Ideal gas

From Wikipedia, the free encyclopedia

An ideal gas is a theoretical gas composed of many randomly moving point particles whose only interactions are perfectly elastic collisions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.

One mole of an ideal gas has a volume of 22.710947(13) litres^[1] at STP (a temperature of 273.15 K and an absolute pressure of exactly 10⁵ Pa) as defined by IUPAC since 1982. (Until 1982, STP was defined as a temperature of 273.15 K and an absolute pressure of exactly 1 atm. The volume of one mole of an ideal gas at this temperature and pressure is 22.413962(13) litres.^[2] IUPAC recommends that the former use of this definition should be discontinued;^[3] however, some textbooks still use these old values.)

At normal conditions such as standard temperature and pressure, most real gases behave qualitatively like an ideal gas. Many gases such as nitrogen, oxygen, hydrogen, noble gases, and some heavier gases like carbon dioxide can be treated like ideal gases within reasonable tolerances.^[4] Generally, a gas behaves more like an ideal gas at higher temperature and lower pressure,^[4] as the potential energy due to intermolecular forces becomes less significant compared with the particles' kinetic energy, and the size of the molecules becomes less significant compared to the empty space between them.

The ideal gas model tends to fail at lower temperatures or higher pressures, when intermolecular forces and molecular size become important. It also fails for most heavy gases, such as many refrigerants,^[4] and for gases with strong intermolecular forces, notably water vapor. At high pressures, the volume of a real gas is often considerably greater than that of an ideal gas. At low temperatures, the pressure of a real gas is often considerably less than that of an ideal gas. At some point of low temperature and high pressure, real gases undergo a phase transition, such as to a liquid or a solid. The model of an ideal gas, however, does not describe or allow phase transitions. These must be modeled by more complex equations of state. The deviation from the ideal gas behaviour can be described by a dimensionless quantity, the compressibility factor, Z.

The ideal gas model has been explored in both the Newtonian dynamics (as in "kinetic theory") and in quantum mechanics (as a "gas in a box"). The ideal gas model has also been used to model the behavior of electrons in a metal (in the Drude model and the free electron model), and it is one of the most important models in statistical mechanics.

Types of ideal gas

There are three basic classes of ideal gas^{[citation needed]}:

• the classical or Maxwell–Boltzmann ideal gas,
• the ideal quantum Bose gas, composed of bosons, and
• the ideal quantum Fermi gas, composed of fermions.

The classical ideal gas can be separated into two types: The classical thermodynamic ideal gas and the ideal quantum Boltzmann gas. Both are essentially the same, except that the classical thermodynamic ideal gas is based on classical statistical mechanics, and certain thermodynamic parameters such as the entropy are only specified to within an undetermined additive constant. The ideal quantum Boltzmann gas overcomes this limitation by taking the limit of the quantum Bose gas and quantum Fermi gas in the limit of high temperature to specify these additive constants. The behavior of a quantum Boltzmann gas is the same as that of a classical ideal gas except for the specification of these constants. The results of the quantum Boltzmann gas are used in a number of cases including the Sackur–Tetrode equation for the entropy of an ideal gas and the Saha ionization equation for a weakly ionized plasma.

Classical thermodynamic ideal gas

Macroscopic account

The ideal gas law is an extension of experimentally discovered gas laws. Real fluids at low density and high temperature approximate the behavior of a classical ideal gas. However, at lower temperatures or a higher density, a real fluid deviates strongly from the behavior of an ideal gas, particularly as it condenses from a gas into a liquid or as it deposits from a gas into a solid. This deviation is expressed as a compressibility factor.

The classical thermodynamic properties of an ideal gas can be described by two equations of state:.^[5][6]

One of them is the well known ideal gas law

${\displaystyle PV=nRT\,}$

where

• P is the pressure
• V is the volume
• n is the amount of substance of the gas (in moles)
• R is the gas constant (8.314 J·K⁻¹·mol⁻¹)
• T is the absolute temperature.

This equation is derived from Boyle's law: V = k/P (at constant T and n); Charles's law: V = bT (at constant P and n); and Avogadro's law: V = an (at constant T and P); where

• k is a constant used in Boyle's law
• b is a proportionality constant; equal to V/T
• a is a proportionality constant; equal to V/n.

Multiplying the equations representing the three laws:

${\displaystyle V*V*V=kba\left({\frac {Tn}{P}}\right)}$

Gives:

${\displaystyle V*V*V=\left({\frac {kba}{3}}\right)\left({\frac {Tn}{P}}\right)}$.

Under ideal conditions,

${\displaystyle V=R\left({\frac {Tn}{P}}\right)}$ ;

that is,

${\displaystyle PV=nRT}$.

The other equation of state of an ideal gas must express Joule's law, that the internal energy of a fixed mass of ideal gas is a function only of its temperature. For the present purposes it is convenient to postulate an exemplary version of this law by writing:

${\displaystyle U={\hat {c}}_{V}nRT}$

where

• U is the internal energy
• ĉ_V is the dimensionless specific heat capacity at constant volume, approximately 3/2 for a monatomic gas, 5/2 for diatomic gas, and 7/2 for more complex molecules.

Microscopic model

In order to switch from macroscopic quantities (left hand side of the following equation) to microscopic ones (right hand side), we use

${\displaystyle nR=Nk_{\mathrm {B} }\ }$

where

• N is the number of gas particles
• k_B is the Boltzmann constant (1.381×10⁻²³ J·K⁻¹).

The probability distribution of particles by velocity or energy is given by the Maxwell speed distribution.

The ideal gas model depends on the following assumptions:

• The molecules of the gas are indistinguishable, small, hard spheres
• All collisions are elastic and all motion is frictionless (no energy loss in motion or collision)
• Newton's laws apply
• The average distance between molecules is much larger than the size of the molecules
• The molecules are constantly moving in random directions with a distribution of speeds
• There are no attractive or repulsive forces between the molecules apart from those that determine their point-like collisions
• The only forces between the gas molecules and the surroundings are those that determine the point-like collisions of the molecules with the walls
• In the simplest case, there are no long-range forces between the molecules of the gas and the surroundings.

The assumption of spherical particles is necessary so that there are no rotational modes allowed, unlike in a diatomic gas. The following three assumptions are very related: molecules are hard, collisions are elastic, and there are no inter-molecular forces. The assumption that the space between particles is much larger than the particles themselves is of paramount importance, and explains why the ideal gas approximation fails at high pressures.

Heat capacity

The heat capacity at constant volume, including an ideal gas is:

${\displaystyle {\hat {c}}_{V}={\frac {1}{nR}}T\left({\frac {\partial S}{\partial T}}\right)_{V}={\frac {1}{nR}}\left({\frac {\partial U}{\partial T}}\right)_{V}}$

where S is the entropy. This is the dimensionless heat capacity at constant volume, which is generally a function of temperature due to intermolecular forces. For moderate temperatures, the constant for a monatomic gas is ĉ_V = 3/2 while for a diatomic gas it is ĉ_V = 5/2. It is seen that macroscopic measurements on heat capacity provide information on the microscopic structure of the molecules.

The heat capacity at constant pressure of 1/R mole of ideal gas is:

${\displaystyle {\hat {c}}_{P}={\frac {1}{nR}}T\left({\frac {\partial S}{\partial T}}\right)_{P}={\frac {1}{nR}}\left({\frac {\partial H}{\partial T}}\right)_{P}={\hat {c}}_{V}+1}$

where H = U + PV is the enthalpy of the gas.

Sometimes, a distinction is made between an ideal gas, where ĉ_V and ĉ_P could vary with temperature, and a perfect gas, for which this is not the case.

The ratio of the constant volume and constant pressure heat capacity is

${\displaystyle \gamma ={\frac {c_{P}}{c_{V}}}}$

For air, which is a mixture of gases, this ratio is 1.4.

Entropy

Using the results of thermodynamics only, we can go a long way in determining the expression for the entropy of an ideal gas. This is an important step since, according to the theory of thermodynamic potentials, if we can express the entropy as a function of U (U is a thermodynamic potential), volume V and the number of particles N, then we will have a complete statement of the thermodynamic behavior of the ideal gas. We will be able to derive both the ideal gas law and the expression for internal energy from it.

Since the entropy is an exact differential, using the chain rule, the change in entropy when going from a reference state 0 to some other state with entropy S may be written as ΔS where:

${\displaystyle \Delta S=\int _{S_{0}}^{S}dS=\int _{T_{0}}^{T}\left({\frac {\partial S}{\partial T}}\right)_{V}\!dT+\int _{V_{0}}^{V}\left({\frac {\partial S}{\partial V}}\right)_{T}\!dV}$

where the reference variables may be functions of the number of particles N. Using the definition of the heat capacity at constant volume for the first differential and the appropriate Maxwell relation for the second we have:

${\displaystyle \Delta S=\int _{T_{0}}^{T}{\frac {C_{V}}{T}}\,dT+\int _{V_{0}}^{V}\left({\frac {\partial P}{\partial T}}\right)_{V}dV.}$

Expressing C_V in terms of ĉ_V as developed in the above section, differentiating the ideal gas equation of state, and integrating yields:

${\displaystyle \Delta S={\hat {c}}_{V}Nk\ln \left({\frac {T}{T_{0}}}\right)+Nk\ln \left({\frac {V}{V_{0}}}\right)}$

which implies that the entropy may be expressed as:

${\displaystyle S=Nk\ln \left({\frac {VT^{{\hat {c}}_{V}}}{f(N)}}\right)}$

where all constants have been incorporated into the logarithm as f(N) which is some function of the particle number N having the same dimensions as VT^ĉ_V in order that the argument of the logarithm be dimensionless. We now impose the constraint that the entropy be extensive. This will mean that when the extensive parameters (V and N) are multiplied by a constant, the entropy will be multiplied by the same constant. Mathematically:

${\displaystyle S(T,aV,aN)=aS(T,V,N).\,}$

From this we find an equation for the function f(N)

${\displaystyle af(N)=f(aN).\,}$

Differentiating this with respect to a, setting a equal to 1, and then solving the differential equation yields f(N):

${\displaystyle f(N)=\Phi N\,}$

where Φ may vary for different gases, but will be independent of the thermodynamic state of the gas. It will have the dimensions of VT^ĉ_V/N. Substituting into the equation for the entropy:

${\displaystyle {\frac {S}{Nk}}=\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right).\,}$

and using the expression for the internal energy of an ideal gas, the entropy may be written:

${\displaystyle {\frac {S}{Nk}}=\ln \left[{\frac {V}{N}}\,\left({\frac {U}{{\hat {c}}_{V}kN}}\right)^{{\hat {c}}_{V}}\,{\frac {1}{\Phi }}\right]}$

Since this is an expression for entropy in terms of U, V, and N, it is a fundamental equation from which all other properties of the ideal gas may be derived.

This is about as far as we can go using thermodynamics alone. Note that the above equation is flawed — as the temperature approaches zero, the entropy approaches negative infinity, in contradiction to the third law of thermodynamics. In the above "ideal" development, there is a critical point, not at absolute zero, at which the argument of the logarithm becomes unity, and the entropy becomes zero. This is unphysical. The above equation is a good approximation only when the argument of the logarithm is much larger than unity — the concept of an ideal gas breaks down at low values of V/N. Nevertheless, there will be a "best" value of the constant in the sense that the predicted entropy is as close as possible to the actual entropy, given the flawed assumption of ideality. A quantum-mechanical derivation of this constant is developed in the derivation of the Sackur–Tetrode equation which expresses the entropy of a monatomic (ĉ_V = 3/2) ideal gas. In the Sackur–Tetrode theory the constant depends only upon the mass of the gas particle. The Sackur–Tetrode equation also suffers from a divergent entropy at absolute zero, but is a good approximation for the entropy of a monatomic ideal gas for high enough temperatures.

Thermodynamic potentials

Expressing the entropy as a function of T, V, and N:

${\displaystyle {\frac {S}{kN}}=\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right)}$

The chemical potential of the ideal gas is calculated from the corresponding equation of state (see thermodynamic potential):

${\displaystyle \mu =\left({\frac {\partial G}{\partial N}}\right)_{T,P}}$

where G is the Gibbs free energy and is equal to U + PV − TS so that:

${\displaystyle \mu (T,V,N)=kT\left({\hat {c}}_{P}-\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right)\right)}$

The thermodynamic potentials for an ideal gas can now be written as functions of T, V, and N as:

${\displaystyle U\,}$		${\displaystyle ={\hat {c}}_{V}NkT\,}$
${\displaystyle A\,}$	${\displaystyle =U-TS\,}$	${\displaystyle =\mu N-NkT\,}$
${\displaystyle H\,}$	${\displaystyle =U+PV\,}$	${\displaystyle ={\hat {c}}_{P}NkT\,}$
${\displaystyle G\,}$	${\displaystyle =U+PV-TS\,}$	${\displaystyle =\mu N\,}$

where, as before,

${\displaystyle {\hat {c}}_{P}={\hat {c}}_{V}+1}$.

The most informative way of writing the potentials is in terms of their natural variables, since each of these equations can be used to derive all of the other thermodynamic variables of the system. In terms of their natural variables, the thermodynamic potentials of a single-species ideal gas are:

${\displaystyle U(S,V,N)={\hat {c}}_{V}Nk\left({\frac {N\Phi }{V}}\,e^{S/Nk}\right)^{1/{\hat {c}}_{V}}}$

${\displaystyle A(T,V,N)=NkT\left({\hat {c}}_{V}-\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right)\right)}$

${\displaystyle H(S,P,N)={\hat {c}}_{P}Nk\left({\frac {P\Phi }{k}}\,e^{S/Nk}\right)^{1/{\hat {c}}_{P}}}$

${\displaystyle G(T,P,N)=NkT\left({\hat {c}}_{P}-\ln \left({\frac {kT^{{\hat {c}}_{P}}}{P\Phi }}\right)\right)}$

In statistical mechanics, the relationship between the Helmholtz free energy and the partition function is fundamental, and is used to calculate the thermodynamic properties of matter; see configuration integral for more details.

Speed of sound

The speed of sound in an ideal gas is given by

${\displaystyle c_{\text{sound}}={\sqrt {\left({\frac {\partial P}{\partial \rho }}\right)_{s}}}={\sqrt {\frac {\gamma P}{\rho }}}={\sqrt {\frac {\gamma RT}{M}}}}$

where

γ is the adiabatic index (ĉ_P/ĉ_V)

s is the entropy per particle of the gas.

ρ is the mass density of the gas.

P is the pressure of the gas.

R is the universal gas constant

T is the temperature

M is the molar mass of the gas.

Ideal quantum gases

In the above-mentioned Sackur–Tetrode equation, the best choice of the entropy constant was found to be proportional to the quantum thermal wavelength of a particle, and the point at which the argument of the logarithm becomes zero is roughly equal to the point at which the average distance between particles becomes equal to the thermal wavelength. In fact, quantum theory itself predicts the same thing. Any gas behaves as an ideal gas at high enough temperature and low enough density, but at the point where the Sackur–Tetrode equation begins to break down, the gas will begin to behave as a quantum gas, composed of either bosons or fermions. (See the gas in a box article for a derivation of the ideal quantum gases, including the ideal Boltzmann gas.)

Gases tend to behave as an ideal gas over a wider range of pressures when the temperature reaches the Boyle temperature.

Ideal Boltzmann gas

The ideal Boltzmann gas yields the same results as the classical thermodynamic gas, but makes the following identification for the undetermined constant Φ:

${\displaystyle \Phi ={\frac {T^{\frac {3}{2}}\Lambda ^{3}}{g}}}$

where Λ is the thermal de Broglie wavelength of the gas and g is the degeneracy of states.

Ideal Bose and Fermi gases

An ideal gas of bosons (e.g. a photon gas) will be governed by Bose–Einstein statistics and the distribution of energy will be in the form of a Bose–Einstein distribution. An ideal gas of fermions will be governed by Fermi–Dirac statistics and the distribution of energy will be in the form of a Fermi–Dirac distribution.

Thermohaline circulation

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A summary of the path of the thermohaline circulation. Blue paths represent deep-water currents, while red paths represent surface currents.

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Thermohaline circulation

Thermohaline circulation (THC) is a part of the large-scale ocean circulation that is driven by global density gradients created by surface heat and freshwater fluxes.^[1][2] The adjective thermohaline derives from thermo- referring to temperature and -haline referring to salt content, factors which together determine the density of sea water. Wind-driven surface currents (such as the Gulf Stream) travel polewards from the equatorial Atlantic Ocean, cooling en route, and eventually sinking at high latitudes (forming North Atlantic Deep Water). This dense water then flows into the ocean basins. While the bulk of it upwells in the Southern Ocean, the oldest waters (with a transit time of around 1000 years)^[3] upwell in the North Pacific.^[4] Extensive mixing therefore takes place between the ocean basins, reducing differences between them and making the Earth's oceans a global system. On their journey, the water masses transport both energy (in the form of heat) and matter (solids, dissolved substances and gases) around the globe. As such, the state of the circulation has a large impact on the climate of the Earth.

The thermohaline circulation is sometimes called the ocean conveyor belt, the great ocean conveyor, or the global conveyor belt. On occasion, it is used to refer to the meridional overturning circulation (often abbreviated as MOC). The term MOC is more accurate and well defined, as it is difficult to separate the part of the circulation which is driven by temperature and salinity alone as opposed to other factors such as the wind and tidal forces.^[5] Moreover, temperature and salinity gradients can also lead to circulation effects that are not included in the MOC itself.

Overview

The global conveyor belt on a continuous-ocean map

The movement of surface currents pushed by the wind is fairly intuitive. For example, the wind easily produces ripples on the surface of a pond. Thus the deep ocean—devoid of wind—was assumed to be perfectly static by early oceanographers. However, modern instrumentation shows that current velocities in deep water masses can be significant (although much less than surface speeds). In general, ocean water velocities range from fractions of centimeters per second (in the depth of the oceans) to sometimes more than 1 m/s in surface currents like the Gulf Stream and Kuroshio.

In the deep ocean, the predominant driving force is differences in density, caused by salinity and temperature variations (increasing salinity and lowering the temperature of a fluid both increase its density). There is often confusion over the components of the circulation that are wind and density driven.^[6][7] Note that ocean currents due to tides are also significant in many places; most prominent in relatively shallow coastal areas, tidal currents can also be significant in the deep ocean. There they are currently thought to facilitate mixing processes, especially diapycnal mixing.^[8]

The density of ocean water is not globally homogeneous, but varies significantly and discretely. Sharply defined boundaries exist between water masses which form at the surface, and subsequently maintain their own identity within the ocean. But these sharp boundaries are not to be imagined spatially but rather in a T-S-diagram where water masses are distinguished. They position themselves above or below each other according to their density, which depends on both temperature and salinity.

Warm seawater expands and is thus less dense than cooler seawater. Saltier water is denser than fresher water because the dissolved salts fill interstices between water molecules, resulting in more mass per unit volume. Lighter water masses float over denser ones (just as a piece of wood or ice will float on water, see buoyancy). This is known as "stable stratification" as opposed to unstable stratification (see Bruunt-Väisälä frequency) where denser waters are located over less dense waters (see convection or deep convection needed for water mass formation). When dense water masses are first formed, they are not stably stratified, so they seek to locate themselves in the correct vertical position according to their density. This motion is called convection, it orders the stratification by gravitation. Driven by the density gradients this sets up the main driving force behind deep ocean currents like the deep western boundary current (DWBC).

The thermohaline circulation is mainly driven by the formation of deep water masses in the North Atlantic and the Southern Ocean caused by differences in temperature and salinity of the water.
The great quantities of dense water sinking at high latitudes must be offset by equal quantities of water rising elsewhere. Note that cold water in polar zones sink relatively rapidly over a small area, while warm water in temperate and tropical zones rise more gradually across a much larger area. It then slowly returns poleward near the surface to repeat the cycle. The continual diffuse upwelling of deep water maintains the existence of the permanent thermocline found everywhere at low and mid-latitudes. This model was described by Henry Stommel and Arnold B. Arons in 1960 and is known as the Stommel-Arons box model for the MOC.^[9] This slow upward movement is approximated to be about 1 centimeter (0.5 inch) per day over most of the ocean. If this rise were to stop, downward movement of heat would cause the thermocline to descend and would reduce its steepness.

Formation of deep water masses

The dense water masses that sink into the deep basins are formed in quite specific areas of the North Atlantic and the Southern Ocean. In the North Atlantic, seawater at the surface of the ocean is intensely cooled by the wind and low ambient air temperatures. Wind moving over the water also produces a great deal of evaporation, leading to a decrease in temperature, called evaporative cooling related to latent heat. Evaporation removes only water molecules, resulting in an increase in the salinity of the seawater left behind, and thus an increase in the density of the water mass along with the decrease in temperature. In the Norwegian Sea evaporative cooling is predominant, and the sinking water mass, the North Atlantic Deep Water (NADW), fills the basin and spills southwards through crevasses in the submarine sills that connect Greenland, Iceland and Great Britain which are known as the Greenland-Scotland-Ridge. It then flows very slowly into the deep abyssal plains of the Atlantic, always in a southerly direction. Flow from the Arctic Ocean Basin into the Pacific, however, is blocked by the narrow shallows of the Bering Strait.

Diagram showing relation between temperature and salinity for sea water density maximum and sea water freezing temperature.

In the Southern Ocean, strong katabatic winds blowing from the Antarctic continent onto the ice shelves will blow the newly formed sea ice away, opening polynyas along the coast. The ocean, no longer protected by sea ice, suffers a brutal and strong cooling (see polynya). Meanwhile, sea ice starts reforming, so the surface waters also get saltier, hence very dense. In fact, the formation of sea ice contributes to an increase in surface seawater salinity; saltier brine is left behind as the sea ice forms around it (pure water preferentially being frozen). Increasing salinity lowers the freezing point of seawater, so cold liquid brine is formed in inclusions within a honeycomb of ice. The brine progressively melts the ice just beneath it, eventually dripping out of the ice matrix and sinking. This process is known as brine rejection.

The resulting Antarctic Bottom Water (AABW) sinks and flows north and east, but is so dense it actually underflows the NADW. AABW formed in the Weddell Sea will mainly fill the Atlantic and Indian Basins, whereas the AABW formed in the Ross Sea will flow towards the Pacific Ocean.

The dense water masses formed by these processes flow downhill at the bottom of the ocean, like a stream within the surrounding less dense fluid, and fill up the basins of the polar seas. Just as river valleys direct streams and rivers on the continents, the bottom topography constrains the deep and bottom water masses.

Note that, unlike fresh water, seawater does not have a density maximum at 4 °C but gets denser as it cools all the way to its freezing point of approximately −1.8 °C. This freezing point is however a function of salinity and pressure and thus -1.8°C is not a general freezing temperature for sea water (see diagram to the right).

Movement of deep water masses

Formation and movement of the deep water masses at the North Atlantic Ocean, creates sinking water masses that fill the basin and flows very slowly into the deep abyssal plains of the Atlantic. This high-latitude cooling and the low-latitude heating drives the movement of the deep water in a polar southward flow. The deep water flows through the Antarctic Ocean Basin around South Africa where it is split into two routes: one into the Indian Ocean and one past Australia into the Pacific.

At the Indian Ocean, some of the cold and salty water from the Atlantic—drawn by the flow of warmer and fresher upper ocean water from the tropical Pacific—causes a vertical exchange of dense, sinking water with lighter water above. It is known as overturning. In the Pacific Ocean, the rest of the cold and salty water from the Atlantic undergoes haline forcing, and becomes warmer and fresher more quickly.

The out-flowing undersea of cold and salty water makes the sea level of the Atlantic slightly lower than the Pacific and salinity or halinity of water at the Atlantic higher than the Pacific. This generates a large but slow flow of warmer and fresher upper ocean water from the tropical Pacific to the Indian Ocean through the Indonesian Archipelago to replace the cold and salty Antarctic Bottom Water. This is also known as 'haline forcing' (net high latitude freshwater gain and low latitude evaporation). This warmer, fresher water from the Pacific flows up through the South Atlantic to Greenland, where it cools off and undergoes evaporative cooling and sinks to the ocean floor, providing a continuous thermohaline circulation.^[10]

Hence, a recent and popular name for the thermohaline circulation, emphasizing the vertical nature and pole-to-pole character of this kind of ocean circulation, is the meridional overturning circulation.

Quantitative estimation

Direct estimates of the strength of the thermohaline circulation have been made at 26.5°N in the North Atlantic since 2004 by the UK-US RAPID programme.^[11] By combining direct estimates of ocean transport using current meters and subsea cable measurements with estimates of the geostrophic current from temperature and salinity measurements, the RAPID programme provides continuous, full-depth, basinwide estimates of the thermohaline circulation or, more accurately, the meridional overturning circulation.

The deep water masses that participate in the MOC have chemical, temperature and isotopic ratio signatures and can be traced, their flow rate calculated, and their age determined. These include ²³¹Pa / ²³⁰Th ratios.

Gulf Stream

Benjamin Franklin's map of the Gulf Stream

The Gulf Stream, together with its northern extension towards Europe, the North Atlantic Drift, is a powerful, warm, and swift Atlantic ocean current that originates at the tip of Florida, and follows the eastern coastlines of the United States and Newfoundland before crossing the Atlantic Ocean. The process of western intensification causes the Gulf Stream to be a northward accelerating current off the east coast of North America.^[12] At about 40°0′N 30°0′W, it splits in two, with the northern stream crossing to northern Europe and the southern stream recirculating off West Africa. The Gulf Stream influences the climate of the east coast of North America from Florida to Newfoundland, and the west coast of Europe. Although there has been recent debate, there is consensus that the climate of Western Europe and Northern Europe is warmer than it would otherwise be due to the North Atlantic drift,^[13][14] one of the branches from the tail of the Gulf Stream. It is part of the North Atlantic Gyre. Its presence has led to the development of strong cyclones of all types, both within the atmosphere and within the ocean. The Gulf Stream is also a significant potential source of renewable power generation.^[15][16]

Upwelling

All these dense water masses sinking into the ocean basins displace the older deep water masses were made less dense by ocean mixing. To maintain a balance, water must be rising elsewhere. However, because this thermohaline upwelling is so widespread and diffuse, its speeds are very slow even compared to the movement of the bottom water masses. It is therefore difficult to measure where upwelling occurs using current speeds, given all the other wind-driven processes going on in the surface ocean. Deep waters have their own chemical signature, formed from the breakdown of particulate matter falling into them over the course of their long journey at depth. A number of scientists have tried to use these tracers to infer where the upwelling occurs.
Wallace Broecker, using box models, has asserted that the bulk of deep upwelling occurs in the North Pacific, using as evidence the high values of silicon found in these waters. Other investigators have not found such clear evidence. Computer models of ocean circulation increasingly place most of the deep upwelling in the Southern Ocean,^[17] associated with the strong winds in the open latitudes between South America and Antarctica. While this picture is consistent with the global observational synthesis of William Schmitz at Woods Hole and with low observed values of diffusion, not all observational syntheses agree. Recent papers by Lynne Talley at the Scripps Institution of Oceanography and Bernadette Sloyan and Stephen Rintoul in Australia suggest that a significant amount of dense deep water must be transformed to light water somewhere north of the Southern Ocean.

Effects on global climate

The thermohaline circulation plays an important role in supplying heat to the polar regions, and thus in regulating the amount of sea ice in these regions, although poleward heat transport outside the tropics is considerably larger in the atmosphere than in the ocean.^[18] Changes in the thermohaline circulation are thought to have significant impacts on the Earth's radiation budget. Insofar as the thermohaline circulation governs the rate at which deep waters are exposed to the surface, it may also play an important role in the concentration of carbon dioxide in the atmosphere. While it is often stated that the thermohaline circulation is the primary reason that Western Europe is so temperate, it has been suggested that this is largely incorrect, and that Europe is warm mostly because it lies downwind of an ocean basin, and because of the effect of atmospheric waves bringing warm air north from the subtropics.^[19] However, the underlying assumptions of this particular analysis have likewise been challenged.^[20]

Large influxes of low-density meltwater from Lake Agassiz and deglaciation in North America are thought to have led to a shifting of deep water formation and subsidence in the extreme North Atlantic and caused the climate period in Europe known as the Younger Dryas.^[21]

Shutdown of thermohaline circulation

In 2005, British researchers noticed that the net flow of the northern Gulf Stream had decreased by about 30% since 1957. Coincidentally, scientists at Woods Hole had been measuring the freshening of the North Atlantic as Earth becomes warmer. Their findings suggested that precipitation increases in the high northern latitudes, and polar ice melts as a consequence. By flooding the northern seas with lots of extra fresh water, global warming could, in theory, divert the Gulf Stream waters that usually flow northward, past the British Isles and Norway, and cause them to instead circulate toward the equator. If this were to happen, Europe's climate would be seriously impacted.^[22][23][24]
Downturn of AMOC (Atlantic meridional overturning circulation), has been tied to extreme regional sea level rise.^[25]

In 2013, an unexpected significant weakening of the THC led to one of the quietest Atlantic hurricane seasons observed since 1994. The main cause of the inactivity was caused by a continuation of the spring pattern across the Atlantic basin.

Tidal force

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Figure 1: Comet Shoemaker-Levy 9 in 1994 after breaking up under the influence of Jupiter's tidal forces during a previous pass in 1992.

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This simulation shows a star getting torn apart by the gravitational tides of a supermassive black hole.

The tidal force is a force that is the secondary effect of the force of gravity; it is responsible for the phenomenon of tides. It arises because the gravitational force exerted by one body on another is not constant across it: the nearest side is attracted more strongly than the farthest side. Thus, the tidal force is differential. Consider the gravitational attraction of the Moon on the oceans nearest to the Moon, the solid Earth and the oceans farthest from the Moon. There is a mutual attraction between the Moon and the solid Earth, which can be considered to act on its centre of mass. However, the near oceans are more strongly attracted and, especially since they are fluid, they approach the Moon slightly, causing a high tide. The far oceans are attracted less. The attraction on the far-side oceans could be expected to cause a low tide, but since the solid Earth is attracted (accelerated) more strongly towards the moon, there is a relative acceleration of those waters in the outwards direction. Viewing the Earth as a whole, we see that all its mass experiences a mutual attraction with that of the Moon, but the near oceans more so than the far oceans, leading to a separation of the two.

In a more general usage in celestial mechanics, the expression "tidal force" can refer to a situation in which a body or material (for example, tidal water) is mainly under the gravitational influence of a second body (for example, the Earth), but is also perturbed by the gravitational effects of a third body (for example, the Moon). The perturbing force is sometimes in such cases called a tidal force^[1] (for example, the perturbing force on the Moon): it is the difference between the force exerted by the third body on the second and the force exerted by the third body on the first.^[2]

Explanation

Figure 2: The Moon's gravity differential field at the surface of the Earth is known (along with another and weaker differential effect due to the Sun) as the Tide Generating Force. This is the primary mechanism driving tidal action, explaining two tidal equipotential bulges, and accounting for two high tides per day. In this figure, the Earth is the central blue circle while the Moon is far off to the right. The outward direction of the arrows on the right and left indicates that where the Moon is overhead (or at the nadir) its perturbing force opposes that between the earth and ocean.

When a body (body 1) is acted on by the gravity of another body (body 2), the field can vary significantly on body 1 between the side of the body facing body 2 and the side facing away from body 2. Figure 2 shows the differential force of gravity on a spherical body (body 1) exerted by another body (body 2). These so-called tidal forces cause strains on both bodies and may distort them or even, in extreme cases, break one or the other apart.^[3] The Roche limit is the distance from a planet at which tidal effects would cause an object to disintegrate because the differential force of gravity from the planet overcomes the attraction of the parts of the object for one another.^[4] These strains would not occur if the gravitational field were uniform, because a uniform field only causes the entire body to accelerate together in the same direction and at the same rate.

Effects of tidal forces

Figure 3: Saturn's rings are inside the orbits of its principal moons. Tidal forces oppose gravitational coalescence of the material in the rings to form moons.^[5]

In the case of an infinitesimally small elastic sphere, the effect of a tidal force is to distort the shape of the body without any change in volume. The sphere becomes an ellipsoid with two bulges, pointing towards and away from the other body. Larger objects distort into an ovoid, and are slightly compressed, which is what happens to the Earth's oceans under the action of the Moon. The Earth and Moon rotate about their common center of mass or barycenter, and their gravitational attraction provides the centripetal force necessary to maintain this motion. To an observer on the Earth, very close to this barycenter, the situation is one of the Earth as body 1 acted upon by the gravity of the Moon as body 2. All parts of the Earth are subject to the Moon's gravitational forces, causing the water in the oceans to redistribute, forming bulges on the sides near the Moon and far from the Moon.^[6]

When a body rotates while subject to tidal forces, internal friction results in the gradual dissipation of its rotational kinetic energy as heat. In the case for the Earth, and Earth's Moon, the loss of rotational kinetic energy results in a gain of about 2 milliseconds per century. If the body is close enough to its primary, this can result in a rotation which is tidally locked to the orbital motion, as in the case of the Earth's moon. Tidal heating produces dramatic volcanic effects on Jupiter's moon Io. Stresses caused by tidal forces also cause a regular monthly pattern of moonquakes on Earth's Moon.^[7]

Tidal forces contribute to ocean currents, which moderate global temperatures by transporting heat energy toward the poles. It has been suggested that in addition to other factors, harmonic beat variations in tidal forcing may contribute to climate changes. However, no strong link has been found to date.^[8]

Tidal effects become particularly pronounced near small bodies of high mass, such as neutron stars or black holes, where they are responsible for the "spaghettification" of infalling matter. Tidal forces create the oceanic tide of Earth's oceans, where the attracting bodies are the Moon and, to a lesser extent, the Sun. Tidal forces are also responsible for tidal locking, tidal acceleration, and tidal heating. Tides may also induce seismicity.

By generating conducting fluids within the interior of the Earth, tidal forces also affect the Earth's magnetic field.^[9]

Mathematical treatment

Tidal force is responsible for the merge of galactic pair MRK 1034.^[10]

Figure 4: Graphic of tidal forces. The top picture shows the gravity field of a body to the right, the lower shows their residual once the field at the centre of the sphere is subtracted; this is the tidal force. See Figure 2 for a more detailed version

For a given (externally generated) gravitational field, the tidal acceleration at a point with respect to a body is obtained by vectorially subtracting the gravitational acceleration at the center of the body (due to the given externally generated field) from the gravitational acceleration (due to the same field) at the given point. Correspondingly, the term tidal force is used to describe the forces due to tidal acceleration. Note that for these purposes the only gravitational field considered is the external one; the gravitational field of the body (as shown in the graphic) is not relevant. (In other words, the comparison is with the conditions at the given point as they would be if there were no externally generated field acting unequally at the given point and at the center of the reference body. The externally generated field is usually that produced by a perturbing third body, often the Sun or the Moon in the frequent example-cases of points on or above the Earth's surface in a geocentric reference frame.)

Tidal acceleration does not require rotation or orbiting bodies; for example, the body may be freefalling in a straight line under the influence of a gravitational field while still being influenced by (changing) tidal acceleration.

By Newton's law of universal gravitation and laws of motion, a body of mass m at distance R from the center of a sphere of mass M feels a force ${\displaystyle {\vec {F}}_{g}}$,

${\displaystyle {\vec {F}}_{g}=-{\hat {r}}~G~{\frac {Mm}{R^{2}}}}$

equivalent to an acceleration ${\displaystyle {\vec {a}}_{g}}$,

${\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}}$

where ${\displaystyle {\hat {r}}}$ is a unit vector pointing from the body M to the body m (here, acceleration from m towards M has negative sign).

Consider now the acceleration due to the sphere of mass M experienced by a particle in the vicinity of the body of mass m. With R as the distance from the center of M to the center of m, let ∆r be the (relatively small) distance of the particle from the center of the body of mass m. For simplicity, distances are first considered only in the direction pointing towards or away from the sphere of mass M. If the body of mass m is itself a sphere of radius ∆r, then the new particle considered may be located on its surface, at a distance (R ± ∆r) from the centre of the sphere of mass M, and ∆r may be taken as positive where the particle's distance from M is greater than R. Leaving aside whatever gravitational acceleration may be experienced by the particle towards m on account of m's own mass, we have the acceleration on the particle due to gravitational force towards M as:

${\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{(R\pm \Delta r)^{2}}}}$

Pulling out the R² term from the denominator gives:

${\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}~{\frac {1}{(1\pm \Delta r/R)^{2}}}}$

The Maclaurin series of ${\displaystyle 1/(1\pm x)^{2}}$ is ${\displaystyle 1\mp 2x+3x^{2}\mp \cdots }$ which gives a series expansion of:

${\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}\pm {\hat {r}}~G~{\frac {2M}{R^{2}}}~{\frac {\Delta r}{R}}+\cdots }$

The first term is the gravitational acceleration due to M at the center of the reference body ${\displaystyle m}$, i.e., at the point where ${\displaystyle \Delta r}$ is zero. This term does not affect the observed acceleration of particles on the surface of m because with respect to M, m (and everything on its surface) is in free fall. When the force on the far particle is subtracted from the force on the near particle, this first term cancels, as do all other even-order terms. The remaining (residual) terms represent the difference mentioned above and are tidal force (acceleration) terms. When ∆r is small compared to R, the terms after the first residual term are very small and can be neglected, giving the approximate tidal acceleration ${\displaystyle {\vec {a}}_{t}}$(axial) for the distances ∆r considered, along the axis joining the centers of m and M:

${\displaystyle {\vec {a}}_{t}}$(axial) ${\displaystyle ~\approx ~\pm ~{\hat {r}}~2\Delta r~G~{\frac {M}{R^{3}}}}$

When calculated in this way for the case where ∆r is a distance along the axis joining the centers of m and M, ${\displaystyle {\vec {a}}_{t}}$ is directed outwards from to the center of m (where ∆r is zero).

Tidal accelerations can also be calculated away from the axis connecting the bodies m and M, requiring a vector calculation. In the plane perpendicular to that axis, the tidal acceleration is directed inwards (towards the center where ∆r is zero), and its magnitude is ${\displaystyle |{\vec {a}}_{t}}$(axial)${\displaystyle |/2}$ in linear approximation as in Figure 2.

The tidal accelerations at the surfaces of planets in the Solar System are generally very small. For example, the lunar tidal acceleration at the Earth's surface along the Moon-Earth axis is about 1.1 × 10⁻⁷ g, while the solar tidal acceleration at the Earth's surface along the Sun-Earth axis is about 0.52 × 10⁻⁷ g, where g is the gravitational acceleration at the Earth's surface. Hence the tide-raising force (acceleration) due to the Sun is about 45% of that due to the Moon.^[11] The solar tidal acceleration at the Earth's surface was first given by Newton in the Principia.^[12]

Valence bond theory

From Wikipedia, the free encyclopedia

In chemistry, valence bond (VB) theory is one of two basic theories, along with molecular orbital (MO) theory, that were developed to use the methods of quantum mechanics to explain chemical bonding. It focuses on how the atomic orbitals of the dissociated atoms combine to give individual chemical bonds when a molecule is formed. In contrast, molecular orbital theory has orbitals that cover the whole molecule.^[1]

History

In 1916, G. N. Lewis proposed that a chemical bond forms by the interaction of two shared bonding electrons, with the representation of molecules as Lewis structures. In 1927 the Heitler–London theory was formulated which for the first time enabled the calculation of bonding properties of the hydrogen molecule H₂ based on quantum mechanical considerations. Specifically, Walter Heitler determined how to use Schrödinger's wave equation (1926) to show how two hydrogen atom wavefunctions join together, with plus, minus, and exchange terms, to form a covalent bond. He then called up his associate Fritz London and they worked out the details of the theory over the course of the night.^[2] Later, Linus Pauling used the pair bonding ideas of Lewis together with Heitler–London theory to develop two other key concepts in VB theory: resonance (1928) and orbital hybridization (1930). According to Charles Coulson, author of the noted 1952 book Valence, this period marks the start of "modern valence bond theory", as contrasted with older valence bond theories, which are essentially electronic theories of valence couched in pre-wave-mechanical terms. Resonance theory was criticized as imperfect by Soviet chemists during the 1950s.^[3]

Theory

According to this theory a covalent bond is formed between the two atoms by the overlap of half filled valence atomic orbitals of each atom containing one unpaired electron. A valence bond structure is similar to a Lewis structure, but where a single Lewis structure cannot be written, several valence bond structures are used. Each of these VB structures represents a specific Lewis structure. This combination of valence bond structures is the main point of resonance theory. Valence bond theory considers that the overlapping atomic orbitals of the participating atoms form a chemical bond. Because of the overlapping, it is most probable that electrons should be in the bond region. Valence bond theory views bonds as weakly coupled orbitals (small overlap). Valence bond theory is typically easier to employ in ground state molecules. The inner-shell orbitals and electrons remain essentially unchanged during the formation of bonds.

σ bond between two atoms: localization of electron density

Two p-orbitals forming a π-bond.

The overlapping atomic orbitals can differ. The two types of overlapping orbitals are sigma and pi. Sigma bonds occur when the orbitals of two shared electrons overlap head-to-head. Pi bonds occur when two orbitals overlap when they are parallel. For example, a bond between two s-orbital electrons is a sigma bond, because two spheres are always coaxial. In terms of bond order, single bonds have one sigma bond, double bonds consist of one sigma bond and one pi bond, and triple bonds contain one sigma bond and two pi bonds. However, the atomic orbitals for bonding may be hybrids. Often, the bonding atomic orbitals have a character of several possible types of orbitals. The methods to get an atomic orbital with the proper character for the bonding is called hybridization.

Valence bond theory today

Modern valence bond theory now complements molecular orbital theory, which does not adhere to the valence bond idea that electron pairs are localized between two specific atoms in a molecule but that they are distributed in sets of molecular orbitals which can extend over the entire molecule. Molecular orbital theory can predict magnetic and ionization properties in a straightforward manner, while valence bond theory gives similar results but is more complicated. Modern valence bond theory views aromatic properties of molecules as due to spin coupling of the π orbitals.^[4][5][6][7] This is essentially still the old idea of resonance between Kekulé and Dewar structures. In contrast, molecular orbital theory views aromaticity as delocalization of the π-electrons. Valence bond treatments are restricted to relatively small molecules, largely due to the lack of orthogonality between valence bond orbitals and between valence bond structures, while molecular orbitals are orthogonal. On the other hand, valence bond theory provides a much more accurate picture of the reorganization of electronic charge that takes place when bonds are broken and formed during the course of a chemical reaction. In particular, valence bond theory correctly predicts the dissociation of homonuclear diatomic molecules into separate atoms, while simple molecular orbital theory predicts dissociation into a mixture of atoms and ions. For example, the molecular orbital function for dihydrogen is an equal mixture of the covalent and ionic valence bond structures and so predicts incorrectly that the molecule would dissociate into an equal mixture of hydrogen atoms and hydrogen positive and negative ions.

Modern valence bond theory replaces the overlapping atomic orbitals by overlapping valence bond orbitals that are expanded over a large number of basis functions, either centered each on one atom to give a classical valence bond picture, or centered on all atoms in the molecule. The resulting energies are more competitive with energies from calculations where electron correlation is introduced based on a Hartree–Fock reference wavefunction. The most recent text is by Shaik and Hiberty.^[8]

Applications of valence bond theory

An important aspect of the VB theory is the condition of maximum overlap, which leads to the formation of the strongest possible bonds. This theory is used to explain the covalent bond formation in many molecules.

For example, in the case of the F₂ molecule, the F−F bond is formed by the overlap of p_z orbitals of the two F atoms, each containing an unpaired electron. Since the nature of the overlapping orbitals are different in H₂ and F₂ molecules, the bond strength and bond lengths differ between H₂ and F₂ molecules.

In an HF molecule the covalent bond is formed by the overlap of the 1s orbital of H and the 2p_z orbital of F, each containing an unpaired electron. Mutual sharing of electrons between H and F results in a covalent bond in HF.