + A(T) * S + B(T) * S^1.5 + C(T) * S²
    • 𝞺_o(T) = Pure Water Density at the desired Temperature
    • A(T) = 8.197247e-01 - 3.779454e-03 * T + 6.821795e-05 * T² - 8.009571e-07 * T³ + 6.158885e-09 * T⁴ - 2.001919e-11 * T⁵
    • B(T) = -5.808305e-03 + 5.354872e-05 * T - 4.714602e-07 * T²
    • C(T) = 5.249266e-04
  • These coefficients are only valid when units are specified as:
    • 𝞺: kg / m³
    • T: ℃
    • S: g / kg
  • Sea Water is typically specified with S = 35.16504

Density Values

• The Suggested Sources have been assembled into an Excel 2019 Spreadsheet with outright data for Pure Water, but with the above formula for Sea Water coded in.  Users only need to specify the Salinity they're interested in and Density and Thermal Expansion Coefficients will be populated for temperatures at 0.01 ℃, 0.1 ℃, and 1.0 ℃ increments.

Water Property Calculator - Excel 2019 spreadsheet to calculate Water Density and Coefficient of Thermal Expansion vs Temperature for a user-specified Salinity.  ~1.4 Mb in size.

We can combine these two Suggested Sources and produce a Surface Plot of Water Density versus Temperature and Salinity.

Water Density vs Temperature and Salinity

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Background

Equation of State

On this page we sometimes reference Water's "Equation of State", and combing through the various articles linked below we found many different varieties, but it's worth explaining what exactly it is.

An Equation of State is an equation that tells you how various physical properties, like density, temperature, volume, vary as the other properties are changed.

For example, when the pressure on a substance is increased its volume will generally decrease, or when the temperature of a substance is increased its volume will generally increase.  These relationships vary from substance to substance, and vary for the same substance depending on the phase of matter (i.e. - solid, liquid, gas).

Scientists arrive at an Equation of State by first conducting experiments to see what values each property has under various conditions.  They might take a sample of water, record its purity/salinity, and see what the density is under different combinations of pressure and temperature.    Even if they performed hundreds or thousands of such experiments, it would not be sufficient for how scientists need to use the data.  There will always be gaps in their dataset (they might only measure properties every 1 ℃), and they can't cover the entire range of possible values (they might only measure up to pressures of 1,000 atmospheres).  Scientists and Engineers may need to know values for which experiments have not been done, yet still have confidence they are accurate.

To fill in the gaps, scientists perform extremely complicated curve-fits that try to get as close as possible to the experimentally measured points while not oscillating too much in between the points.  The most universally agreed upon models for Water have over 60 parameters and involve polynomials up to the fifth (x⁵) or sixth (x⁶) degrees.  As complicated as they are, there are two major limitations to such methods:

1. Interpolation Accuracy - if the real behavior of the substance varies too rapidly or suddenly changes direction, then the fitted curve may not be able to keep up with the changes.
2. Extrapolation Accuracy - the fitted curve is only guaranteed to be accurate around the experimental points used to build the curve.  Once you leave the highest/lowest values used when training, the fitted curve can diverge quite strongly from experiment.

Both of these sources of error came up when investigating the Density of Water using internet sources.

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Methodology

We googled things like "density of water versus temperature" and got a variety of sites, several run by Universities or U.S. government departments.  These sources are of varying accuracy and completeness, often provide values that seem statistically different from each other, and can be difficult to reproduce due to complicated underlying equations of state.

Pure Water

For Pure Water you can use the National Institute of Standards and Technology (NIST) site:

• NIST: https://webbook.nist.gov/chemistry/fluid/
  • You fill out a form of desired substance, units, temperature range, and temperature increment, and it returns a table of properties that includes water density.
  • The site won't return more than 601 rows of values, and they automatically adjust the temperature increment upwards to enforce that limit.  e.g. - if you request water properties for T = (1, 100) ℃ in 0.1 ℃ increments, then you'd like to receive NumRows = 1 + (MaxTemp - MinTemp) / IncrTemp = 1 + (100 - 1)/0.1 = 991 rows of values.  Instead, it will reverse the formula to cap NumRows at 601, so 601 = 1 + (100 - 1)/IncrTemp,  IncrTemp = (100 - 1)/600 = 0.165 ℃.
  • For Pressure of 1 atm = 0.101325 MPa = 1.01325 bar, values are tabulated at (we verified as identical with those from NIST):
    • https://www.internetchemistry.com/chemical-data/water-density-table.php

The NIST site uses the formulated Equation of State from the 1995 International Association for the Properties of Water and Steam (IAPWS):

• IAPWS: http://www.iapws.org/relguide/IAPWS95-2018.pdf
  • They gathered experimental results and performed a 60+ parameter curve-fit, then used thermodynamic equations to convert the fitted equation into a variety of physical properties like density, speed of sound, heat capacity, etc.
  • It is considered accurate from the Melting Point (which differs based on Pressure) up to 1273 K, and pressures up to 1000 MPa, with accuracy around 0.01% except when approaching the Critical Point.

Because they provide a fairly straightforward web interface to get density values for Pure Water down to 0.01 ℃ accuracy, we consider the NIST site to be both the Canonical and Suggested source.  We can compare the NIST values to other sources we found on the internet:

1. USGS.PURE - U.S. Geological Survey
   • https://www.usgs.gov/special-topics/water-science-school/science/water-density
2. TOOLBOX.PURE - The Engineering ToolBox
   • https://www.engineeringtoolbox.com/water-density-specific-weight-d_595.html
3. PLYMOUTH.PURE - Plymouth State University
   • http://jupiter.plymouth.edu/~jsduncan/courses/2012_Spring/Techniques/Exams/DensityOfWater-vs-Temp.pdf
4. UIUC.PURE - University of Illinois Urbana-Champaign
   • http://butane.chem.uiuc.edu/pshapley/GenChem1/L21/2.html
5. CSUS.PURE - California State University Sacramento
   • https://www.csus.edu/indiv/m/mackj/chem1a/docs/h2oden.pdf
6. VALVES.PURE - Valves Instruments Plus, Ltd.
   • https://www.vip-ltd.co.uk/Expansion/Density_Of_Water_Tables.pdf
7. COASTAL.PURE - Coastal Wiki
   • http://www.coastalwiki.org/wiki/Seawater_density
8. MIT.PURE - Massachusetts Institute of Technology
   • http://web.mit.edu/seawater/2017_MIT_Seawater_Property_Tables_r2b.pdf
9. NIST.PURE (Canonical, Suggested) - National Institute of Standards and Technology
   • https://webbook.nist.gov/chemistry/fluid/

Pure Water Density vs Temperature (0-100) ℃

It looks like there's pretty good agreement between the various websites, so good that we can't really see the non-canonical lines as they fit within the width of the green NIST line.  However, we know that Water behaves strangely around 4 ℃, and we'd like to see how the sources compare in that region:

Pure Water Density vs Temperature (0-10) ℃

Things are looking much less good now.  The MIT.PURE data, which one might expect to be high quality as it comes from the Massachusetts Institute of Technology and uses its own Equation of State, misses the temperature of maximum Density by several ℃.  Arguably that's an important feature that we would want our Equation of State to honor.

Overall, the NIST data is both the canonical source and user-accessible, so there's little reason to use anything else.

Sea Water

For Sea Water the issue is more complicated, as we couldn't find an online resource that provided values like we could for Pure Water.  The canonical Equation of State for Sea Water has gone through multiple versions over the years, relatively recently due to a transition from Practical Salinity to Absolute Salinity.  Two good sources of data on the canonical version are the International Association for the Properties of Water and Steam (IAPWS) site and Thermodynamic Equation of Seawater - 2010 (TEOS) site:

• IAPWS: http://www.iapws.org/relguide/seawater.pdf
• TEOS: https://www.teos-10.org/

We were able to locate a website that provides IAPWS determined values for Sea Water in the Temperature Range (0 - 50) ℃, in 0.1 ℃ increments at the International Towing Tank Conference (ITTC) site:

• ITTC: https://ittc.info/media/4048/75-02-01-03.pdf

Unfortunately, this only provides values for a specific Salinity, and only with limited Temperature Range and Temperature Increment.  To get wider coverage, the best we could find was a paper by Millero and Huang 2009, which summarized a series of experimental measurements of density and provided their own Equation of State (see Conclusions).  Finding a random paper on the internet required us to check it for accuracy, so we compared the values produced by that model for S = 35.16504 to those sources of Sea Water density values that we could find on the internet.  Unfortunately, there were far fewer sources of Sea Water density than Pure Water density:

1. MILLERO.SEA (Suggested) - a paper by F. J. Millero and F. Huang from 2009
   • https://os.copernicus.org/articles/5/91/2009/os-5-91-2009.pdf
2. TOOLBOX.SEA - The Engineering ToolBox
   • https://www.engineeringtoolbox.com/sea-water-properties-d_840.html
3. COASTAL.SEA - Coastal Wiki
   • http://www.coastalwiki.org/wiki/Seawater_density
4. MIT.SEA - Massachusetts Institute of Technology
   • http://web.mit.edu/seawater/2017_MIT_Seawater_Property_Tables_r2b.pdf
5. ITTC.SEA (Canonical) - International Towing Tank Conference, sourced from IAPWS/TEOS
   • https://ittc.info/media/4048/75-02-01-03.pdf

Sea Water Density vs Temperature (0-100) ℃

While not as tightly packed as Pure Water vs Temperature (0-100) ℃, the results for everything except TOOLBOX.SEA are quite close, and TOOLBOX.SEA is only bad because it offers so few digits of precision.  But as we did for Pure Water, we will zoom in around the critical temperature of 4 ℃ to see how each dataset performs:

Sea Water Density vs Temperature (0-10) ℃

Here we see that our Suggested source, MILLERO.SEA, does differ from the Canonical source, ITTC.SEA.  The reason we still suggest using MILLERO.SEA is because the divergence is quite small, and, importantly, it is consistent in magnitude and doesn't distort the shape of the canonical curve.

One final thing we can examine is the Temperature of Maximum Density as a function of Salinity, using the Millero, Huang 2009 Equation of State.  We expect this to decrease as Salinity is increased.

Water - Temperature of Maximum Density vs Salinity

This has exactly the shape and behavior we expect from other sources on the internet, so even though we didn't do numerical comparisons it is still encouraging that we aren't horribly off base.

Related Pages

• Water Main Page
  • Description of Full Research Project
• Water: Thermal Expansion vs Temperature and Salinity
  • How quickly does Water's volume change as a function of Temperature and Salinity?
• Water Property Calculator
  • Excel 2019 spreadsheet to calculate Water Density and Coefficient of Thermal Expansion vs Temperature for a user-specified Salinity.  ~1.4 Mb in size.