Clausius–Clapeyron relation

From Wikipedia, the free encyclopedia

 

The Clausius–Clapeyron relation, named after Rudolf Clausius^[1] and Benoît Paul Émile Clapeyron,^[2] is a way of characterizing a discontinuous phase transition between two phases of matter of a single constituent. On a pressure–temperature (P–T) diagram, the line separating the two phases is known as the coexistence curve. The Clausius–Clapeyron relation gives the slope of the tangents to this curve. Mathematically,

where is the slope of the tangent to the coexistence curve at any point, is the specific latent heat, is the temperature, is the specific volume change of the phase transition and is the entropy change of the phase transition.

Derivations

A typical phase diagram. The dotted green line gives the anomalous behavior of water. The Clausius–Clapeyron relation can be used to find the relationship between pressure and temperature along phase boundaries.

Derivation from state postulate

Using the state postulate, take the specific entropy for a homogeneous substance to be a function of specific volume and temperature .^[3]:508

The Clausius–Clapeyron relation characterizes behavior of a closed system during a phase change, during which temperature and pressure are constant by definition. Therefore,^[3]:508

Using the appropriate Maxwell relation gives^[3]:508

where is the pressure. Since pressure and temperature are constant, by definition the derivative of pressure with respect to temperature does not change.^{[4][5]:57, 62 & 671} Therefore the partial derivative of specific entropy may be changed into a total derivative

and the total derivative of pressure with respect to temperature may be factored out when integrating from an initial phase to a final phase ,^[3]:508 to obtain

where and are respectively the change in specific entropy and specific volume. Given that a phase change is an internally reversible process, and that our system is closed, the first law of thermodynamics holds

where is the internal energy of the system. Given constant pressure and temperature (during a phase change) and the definition of specific enthalpy , we obtain

Given constant pressure and temperature (during a phase change), we obtain^[3]:508

Substituting the definition of specific latent heat gives

Substituting this result into the pressure derivative given above (), we obtain^[3]:508[6]

This result (also known as the Clapeyron equation) equates the slope of the tangent to the coexistence curve , at any given point on the curve, to the function of the specific latent heat , the temperature , and the change in specific volume .

Derivation from Gibbs–Duhem relation

Suppose two phases, and , are in contact and at equilibrium with each other. Their chemical potentials are related by

Furthermore, along the coexistence curve,

One may therefore use the Gibbs–Duhem relation

(where is the specific entropy, is the specific volume, and is the molar mass) to obtain

Rearrangement gives

from which the derivation of the Clapeyron equation continues as in the previous section.

Ideal gas approximation at low temperatures

When the phase transition of a substance is between a gas phase and a condensed phase (liquid or solid), and occurs at temperatures much lower than the critical temperature of that substance, the specific volume of the gas phase greatly exceeds that of the condensed phase . Therefore one may approximate

at low temperatures. If pressure is also low, the gas may be approximated by the ideal gas law, so that

where is the pressure, is the specific gas constant, and is the temperature. Substituting into the Clapeyron equation

we can obtain the Clausius–Clapeyron equation^[3]:509

for low temperatures and pressures,^[3]:509 where is the specific latent heat of the substance.
Let and be any two points along the coexistence curve between two phases and . In general, varies between any two such points, as a function of temperature. But if is constant,

or^[5]:672

These last equations are useful because they relate equilibrium or saturation vapor pressure and temperature to the latent heat of the phase change, without requiring specific volume data.

Applications

Chemistry and chemical engineering

For transitions between a gas and a condensed phase with the approximations described above, the expression may be rewritten as

where is a constant. For a liquid-gas transition, is the specific latent heat (or specific enthalpy) of vaporization; for a solid-gas transition, is the specific latent heat of sublimation. If the latent heat is known, then knowledge of one point on the coexistence curve determines the rest of the curve. Conversely, the relationship between and is linear, and so linear regression is used to estimate the latent heat.

Meteorology and climatology

Atmospheric water vapor drives many important meteorologic phenomena (notably precipitation), motivating interest in its dynamics. The Clausius–Clapeyron equation for water vapor under typical atmospheric conditions (near standard temperature and pressure) is

where:

• is saturation vapor pressure
• is temperature
• is the specific latent heat of evaporation of water
• is the gas constant of water vapor

The temperature dependence of the latent heat , and therefore of the saturation vapor pressure , cannot be neglected in this application. Fortunately, the August-Roche-Magnus formula provides a very good approximation, using pressure in hPa and temperature in Celsius:

^[7][8]

(This is also sometimes called the Magnus or Magnus-Tetens approximation, though this attribution is historically inaccurate.^[9])

Under typical atmospheric conditions, the denominator of the exponent depends weakly on (for which the unit is Celsius). Therefore, the August-Roche-Magnus equation implies that saturation water vapor pressure changes approximately exponentially with temperature under typical atmospheric conditions, and hence the water-holding capacity of the atmosphere increases by about 7% for every 1 °C rise in temperature.^[10]

Example

One of the uses of this equation is to determine if a phase transition will occur in a given situation. Consider the question of how much pressure is needed to melt ice at a temperature below 0 °C. Note that water is unusual in that its change in volume upon melting is negative. We can assume

and substituting in

= 3.34×10⁵ J/kg (latent heat of fusion for water),

= 273 K (absolute temperature), and

= −9.05×10⁻⁵ m³/kg (change in specific volume from solid to liquid),

we obtain

= −13.5 MPa/K.

To provide a rough example of how much pressure this is, to melt ice at −7 °C (the temperature many ice skating rinks are set at) would require balancing a small car (mass = 1000 kg^[11]) on a thimble (area = 1 cm²).

Second derivative

While the Clausius–Clapeyron relation gives the slope of the coexistence curve, it does not provide any information about its curvature or second derivative. The second derivative of the coexistence curve of phases 1 and 2 is given by ^[12]

where subscripts 1 and 2 denote the different phases, is the specific heat capacity at constant pressure, is the thermal expansion coefficient, and is the isothermal compressibility.

Albedo

From Wikipedia, the free encyclopedia

Percentage of diffusely reflected sunlight in relation to various surface conditions

Albedo (/ælˈbiːdoʊ/), or reflection coefficient, derived from Latin albedo "whiteness" (or reflected sunlight) in turn from albus "white", is the diffuse reflectivity or reflecting power of a surface. It is the ratio of reflected radiation from the surface to incident radiation upon it. Its dimensionless nature lets it be expressed as a percentage and is measured on a scale from zero for no reflection of a perfectly black surface to 1 for perfect reflection of a white surface.

Albedo depends on the frequency of the radiation. When quoted unqualified, it usually refers to some appropriate average across the spectrum of visible light. In general, the albedo depends on the directional distribution of incident radiation, except for Lambertian surfaces, which scatter radiation in all directions according to a cosine function and therefore have an albedo that is independent of the incident distribution. In practice, a bidirectional reflectance distribution function (BRDF) may be required to accurately characterize the scattering properties of a surface, but albedo is very useful as a first approximation.

The albedo is an important concept in climatology, astronomy, and calculating reflectivity of surfaces in LEED sustainable-rating systems for buildings. The average overall albedo of Earth, its planetary albedo, is 30 to 35% because of cloud cover, but widely varies locally across the surface because of different geological and environmental features.^[1]

The term was introduced into optics by Johann Heinrich Lambert in his 1760 work Photometria.

Terrestrial albedo

Sample albedos

Surface	Typical albedo
Fresh asphalt	0.04[2]
Worn asphalt	0.12[2]
Conifer forest (Summer)	0.08,[3] 0.09 to 0.15[4]
Deciduous trees	0.15 to 0.18[4]
Bare soil	0.17[5]
Green grass	0.25[5]
Desert sand	0.40[6]
New concrete	0.55[5]
Ocean ice	0.5–0.7[5]
Fresh snow	0.80–0.90[5]

Albedos of typical materials in visible light range from up to 0.9 for fresh snow to about 0.04 for charcoal, one of the darkest substances. Deeply shadowed cavities can achieve an effective albedo approaching the zero of a black body. When seen from a distance, the ocean surface has a low albedo, as do most forests, whereas desert areas have some of the highest albedos among landforms. Most land areas are in an albedo range of 0.1 to 0.4.^[7] The average albedo of Earth is about 0.3.^[8] This is far higher than for the ocean primarily because of the contribution of clouds.

2003–2004 mean annual clear-sky and total-sky albedo

Earth's surface albedo is regularly estimated via Earth observation satellite sensors such as NASA's MODIS instruments on board the Terra and Aqua satellites. As the total amount of reflected radiation cannot be directly measured by satellite, a mathematical model of the BRDF is used to translate a sample set of satellite reflectance measurements into estimates of directional-hemispherical reflectance and bi-hemispherical reflectance (e.g.^[9]).

Earth's average surface temperature due to its albedo and the greenhouse effect is currently about 15 °C. If Earth were frozen entirely (and hence be more reflective) the average temperature of the planet would drop below −40 °C.^[10] If only the continental land masses became covered by glaciers, the mean temperature of the planet would drop to about 0 °C.^[11] In contrast, if the entire Earth is covered by water—a so-called aquaplanet—the average temperature on the planet would rise to just under 27 °C.^[12]

White-sky and black-sky albedo

It has been shown that for many applications involving terrestrial albedo, the albedo at a particular solar zenith angle θ_i can reasonably be approximated by the proportionate sum of two terms: the directional-hemispherical reflectance at that solar zenith angle, , and the bi-hemispherical reflectance, the proportion concerned being defined as the proportion of diffuse illumination .
Albedo can then be given as:

Directional-hemispherical reflectance is sometimes referred to as black-sky albedo and bi-hemispherical reflectance as white-sky albedo. These terms are important because they allow the albedo to be calculated for any given illumination conditions from a knowledge of the intrinsic properties of the surface.^[13]

Astronomical albedo

The albedos of planets, satellites and asteroids can be used to infer much about their properties. The study of albedos, their dependence on wavelength, lighting angle ("phase angle"), and variation in time comprises a major part of the astronomical field of photometry. For small and far objects that cannot be resolved by telescopes, much of what we know comes from the study of their albedos. For example, the absolute albedo can indicate the surface ice content of outer Solar System objects, the variation of albedo with phase angle gives information about regolith properties, whereas unusually high radar albedo is indicative of high metal content in asteroids.

Enceladus, a moon of Saturn, has one of the highest known albedos of any body in the Solar System, with 99% of EM radiation reflected. Another notable high-albedo body is Eris, with an albedo of 0.96.^[14] Many small objects in the outer Solar System^[15] and asteroid belt have low albedos down to about 0.05.^[16] A typical comet nucleus has an albedo of 0.04.^[17] Such a dark surface is thought to be indicative of a primitive and heavily space weathered surface containing some organic compounds.

The overall albedo of the Moon is around 0.12, but it is strongly directional and non-Lambertian, displaying also a strong opposition effect.^[18] Although such reflectance properties are different from those of any terrestrial terrains, they are typical of the regolith surfaces of airless Solar System bodies.

Two common albedos that are used in astronomy are the (V-band) geometric albedo (measuring brightness when illumination comes from directly behind the observer) and the Bond albedo (measuring total proportion of electromagnetic energy reflected). Their values can differ significantly, which is a common source of confusion.

In detailed studies, the directional reflectance properties of astronomical bodies are often expressed in terms of the five Hapke parameters which semi-empirically describe the variation of albedo with phase angle, including a characterization of the opposition effect of regolith surfaces.

The correlation between astronomical (geometric) albedo, absolute magnitude and diameter is:^[19]

  ,

where is the astronomical albedo, is the diameter in kilometers, and is the absolute magnitude.

Examples of terrestrial albedo effects

Illumination

Although the albedo–temperature effect is best known in colder, whiter regions on Earth, the maximum albedo is actually found in the tropics where year-round illumination is greater. The maximum is additionally in the northern hemisphere, varying between three and twelve degrees north.^[20] The minima are found in the subtropical regions of the northern and southern hemispheres, beyond which albedo increases without respect to illumination.^[20]

Insolation effects

The intensity of albedo temperature effects depend on the amount of albedo and the level of local insolation; high albedo areas in the arctic and antarctic regions are cold due to low insolation, where areas such as the Sahara Desert, which also have a relatively high albedo, will be hotter due to high insolation. Tropical and sub-tropical rain forest areas have low albedo, and are much hotter than their temperate forest counterparts, which have lower insolation. Because insolation plays such a big role in the heating and cooling effects of albedo, high insolation areas like the tropics will tend to show a more pronounced fluctuation in local temperature when local albedo changes.^{[citation needed]}

Climate and weather

Albedo affects climate and drives weather. All weather is a result of the uneven heating of Earth caused by different areas of the planet having different albedos. Essentially, for the driving of weather, there are two types of albedo regions on Earth: Land and ocean. Land and ocean regions produce the four basic different types of air masses, depending on latitude and therefore insolation: Warm and dry, which form over tropical and sub-tropical land masses; warm and wet, which form over tropical and sub-tropical oceans; cold and dry which form over temperate, polar and sub-polar land masses; and cold and wet, which form over temperate, polar and sub-polar oceans. Different temperatures between the air masses result in different air pressures, and the masses develop into pressure systems. High pressure systems flow toward lower pressure, driving weather from north to south in the northern hemisphere, and south to north in the lower; however due to the spinning of Earth, the Coriolis effect further complicates flow and creates several weather/climate bands and the jet streams.

Albedo–temperature feedback

When an area's albedo changes due to snowfall, a snow–temperature feedback results. A layer of snowfall increases local albedo, reflecting away sunlight, leading to local cooling. In principle, if no outside temperature change affects this area (e.g. a warm air mass), the raised albedo and lower temperature would maintain the current snow and invite further snowfall, deepening the snow–temperature feedback. However, because local weather is dynamic due to the change of seasons, eventually warm air masses and a more direct angle of sunlight (higher insolation) cause melting. When the melted area reveals surfaces with lower albedo, such as grass or soil, the effect is reversed: the darkening surface lowers albedo, increasing local temperatures, which induces more melting and thus reducing the albedo further, resulting in still more heating.

Snow

Snow albedo is highly variable, ranging from as high as 0.9 for freshly fallen snow, to about 0.4 for melting snow, and as low as 0.2 for dirty snow.^[21] Over Antarctica they average a little more than 0.8. If a marginally snow-covered area warms, snow tends to melt, lowering the albedo, and hence leading to more snowmelt because more radiation is being absorbed by the snowpack (the ice–albedo positive feedback). Cryoconite, powdery windblown dust containing soot, sometimes reduces albedo on glaciers and ice sheets.^[22] Hence, small errors in albedo can lead to large errors in energy estimates, which is why it is important to measure the albedo of snow-covered areas through remote sensing techniques rather than applying a single value over broad regions.

Small-scale effects

Albedo works on a smaller scale, too. In sunlight, dark clothes absorb more heat and light-coloured clothes reflect it better, thus allowing some control over body temperature by exploiting the albedo effect of the colour of external clothing.^[23]

Solar photovoltaic effects

Albedo can affect the electrical energy output of solar photovoltaic devices. For example, the effects of a spectrally responsive albedo are illustrated by the differences between the spectrally weighted albedo of solar photovoltaic technology based on hydrogenated amorphous silicon (a-Si:H) and crystalline silicon (c-Si)-based compared to traditional spectral-integrated albedo predictions.
Research showed impacts of over 10%.^[24] More recently, the analysis was extended to the effects of spectral bias due to the specular reflectivity of 22 commonly occurring surface materials (both human-made and natural) and analyzes the albedo effects on the performance of seven photovoltaic materials covering three common photovoltaic system topologies: industrial (solar farms), commercial flat rooftops and residential pitched-roof applications.^[25]

Trees

Because forests generally have a low albedo, (the majority of the ultraviolet and visible spectrum is absorbed through photosynthesis), some scientists have suggested that greater heat absorption by trees could offset some of the carbon benefits of afforestation (or offset the negative climate impacts of deforestation). In the case of evergreen forests with seasonal snow cover albedo reduction may be great enough for deforestation to cause a net cooling effect.^[26] Trees also impact climate in extremely complicated ways through evapotranspiration. The water vapor causes cooling on the land surface, causes heating where it condenses, acts a strong greenhouse gas, and can increase albedo when it condenses into clouds^[27] Scientists generally treat evapotranspiration as a net cooling impact, and the net climate impact of albedo and evapotranspiration changes from deforestation depends greatly on local climate ^[28]

In seasonally snow-covered zones, winter albedos of treeless areas are 10% to 50% higher than nearby forested areas because snow does not cover the trees as readily. Deciduous trees have an albedo value of about 0.15 to 0.18 whereas coniferous trees have a value of about 0.09 to 0.15.^[4]
Studies by the Hadley Centre have investigated the relative (generally warming) effect of albedo change and (cooling) effect of carbon sequestration on planting forests. They found that new forests in tropical and midlatitude areas tended to cool; new forests in high latitudes (e.g. Siberia) were neutral or perhaps warming.^[29]

Water

Water reflects light very differently from typical terrestrial materials. The reflectivity of a water surface is calculated using the Fresnel equations (see graph).

Reflectivity of smooth water at 20 °C (refractive index=1.333)

At the scale of the wavelength of light even wavy water is always smooth so the light is reflected in a locally specular manner (not diffusely). The glint of light off water is a commonplace effect of this. At small angles of incident light, waviness results in reduced reflectivity because of the steepness of the reflectivity-vs.-incident-angle curve and a locally increased average incident angle.^[30]

Although the reflectivity of water is very low at low and medium angles of incident light, it becomes very high at high angles of incident light such as those that occur on the illuminated side of Earth near the terminator (early morning, late afternoon, and near the poles). However, as mentioned above, waviness causes an appreciable reduction. Because light specularly reflected from water does not usually reach the viewer, water is usually considered to have a very low albedo in spite of its high reflectivity at high angles of incident light.

Note that white caps on waves look white (and have high albedo) because the water is foamed up, so there are many superimposed bubble surfaces which reflect, adding up their reflectivities. Fresh 'black' ice exhibits Fresnel reflection.

Clouds

Cloud albedo has substantial influence over atmospheric temperatures. Different types of clouds exhibit different reflectivity, theoretically ranging in albedo from a minimum of near 0 to a maximum approaching 0.8. "On any given day, about half of Earth is covered by clouds, which reflect more sunlight than land and water. Clouds keep Earth cool by reflecting sunlight, but they can also serve as blankets to trap warmth."^[31]

Albedo and climate in some areas are affected by artificial clouds, such as those created by the contrails of heavy commercial airliner traffic.^[32] A study following the burning of the Kuwaiti oil fields during Iraqi occupation showed that temperatures under the burning oil fires were as much as 10 °C colder than temperatures several miles away under clear skies.^[33]

Aerosol effects

Aerosols (very fine particles/droplets in the atmosphere) have both direct and indirect effects on Earth's radiative balance. The direct (albedo) effect is generally to cool the planet; the indirect effect (the particles act as cloud condensation nuclei and thereby change cloud properties) is less certain.^[34] As per ^[35] the effects are:

• Aerosol direct effect. Aerosols directly scatter and absorb radiation. The scattering of radiation causes atmospheric cooling, whereas absorption can cause atmospheric warming.
• Aerosol indirect effect. Aerosols modify the properties of clouds through a subset of the aerosol population called cloud condensation nuclei. Increased nuclei concentrations lead to increased cloud droplet number concentrations, which in turn leads to increased cloud albedo, increased light scattering and radiative cooling (first indirect effect), but also leads to reduced precipitation efficiency and increased lifetime of the cloud (second indirect effect).

Black carbon

Another albedo-related effect on the climate is from black carbon particles. The size of this effect is difficult to quantify: the Intergovernmental Panel on Climate Change estimates that the global mean radiative forcing for black carbon aerosols from fossil fuels is +0.2 W m⁻², with a range +0.1 to +0.4 W m⁻².^[36] Black carbon is a bigger cause of the melting of the polar ice cap in the Arctic than carbon dioxide due to its effect on the albedo.^[37]

Human activities

Human activities (e.g. deforestation, farming, and urbanization) change the albedo of various areas around the globe. However, quantification of this effect on the global scale is difficult.^{[citation needed]}

Other types of albedo

Single-scattering albedo is used to define scattering of electromagnetic waves on small particles. It depends on properties of the material (refractive index); the size of the particle or particles; and the wavelength of the incoming radiation.

Wave function gets real in quantum experiment

16:00 02 February 2015 by Michael Slezak

Original link:  http://www.newscientist.com/article/dn26893-wave-function-gets-real-in-quantum-experiment.html#.VNpoYS5avDe

It underpins the whole theory of quantum mechanics, but does it exist? For nearly a century physicists have argued about whether the wave function is a real part of the world or just a mathematical tool. Now, the first experiment in years to draw a line in the quantum sand suggests we should take it seriously.

The wave function helps predict the results of quantum experiments with incredible accuracy. But it describes a world where particles have fuzzy properties – for example, existing in two places at the same time. Erwin Schrödinger argued in 1935 that treating the wave function as a real thing leads to the perplexing situation where a cat in a box can be both dead and alive, until someone opens the box and observes it.

Those who want an objective description of the world – one that doesn't depend on how you're looking at it – have two options. They can accept that the wave function is real and that the cat is both dead and alive. Or they can argue that the wave function is just a mathematical tool, which represents our lack of knowledge about the status of the poor cat, sometimes called the "epistemic interpretation". This was the interpretation favoured by Albert Einstein, who allegedly asked, "Do you really believe the moon exists only when you look at it?"

The trouble is, very few experiments have been performed that can rule versions of quantum mechanics in or out. Previous work that claimed to propose a way to test whether the wave function is real made a splash in the physics community, but turned out to be based on improper assumptions, and no one ever ran the experiment.

What a state

Now, Eric Cavalcanti at the University of Sydney and Alessandro Fedrizzi at the University of Queensland, both in Australia, and their colleagues have made a measurement of the reality of the quantum wave function. Their results rule out a large class of interpretations of quantum mechanics and suggest that if there is any objective description of the world, the famous wave function is part of it: Schrödinger's cat actually is both dead and alive.

"In my opinion, this is the first experiment to place significant bounds on the viability of an epistemic interpretation of the quantum state," says Matthew Leifer at the Perimeter Institute in Waterloo, Canada.

The experiment relies on the quantum properties of something that could be in one of two states, as long as the states are not complete opposites of each other: like a photon that is polarised vertically or on a diagonal, but not horizontally. If the wave function is real, then a single experiment should not be able to determine its polarisation – it can have both until you take more measurements.

Alternatively, if the wave function is not real, then there is no fuzziness and the photon is in a single polarisation state all along. The researchers published a mathematical proof last year showing that, in this case, each measurement you make reveals some information about the polarisation.

Get real

In a complicated setup that involved pairs of photons and hundreds of very accurate measurements, the team showed that the wave function must be real: not enough information could be gained about the polarisation of the photons to imply they were in particular states before measurement.

There are a few ways to save the epistemic view, the team says, but they invite other exotic interpretations. Killing the wave function could mean leaving open the door to many interacting worlds and retrocausality – the idea that things that happen in the future can influence the past.

The results leave some wiggle room, though, because they didn't completely rule out the possibility of some underlying non-fuzzy reality. There may still be a way to distinguish quantum states from each other that their experiment didn't capture. But Howard Wiseman from Griffith University in Brisbane, Australia, says that shouldn't weaken the results. "It's saying there's definitely some reality to the wave function," he says. "You have to admit that to some extent there's some reality to the wave function, so if you've gone that far, why don't you just go the whole way?"

Journal reference: Nature Physics, DOI: 10.1038/nphys3233