Density of Fresh Water from Temperature (Kell)

Water is usually taken as 1000 kg/m³, and that is fine until the answer has to be right to better than a few tenths of a per cent. This formula gives the density of air-free fresh water at one atmosphere from its temperature, which is the number to use when a pressure from a depth, a mass from a volume or a power from a flow has to be that accurate. The temperature goes in as $T$; the formula turns it into the Celsius value $t = T - 273.15$ first, because the polynomial is written in degrees Celsius.

The numerator is a fifth-degree polynomial in $t$, and the denominator, $1 + 16.879850\times10^{-3}\,t$, belongs to the same fit; the coefficients come from fitting measurements, so the terms have no separate physical meaning. What matters is the curve they draw. Water is densest at 3.98 °C (999.97 kg/m³), about 0.18 % lighter at 20 °C and 4 % lighter at 100 °C. Below that maximum the density actually rises as the water warms, which is why the page's 0 °C example reads 999.84 kg/m³ and its 4 °C example reads 999.97 kg/m³.

Kell's 1975 paper fitted measured values of the density of liquid water and put them on the 1968 temperature scale; the page uses the fit between 0 and 100 °C, where it is good to a few parts per million. It was checked against other tables of the same quantity: the CRC Handbook table, which follows the modern reference scale, and the IAPWS steam tables. Kell's scale reads about 3 ppm lower, so the tolerances are a few parts per million wide, and no wider than a part in 30,000. A different fit of the same quantity, the pure-water polynomial of the UNESCO 1981 seawater equation, gives 999.7021 kg/m³ at 10 °C where Kell gives 999.6996.

Other points on the curve: 20 °C gives 998.207 kg/m³ and 25 °C gives 997.05 kg/m³. The formula is for air-free fresh water at one atmosphere; salt in the water raises the density by about 0.75 to 0.8 kg/m³ for each unit of salinity, so seawater needs its own equation. Solving for temperature has a catch: because the curve peaks, a density below the maximum can belong to a cold and a warm temperature, and the page takes the warmer solution.