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# Calculus

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1. Physical Properties of Water and Ice
1. Molecular Weight:
A. 18.01528 g/mol
Water, Molar mass
Triple Point
The temperature and pressure at which solid, liquid, and gaseous water coexist in equilibrium is called the triple point of water. This point is used to define the units of temperature (the kelvin, the SI unit of thermodynamic temperature and, indirectly, the degree Celsius and even the degree Fahrenheit).
As a consequence, water's triple point temperature is a prescribed value rather than a measured quantity.
The triple point is at a temperature of 273.16 K (0.01 °C) by convention, and at a pressure of 611.73 Pa. This pressure is quite low, about 1⁄166 of the normal sea level barometric pressure of 101,325 Pa. The atmospheric surface pressure on planet Mars is 610.5 Pa, which is remarkably close to the triple point pressure. The altitude of this surface pressure was used to define zero-elevation or "sea level" on that planet.[33]
Density of water and ice

Density of ice and water as a function of temperature
Density of liquid water
Temp (°C)
Density (kg/m3)[20][21]
+100
958.4
+80
971.8
+60
983.2
+40
992.2
+30
995.6502
+25
997.0479
+22
997.7735
+20
998.2071
+15
999.1026
+10
999.7026
+4
999.9720
0
999.8395
−10
998.117
−20
993.547
−30
983.854
The values below 0 °C refer to supercooled water.
The density of water is approximately one gram per cubic centimeter. It is dependent on its temperature, but the relation is not linear and is unimodal rather than monotonic (see table at left). When cooled from room temperature liquid water becomes increasingly dense, as with other substances, but at approximately 4 °C (39 °F), pure water reaches its maximum density. As it is cooled further, it expands to become less dense. This unusual negative thermal expansion is attributed to strong, orientation-dependent, intermolecular interactions and is also observed in molten silica.[22]
The solid form of most substances is denser than the liquid phase; thus, a block of most solids will sink in the liquid. However, a block of ice floats in liquid water because ice is less dense. Upon freezing, the density of water decreases by about 9%.[23] This is due to the 'cooling' of intermolecular vibrations allowing the molecules to form steady hydrogen bonds with their neighbors and thereby gradually locking into positions reminiscent of the hexagonal packing achieved upon freezing to ice Ih. Whereas the hydrogen bonds are shorter in the crystal than in the liquid, this locking effect reduces the average coordination number of molecules as the liquid approaches nucleation. Other substances that expand on freezing are silicon, gallium, germanium, antimony, bismuth, plutonium and also chemical compounds that form spacious crystal lattices with tetrahedral coordination.
Only ordinary hexagonal ice is less dense than the liquid. Under increasing pressure, ice undergoes a number of transitions to other allotropic forms with higher density than liquid water, such as ice II, ice III, high-density amorphous ice (HDA), and very-high-density amorphous ice (VHDA).

Temperature distribution in a lake in summer and winter
Water also expands significantly as the temperature increases. Water near the boiling point is about 96% as dense as water at 4 °C.
The melting point of ice is 0 °C (32 °F, 273.15 K) at standard pressure, however, pure liquid water can be supercooled well below that temperature without freezing if the liquid is not mechanically disturbed. It can remain in a fluid state down to its homogeneous nucleation point of approximately 231 K (−42 °C).[24] The melting point of ordinary hexagonal ice falls slightly under moderately high pressures, but as ice transforms into its allotropes (see crystalline states of ice) above 209.9 MPa (2,072 atm), the melting point increases markedly with pressure, i.e., reaching 355 K (82 °C) at 2.216 GPa (21,870 atm) (triple point of Ice VII[25]).
A significant increase of pressure is required to lower the melting point of ordinary ice—the pressure exerted by an ice skater on the ice only reduces the melting point by approximately 0.09 °C (0.16 °F).[citation needed]
These properties of water have important consequences in its role in Earth's ecosystem. Water at a temperature of 4 °C will always accumulate at the bottom of freshwater lakes, irrespective of the temperature in the atmosphere. Since water and ice are poor conductors of heat[26] (good insulators) it is unlikely that sufficiently deep lakes will freeze completely, unless stirred by strong currents that mix cooler and warmer water and accelerate the cooling. In warming weather, chunks of ice float, rather than sink to the bottom where they might melt extremely slowly. These properties therefore allow aquatic life in the lake to survive during the winter.
The enthalpy of fusion or heat of fusion is the change in enthalpy resulting from heating a given quantity of a substance to change its state from a solid to a liquid. The temperature at which this occurs is the melting point.
The 'enthalpy' of fusion is a latent heat, because during melting the introduction of heat cannot be observed as a temperature change, as the temperature remains constant during the process. The latent heat of fusion is the enthalpy change of any amount of substance when it melts. When the heat of fusion is referenced to a unit of mass, it is usually called the specific heat of fusion, while the molar heat of fusion refers to the enthalpy change per amount of substance in moles.
The liquid phase has a higher internal energy than the solid phase. This means energy must be supplied to a solid in order to melt it and energy is released from a liquid when it freezes, because the molecules in the liquid experience weaker intermolecular forces and so have a higher potential energy (a kind of bond-dissociation energy for intermolecular forces).
When liquid water is cooled, its temperature falls steadily until it drops just below the freezing point at 0 °C. The temperature then remains constant at the freezing point while the water crystallizes. Once the water is completely frozen, its temperature continues to fall.
The enthalpy of fusion is almost always a positive quantity; helium is the only known exception.[1] Helium-3 has a negative enthalpy of fusion at temperatures below 0.3 K. Helium-4 also has a very slightly negative enthalpy of fusion below 0.8 K. This means that, at appropriate constant pressures, these substances freeze with the addition of heat.[2]
The enthalpy of vaporization, (symbol ), also known as the (latent) heat of vaporization or heat of evaporation, is the enthalpy change required to transform a given quantity of a substance from a liquid into a gas at a given pressure (often atmospheric pressure, as in STP).
It is often measured at the normal boiling point of a substance; although tabulated values are usually corrected to 298 K, the correction is often smaller than the uncertainty in the measured value.
The heat of vaporization is temperature-dependent, though a constant heat of vaporization can be assumed for small temperature ranges and for reduced temperature Tr

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