What is Density – Physics

Density is defined as the mass per unit volume. It is an intensive property, which is mathematically defined as mass divided by volume:

ρ = m/V

In other words, the density (ρ) of a substance is the total mass (m) of that substance divided by the total volume (V) occupied by that substance. The standard SI unit is kilograms per cubic meter (kg/m3). The Standard English unit is pounds mass per cubic foot (lbm/ft3).

Density - Gas - Liquid - Solid
Typical densities of various substances at atmospheric pressure.

The density (ρ) of a substance is the reciprocal of its specific volume (ν).

ρ = m/V = 1/ρ

Specific volume is an intensive variable, whereas volume is an extensive variable.

The SI system’s standard unit for specific volumes is cubic meters per kilogram (m3/kg). The standard unit in the English system is cubic feet per pound-mass (ft3/lbm).

“Changes
In general, density can be changed by changing either the pressure or the temperature. Increasing the pressure always increases the density of a material. The effect of pressure on the densities of liquids and solids is very, very small. On the other hand, the density of gases is strongly affected by pressure. This is expressed by compressibility. Compressibility measures the relative volume change of a fluid or solid as a response to a pressure change.

The effect of temperature on the densities of liquids and solids is also very important. Most substances expand when heated and contract when cooled. However, the amount of expansion or contraction varies, depending on the material. This phenomenon is known as thermal expansion. The following relation gives the change in volume of a material that undergoes a temperature change:

thermal-expansion

where ∆T is the temperature change, V is the original volume, ∆V is the volume change, and αV is the coefficient of volume expansion.

We must note that there are exceptions to this rule. For example, water differs from most liquids in that it becomes less dense as it freezes. It has a maximum density of 3.98 °C (1000 kg/m3), whereas the density of ice is 917 kg/m3. It differs by about 9% and therefore ice floats on liquid water

Coolant acceleration in a reactor core
See also: Fluid Acceleration – Pressure Loss

It is an illustrative example. The following data do not correspond to any reactor design.

Continuity Equation - Flow Rates through Reactor
Example of flow rates in a reactor. It is an illustrative example, and the data do not represent any reactor design.

Pressurized water reactors are cooled and moderated by high-pressure liquid water (e.g.,, 16MPa). At this pressure, water boils at approximately 350°C (662°F).  The inlet temperature of the water is about 290°C (⍴ ~ 720 kg/m3). The water (coolant) is heated in the reactor core to approximately 325°C (⍴ ~ 654 kg/m3) as the water flows through the core.

The primary circuit of typical PWRs is divided into four independent loops (piping diameter ~ 700mm). Each loop comprises a steam generator and one main coolant pump. Inside the reactor pressure vessel (RPV), the coolant first flows down outside the reactor core (through the downcomer). The flow is reversed up through the core from the bottom of the pressure vessel, where the coolant temperature increases as it passes through the fuel rods and the assemblies formed by them.

Calculate:

  • Pressure loss due to the coolant acceleration in an isolated fuel channel

 when

  • channel inlet flow velocity is equal to  5.17 m/s
  • channel outlet flow velocity is equal to  5.69 m/s

Solution:

The pressure loss due to the coolant acceleration in an isolated fuel channel is then:

coolant acceleration - example

This fact has important consequences. Due to the different relative power of fuel assemblies in a core, these fuel assemblies have different hydraulic resistance, which may induce the local lateral flow of primary coolant. It must be considered in thermal-hydraulic calculations.

See also: How density influences reactor reactivity

Density in the periodic table

For full interactivity, please visit material-properties.org.

densities - elements - periodic table

Density – Atomic Mass and Atomic Number Density

Since the density (ρ) of a substance is the total mass (m) of that substance divided by the total volume (V) occupied by that substance, it is obvious, the density of a substance strongly depends on its atomic mass and also on the atomic number density (N; atoms/cm3),

  • Atomic Weight. The atomic mass is carried by the atomic nucleus, which occupies only about 10-12 of the total volume of the atom or less, but it contains all the positive charge and at least 99.95% of the total mass of the atom. Therefore it is determined by the mass number (number of protons and neutrons).
  • Atomic Number Density. The atomic number density (N; atoms/cm3), which is associated with atomic radii, is the number of atoms of a given type per unit volume (V; cm3) of the material. The atomic number density (N; atoms/cm3) of a pure material having an atomic or molecular weight (M; grams/mol) and the material density (⍴; gram/cm3) is easily computed from the following equation using Avogadro’s number (NA = 6.022×1023 atoms or molecules per mole):Atomic Number Density

Densest Materials on the Earth

Since nucleons (protons and neutrons) make up most of the mass of ordinary atoms, the density of normal matter tends to be limited by how closely we can pack these nucleons and depends on the internal atomic structure. The densest material found on earth is the metal osmium. Still, its density pales by comparison to the densities of exotic astronomical objects such as white dwarf stars and neutron stars.

List of densest materials:

  1. Osmium – 22.6 x 103 kg/m3
  2. Iridium – 22.4 x 103 kg/m3
  3. Platinum – 21.5 x 103 kg/m3
  4. Rhenium – 21.0 x 103 kg/m3
  5. Plutonium – 19.8 x 103 kg/m3
  6. Gold – 19.3 x 103 kg/m3
  7. Tungsten – 19.3 x 103 kg/m3
  8. Uranium – 18.8 x 103 kg/m3
  9. Tantalum – 16.6 x 103 kg/m3
  10. Mercury – 13.6 x 103 kg/m3
  11. Rhodium – 12.4 x 103 kg/m3
  12. Thorium – 11.7 x 103 kg/m3
  13. Lead – 11.3 x 103 kg/m3
  14. Silver – 10.5 x 103 kg/m3

It must be noted, plutonium is a manufactured isotope and is created from uranium in nuclear reactors. But scientists have found trace amounts of naturally-occurring plutonium.

If we include manufactured elements, the densest so far is Hassium. Hassium is a chemical element with the symbol Hs and atomic number 108.  It is a synthetic element (first synthesized at Hasse in Germany) and radioactive. The most stable known isotope, 269Hs, has a half-life of approximately 9.7 seconds. It has an estimated density of 40.7 x 103 kg/m3.  The density of Hassium results from its high atomic weight and the significant decrease in ionic radii of the elements in the lanthanide series, known as lanthanide and actinide contraction.

The density of Hassium is followed by Meitnerium (element 109, named after the physicist Lise Meitner), which has an estimated density of 37.4 x 103 kg/m3.

Density - Important property in gamma rays shielding
See also: Shielding of Gamma Radiation

In short, effective shielding of gamma radiation is in most cases based on the use of materials with two following material properties:

  • the high-density of material. 
  • the high atomic number of material  (high Z materials)

However, low-density materials and low Z materials can be compensated with increased thickness, which is as significant as density and atomic number in shielding applications.

A lead is widely used as a gamma shield.  The major advantage of the lead shield is its compactness due to its higher density. On the other hand, depleted uranium is much more effective due to its higher Z.  Depleted uranium shields in portable gamma-ray sources.

In nuclear power plants, shielding of a reactor core can be provided by materials of reactor pressure vessel, reactor internals (neutron reflector). Also, heavy concrete is usually used to shield both neutrons and gamma radiation.

The density of various Materials – Examples

Density of Water - Specific Volume
Density of Water - Specific VolumePure water has its highest density 1000 kg/m3 at temperature 3.98oC (39.2oF). Water differs from most liquids in that it becomes less dense as it freezes. It has a maximum density of 3.98 °C (1000 kg/m3), whereas the density of ice is 917 kg/m3. It differs by about 9%, and therefore ice floats on liquid water. It must be noted, the change in density is not linear with temperature because the volumetric thermal expansion coefficient for water is not constant over the temperature range. The density of water (1 gram per cubic centimeter) was originally used to define the gram. The density (⍴) of a substance is the reciprocal of its specific volume (ν).

ρ = m/V = 1/ν

The specific volume (ν) of a substance is the total volume (V) of that substance divided by the total mass (m) of that substance (volume per unit mass). It has units of cubic meters per kilogram (m3/kg).

“Density
Pure heavy water (D2O) has a density about 11% greater than water but is otherwise physically and chemically similar.

This difference is caused by the fact that the deuterium nucleus is twice as heavy as the hydrogen nucleus. Since about 89% of the molecular weight of water comes from the single oxygen atom rather than the two hydrogen atoms, the weight of a heavy water molecule is not substantially different from that of a normal water molecule. The molar mass of water is M(H2O) = 18.02, and the molar mass of heavy water is M(D2O) = 20.03 (each deuterium nucleus contains one neutron in contrast to the hydrogen nucleus). Therefore heavy water (D2O) has a density about 11% greater (20.03/18.03 = 1.112).

Pure heavy water (D2O) has its highest density of 1106 kg/m3 at a temperature of 11.6oC (52.9oF). Also, heavy water differs from most liquids in that it becomes less dense as it freezes. It has a maximum density at 11.6oC (1106 kg/m3), whereas its solid form ice density is 1017 kg/m3. It must be noted, the change in density is not linear with temperature because the volumetric thermal expansion coefficient for water is not constant over the temperature range.

“Density
steam properties - steam tables water and steam are a common medium because their properties are very well known. Their properties are tabulated in so-called Steam Tables.” In these tables, the basic and key properties, such as pressure, temperature, enthalpy, density, and specific heat, are tabulated along the vapor-liquid saturation curve as a function of both temperature and pressure.

The density (⍴) of any substance is the reciprocal of its specific volume ().

ρ = m/V = 1/

The specific volume () of a substance is the total volume (V) of that substance divided by the total mass (m) of that substance (volume per unit mass). It has units of cubic meters per kilogram (m3/kg).

Density of Steel
The density of steel varies based on the alloying constituents but usually ranges between 7.5 x 103 kg/m3 and 8 x 103 kg/m3.
Density of Zirconium
In general, zirconium has very low absorption cross-section of thermal neutrons, high hardness, ductility and corrosion resistance. One of the main uses of zirconium alloys is nuclear technology, as cladding of fuel rods in nuclear reactors, due to its very low absorption cross-section (unlike the stainless steel). The density of typical zirconium alloy is about 6.6 x 103 kg/m3.
“Density
uranium is a naturally-occurring chemical element with atomic number 92, which means there are 92 protons and 92 electrons in the atomic structure. Natural uranium consists primarily of isotope 238U (99.28%). Therefore the atomic mass of the uranium element is close to the atomic mass of the 238U isotope (238.03u).  Natural uranium also consists of two other isotopes: 235U (0.71%) and 234U (0.0054%). Uranium has the highest atomic weight of the primordially occurring elements. Uranium metal has a very high density of 19.1 g/cm3, denser than lead (11.3 g/cm3), but slightly less dense than tungsten and gold (19.3 g/cm3).

Uranium metal is one of the densest materials found on earth:

  1. Osmium – 22.6 x 103 kg/m3
  2. Iridium – 22.4 x 103 kg/m3
  3. Platinum – 21.5 x 103 kg/m3
  4. Rhenium – 21.0 x 103 kg/m3
  5. Plutonium – 19.8 x 103 kg/m3
  6. Gold – 19.3 x 103 kg/m3
  7. Tungsten – 19.3 x 103 kg/m3
  8. Uranium – 18.8 x 103 kg/m3
  9. Tantalum – 16.6 x 103 kg/m3
  10. Mercury – 13.6 x 103 kg/m3
  11. Rhodium – 12.4 x 103 kg/m3
  12. Thorium – 11.7 x 103 kg/m3
  13. Lead – 11.3 x 103 kg/m3
  14. Silver – 10.5 x 103 kg/m3

But most of LWRs use uranium fuel, which is in the form of uranium dioxide. Uranium dioxide is a black semiconducting solid with very low thermal conductivity. On the other hand, uranium dioxide has a very high melting point and has well-known behavior.

Uranium dioxide has a significantly lower density than uranium in metal form. Uranium dioxide has a density of 10.97 g/cm3, but this value may vary with fuel burnup because, at low burnup, densification of pellets can occur, and at higher burnup, swelling occurs.

The density of Nuclear Matter

Nuclear density is the density of the nucleus of an atom. It is the ratio of mass per unit volume inside the nucleus. Since the atomic nucleus carries most of the atom’s mass and the atomic nucleus is very small compared to the entire atom, the nuclear density is very high.

The nuclear density for a typical nucleus can be approximately calculated from the size of the nucleus and its mass. Typical nuclear radii are of the order 10−14 m. Nuclear radii can be calculated according to the following formula assuming spherical shape:

r = r0 . A1/3

where r0 = 1.2 x 10-15 m = 1.2 fm

For example, natural uranium consists primarily of isotope 238U (99.28%). Therefore the atomic mass of the uranium element is close to the atomic mass of the 238U isotope (238.03u). The radius of this nucleus will be:

r = r0 . A1/3 = 7.44 fm.

Assuming it is spherical, its volume will be:

V = 4πr3/3 = 1.73 x 10-42 m3.

The usual definition of nuclear density gives for its density:

ρnucleus = m / V = 238 x 1.66 x 10-27 / (1.73 x 10-42) = 2.3 x 1017 kg/m3.

Thus, the density of nuclear material is more than 2.1014 times greater than that of water. It is an immense density. The descriptive term nuclear density is also applied to situations where similarly high densities occur, such as within neutron stars. Such immense densities are also found in neutron stars.

“Density
The densest material found on earth is the metal osmium, but its density pales by comparison to the densities of exotic astronomical objects such as white dwarf stars and neutron stars.

A neutron star is the collapsed core of a large star (usually of a red giant). Neutron stars are the smallest and densest stars known to exist, and they rotate extremely rapidly. A neutron star is a giant atomic nucleus about 11 km in diameter made especially of neutrons. It is believed that under the immense pressures of collapsing massive stars going supernova, the electrons and protons can combine to form neutrons via electron capture, releasing a huge amount of neutrinos.

They are so dense that one teaspoon of its material would have a mass over 5.5×1012 kg. It is assumed they have densities of 3.7 × 1017 to 6 × 1017 kg/m3, which is comparable to the approximate density of an atomic nucleus of 2.3 × 1017 kg/m3.

 
References:
Reactor Physics and Thermal Hydraulics:
  1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
  2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
  3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
  4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
  5. Todreas Neil E., Kazimi Mujid S. Nuclear Systems Volume I: Thermal Hydraulic Fundamentals, Second Edition. CRC Press; 2 edition, 2012, ISBN: 978-0415802871
  6. Zohuri B., McDaniel P. Thermodynamics in Nuclear Power Plant Systems. Springer; 2015, ISBN: 978-3-319-13419-2
  7. Moran Michal J., Shapiro Howard N. Fundamentals of Engineering Thermodynamics, Fifth Edition, John Wiley & Sons, 2006, ISBN: 978-0-470-03037-0
  8. Kleinstreuer C. Modern Fluid Dynamics. Springer, 2010, ISBN 978-1-4020-8670-0.
  9. U.S. Department of Energy, THERMODYNAMICS, HEAT TRANSFER, AND FLUID FLOW. DOE Fundamentals Handbook, Volume 1, 2, and 3. June 1992.

See above:

Thermodynamic Properties