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Tuesday, December 22, 2015

Gas laws


From Wikipedia, the free encyclopedia

The gas laws were developed at the end of the 18th century, when scientists began to realize that relationships between the pressure, volume and temperature of a sample of gas could be obtained which would hold to a good approximation for all gases. Gases behave in a similar way over a wide variety of conditions because they all have molecules which are widely spaced, and the equation of state for an ideal gas is derived from kinetic theory. The earlier gas laws are now considered as special cases of the ideal gas equation, with one or more of the variables held constant.

Boyle's Law

Boyle's Law, published in 1662, states that, at constant temperature, the product of the pressure and volume of a given mass of an ideal gas in a closed system is always constant. It can be verified experimentally using a pressure gauge and a variable volume container. It can also be derived from the kinetic theory of gases: if a container, with a fixed number of molecules inside, is reduced in volume, more molecules will strike a given area of the sides of the container per unit time, causing a greater pressure.

As a mathematical equation, Boyle's Law is written as either:
P \propto \frac{1}{V}, or
PV=k_1, or
P_1 V_1=P_2 V_2\,
where P is the pressure, and V is the volume of a gas, and k1 is the constant in this equation (and is not the same as the proportionality constants in the other equations below). The statement of Boyle 's law is as follows:

The volume of a given mass of a gas is inversely related to the pressure exerted on it at a given temperature and given number of moles.

Charles' Law

Charles' Law, or the law of volumes, was found in 1787 by Jacques Charles. It states that, for a given mass of an ideal gas at constant pressure, the volume is directly proportional to its absolute temperature, assuming in a closed system.
As a mathematical equation, Charles' Law is written as either:
V \propto T\,, or
V/T=k_2, or
V_1/T_1=V_2/T_2
where V is the volume of a gas, T is the absolute temperature and k2 is a proportionality constant (which is not the same as the proportionality constants in the other equations in this article).

Gay-Lussac's Law

Gay-Lussac's Law, or the Pressure Law, was found by Joseph Louis Gay-Lussac in 1809. It states that, for a given mass and constant volume of an ideal gas, the pressure exerted on the sides of its container is directly proportional to its absolute temperature.
As a mathematical equation, Gay-Lussac's Law is written as either:
P \propto T\,, or
P/T=k_3, or
P_1/T_1=P_2/T_2
where P is the pressure, T is the absolute temperature, and k3 is another proportionality constant.

Avogadro's Law

Avogadro's Law states that the volume occupied by an ideal gas is directly proportional to the number of molecules of the gas present in the container. This gives rise to the molar volume of a gas, which at STP is 22.4 dm3 (or litres). The relation is given by
\frac{V_1}{n_1}=\frac{V_2}{n_2} \,
where n is equal to the number of molecules of gas (or the number of moles of gas).

Combined and Ideal Gas Laws

The Combined Gas Law or General Gas Equation is obtained by combining Boyle's Law, Charles' Law, and Gay-Lussac's Law. It shows the relationship between the pressure, volume, and temperature for a fixed mass (quantity) of gas:
pV = k_5T \,
This can also be written as:
\qquad \frac {p_1V_1}{T_1}= \frac {p_2V_2}{T_2}
With the addition of Avogadro's Law, the combined gas law develops into the Ideal Gas Law:
pV = nRT \,
where
p is pressure
V is volume
n is the number of moles
R is the universal gas constant
T is temperature (K)
where the proportionality constant, now named R, is the universal gas constant with a value of 0.08206 (atm∙L)/(mol∙K). An equivalent formulation of this Law is:
pV = kNT \,
where
p is the pressure
V is the volume
N is the number of gas molecules
k is the Boltzmann constant (1.381×10−23 J·K−1 in SI units)
T is the absolute temperature
These equations are exact only for an ideal gas, which neglects various intermolecular effects (see real gas). However, the ideal gas law is a good approximation for most gases under moderate pressure and temperature.

This law has the following important consequences:
  1. If temperature and pressure are kept constant, then the volume of the gas is directly proportional to the number of molecules of gas.
  2. If the temperature and volume remain constant, then the pressure of the gas changes is directly proportional to the number of molecules of gas present.
  3. If the number of gas molecules and the temperature remain constant, then the pressure is inversely proportional to the volume.
  4. If the temperature changes and the number of gas molecules are kept constant, then either pressure or volume (or both) will change in direct proportion to the temperature.

Other gas laws

  • Graham's law states that the rate at which gas molecules diffuse is inversely proportional to the square root of its density. Combined with Avogadro's law (i.e. since equal volumes have equal number of molecules) this is the same as being inversely proportional to the root of the molecular weight.
P_{total} = P_1 + P_2 + P_3 + ... + P_n \equiv \sum_{i=1}^n P_i \,,
OR
P_\mathrm{total} = P_\mathrm{gas} + P_\mathrm{H_2 O} \,
where PTotal is the total pressure of the atmosphere, PGas is the pressure of the gas mixture in the atmosphere, and PH2O is the water pressure at that temperature.
At constant temperature, the amount of a given gas dissolved in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid.
p = k_{\rm H}\, c
at

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. NOTE: Since it is the ratio of all reflected radiation to incident radiation it will include the diffuse AND the specular radiation reflected. It is, however, common to assume a surface reflects in either a totally specular manner or a totally diffuse manner, as this can simplify calculations.

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 θi can reasonably be approximated by the proportionate sum of two terms: the directional-hemispherical reflectance at that solar zenith angle, {\bar \alpha(\theta_i)}, and the bi-hemispherical reflectance, \bar{ \bar \alpha} the proportion concerned being defined as the proportion of diffuse illumination {D}.
Albedo {\alpha} can then be given as:
{\alpha}= (1-D) \bar \alpha(\theta_i) + D \bar{ \bar \alpha}.
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] A =\left ( \frac{1329\times10^{-H/5}}{D} \right ) ^2,
where A is the astronomical albedo, D is the diameter in kilometers, and H 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 (solar irradiance); 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.

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Saturday, December 19, 2015

Stefan–Boltzmann law


From Wikipedia, the free encyclopedia
  (Redirected from Stephan-boltzmann law)


Graph of a function of total emitted energy of a black body j^{\star } proportional to its thermodynamic temperature T\,. In blue is a total energy according to the Wien approximation, j_{W}^{\star }=j^{\star }/\zeta (4)\approx 0.924\,\sigma T^{4}\!\,

The Stefan–Boltzmann law, also known as Stefan's law, describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (also known as the black-body radiant exitance or emissive power), j^{\star }, is directly proportional to the fourth power of the black body's thermodynamic temperature T:
j^{\star }=\sigma T^{4}.
The constant of proportionality σ, called the Stefan–Boltzmann constant or Stefan's constant, derives from other known constants of nature. The value of the constant is
\sigma ={\frac {2\pi ^{5}k^{4}}{15c^{2}h^{3}}}=5.670373\times 10^{-8}\,\mathrm {W\,m^{-2}K^{-4}} ,
where k is the Boltzmann constant, h is Planck's constant, and c is the speed of light in a vacuum. Thus at 100 K the energy flux is 5.67 W/m2, at 1000 K 56,700 W/m2, etc. The radiance (watts per square metre per steradian) is given by
L={\frac {j^{\star }}{\pi }}={\frac {\sigma }{\pi }}T^{4}.
A body that does not absorb all incident radiation (sometimes known as a grey body) emits less total energy than a black body and is characterized by an emissivity, \varepsilon <1:
j^{\star }=\varepsilon \sigma T^{4}.
The irradiance j^{\star } has dimensions of energy flux (energy per time per area), and the SI units of measure are joules per second per square metre, or equivalently, watts per square metre. The SI unit for absolute temperature T is the kelvin. \varepsilon is the emissivity of the grey body; if it is a perfect blackbody, \varepsilon =1. In the still more general (and realistic) case, the emissivity depends on the wavelength, \varepsilon =\varepsilon (\lambda ).

To find the total power radiated from an object, multiply by its surface area, A:
P=Aj^{\star }=A\varepsilon \sigma T^{4}.
Wavelength- and subwavelength-scale particles,[1] metamaterials,[2] and other nanostructures are not subject to ray-optical limits and may be designed to exceed the Stefan–Boltzmann law.

History

The law was deduced by Josef Stefan (1835–1893) in 1879 on the basis of experimental measurements made by John Tyndall and was derived from theoretical considerations, using thermodynamics, by Ludwig Boltzmann (1844–1906) in 1884. Boltzmann considered a certain ideal heat engine with light as a working matter instead of gas. The law is highly accurate only for ideal black objects, the perfect radiators, called black bodies; it works as a good approximation for most "grey" bodies. Stefan published this law in the article Über die Beziehung zwischen der Wärmestrahlung und der Temperatur (On the relationship between thermal radiation and temperature) in the Bulletins from the sessions of the Vienna Academy of Sciences.

Examples

Temperature of the Sun

With his law Stefan also determined the temperature of the Sun's surface. He learned from the data of Charles Soret (1854–1904) that the energy flux density from the Sun is 29 times greater than the energy flux density of a certain warmed metal lamella (a thin plate). A round lamella was placed at such a distance from the measuring device that it would be seen at the same angle as the Sun. Soret estimated the temperature of the lamella to be approximately 1900 °C to 2000 °C. Stefan surmised that ⅓ of the energy flux from the Sun is absorbed by the Earth's atmosphere, so he took for the correct Sun's energy flux a value 3/2 times greater than Soret's value, namely 29 × 3/2 = 43.5.
Precise measurements of atmospheric absorption were not made until 1888 and 1904. The temperature Stefan obtained was a median value of previous ones, 1950 °C and the absolute thermodynamic one 2200 K. As 2.574 = 43.5, it follows from the law that the temperature of the Sun is 2.57 times greater than the temperature of the lamella, so Stefan got a value of 5430 °C or 5700 K (the modern value is 5778 K[3]). This was the first sensible value for the temperature of the Sun. Before this, values ranging from as low as 1800 °C to as high as 13,000,000 °C were claimed. The lower value of 1800 °C was determined by Claude Servais Mathias Pouillet (1790–1868) in 1838 using the Dulong-Petit law. Pouillet also took just half the value of the Sun's correct energy flux.

Temperature of stars

The temperature of stars other than the Sun can be approximated using a similar means by treating the emitted energy as a black body radiation.[4] So:
L=4\pi R^{2}\sigma {T_{e}}^{4}
where L is the luminosity, σ is the Stefan–Boltzmann constant, R is the stellar radius and T is the effective temperature. This same formula can be used to compute the approximate radius of a main sequence star relative to the sun:
{\frac {R}{R_{\odot }}}\approx \left({\frac {T_{\odot }}{T}}\right)^{2}\cdot {\sqrt {\frac {L}{L_{\odot }}}}
where R_{\odot }, is the solar radius, and so forth.
With the Stefan–Boltzmann law, astronomers can easily infer the radii of stars. The law is also met in the thermodynamics of black holes in so-called Hawking radiation.

Temperature of the Earth

Similarly we can calculate the effective temperature of the Earth TE by equating the energy received from the Sun and the energy radiated by the Earth, under the black-body approximation. The amount of power, ES, emitted by the Sun is given by:
E_{S}=4\pi r_{S}^{2}\sigma T_{S}^{4}
At Earth, this energy is passing through a sphere with a radius of a0, the distance between the Earth and the Sun, and the energy passing through each square metre of the sphere is given by
E_{a_{0}}={\frac {E_{S}}{4\pi {a_{0}}^{2}}}
The Earth has a radius of rE, and therefore has a cross-section of \pi r_{E}^{2}. The amount of solar power absorbed by the Earth is thus given by:
E_{abs}=\pi r_{E}^{2}\times E_{a_{0}}:
Assuming the exchange is in a steady state, the amount of energy emitted by Earth must equal the amount absorbed, and so:
{\begin{aligned}4\pi r_{E}^{2}\sigma T_{E}^{4}&=\pi r_{E}^{2}\times E_{a_{0}}\\&=\pi r_{E}^{2}\times {\frac {4\pi r_{S}^{2}\sigma T_{S}^{4}}{4\pi a_{0}^{2}}}\\\end{aligned}}
TE can then be found:
{\begin{aligned}T_{E}^{4}&={\frac {r_{S}^{2}T_{S}^{4}}{4a_{0}^{2}}}\\T_{E}&=T_{S}\times {\sqrt {\frac {r_{S}}{2a_{0}}}}\\&=5780\;{\rm {K}}\times {\sqrt {696\times 10^{6}\;{\rm {m}} \over 2\times 149.598\times 10^{9}\;{\rm {m}}}}\\&\approx 279\;{\rm {K}}\end{aligned}}
where TS is the temperature of the Sun, rS the radius of the Sun, and a0 is the distance between the Earth and the Sun. This gives an effective temperature of 6 °C on the surface of the Earth, assuming that it perfectly absorbs all emission falling on it and has no atmosphere.

The Earth has an albedo of 0.3, meaning that 30% of the solar radiation that hits the planet gets scattered back into space without absorption. The effect of albedo on temperature can be approximated by assuming that the energy absorbed is multiplied by 0.3, but that the planet still radiates as a black body (the latter by definition of effective temperature, which is what we are calculating). This approximation reduces the temperature by a factor of 0.71/4, giving 255 K (−18 °C).[5][6]

However, long-wave radiation from the surface of the earth is partially absorbed and re-radiated back down by greenhouse gases, namely water vapor, carbon dioxide and methane.[7][8] Since the emissivity with greenhouse effect (weighted more in the longer wavelengths where the Earth radiates) is reduced more than the absorptivity (weighted more in the shorter wavelengths of the Sun's radiation) is reduced, the equilibrium temperature is higher than the simple black-body calculation estimates. As a result, the Earth's actual average surface temperature is about 288 K (15 °C), which is higher than the 255 K effective temperature, and even higher than the 279 K temperature that a black body would have.

Origination

Thermodynamic derivation of the energy density


The fact that the energy density of the box containing radiation is proportional to T^{4} can be derived using thermodynamics. It follows from the Maxwell stress tensor of classical electrodynamics that the radiation pressure p is related to the internal energy density u:
p={\frac {u}{3}}.
From the fundamental thermodynamic relation
dU=TdS-pdV,
we obtain the following expression, after dividing by dV and fixing T :
\left({\frac {\partial U}{\partial V}}\right)_{T}=T\left({\frac {\partial S}{\partial V}}\right)_{T}-p=T\left({\frac {\partial p}{\partial T}}\right)_{V}-p.
The last equality comes from the following Maxwell relation:
\left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial p}{\partial T}}\right)_{V}.
From the definition of energy density it follows that
U=uV
where the energy density of radiation only depends on the temperature, therefore
\left({\frac {\partial U}{\partial V}}\right)_{T}=u\left({\frac {\partial V}{\partial V}}\right)_{T}=u.
Now, the equality
\left({\frac {\partial U}{\partial V}}\right)_{T}=T\left({\frac {\partial p}{\partial T}}\right)_{V}-p,
after substitution of \left({\frac {\partial U}{\partial V}}\right)_{T} and p for the corresponding expressions, can be written as
u={\frac {T}{3}}\left({\frac {\partial u}{\partial T}}\right)_{V}-{\frac {u}{3}}.
Since the partial derivative \left({\frac {\partial u}{\partial T}}\right)_{V} can be expressed as a relationship between only u and T (if one isolates it on one side of the equality), the partial derivative can be replaced by the ordinary derivative. After separating the differentials the equality becomes
{\frac {du}{4u}}={\frac {dT}{T}},
which leads immediately to u=AT^{4}, with A as some constant of integration.

Stefan–Boltzmann's law in n-dimensional space[edit]

It can be shown that the radiation pressure in n-dimensional space is given by[10]
P={\frac {u}{n}}
So in n-dimensional space,
dQ=dU+PdV\,=d(uV)+{\frac {u}{n}}dV=Vdu+({\frac {n+1}{n}})udV
thus using 2 nd law of thermodynamics,we can write,
dS={\frac {Vdu}{T}}+{\frac {n+1}{n}}{\frac {u}{T}}dV
Hence
\left({\frac {\partial S}{\partial u}}\right)_{V}={\frac {V}{T}}
and
\left({\frac {\partial S}{\partial V}}\right)_{u}={\frac {n+1}{n}}{\frac {u}{T}}
or
\left({\frac {\partial {\frac {V}{T}}}{\partial V}}\right)={\frac {n+1}{n}}\left({\frac {\partial {\frac {u}{T}}}{\partial T}}\right)
yielding
{\frac {1}{T}}={\frac {n+1}{n}}({\frac {1}{T}}-{\frac {u}{T^{2}}}{\frac {dT}{du}})
or,
(n+1){\frac {dT}{T}}={\frac {du}{u}}
yielding,
u\propto T^{n+1}
implying
{\frac {dQ}{dt}}\propto T^{n+1}
The same result is obtained as the integral over frequency of Planck's law for n-dimensional space, albeit with a different value for the Stefan-Boltzmann constant at each dimension. In general this constant is
\sigma _{n}=2n(n-1){\sqrt {4\pi }}^{\,n-2}\Gamma {\Big (}{\frac {n}{2}}{\Big )}\zeta (n+1){\frac {k^{n+1}}{c^{n-1}h^{n}}} [11]
where \zeta (x) is Riemann's zeta function and \Gamma (x) is the Gamma function, h denotes the Planck's constant, c the speed of light in vacuum, and k the Boltzmann's constant.

Derivation from Planck's law

The law can be derived by considering a small flat black body surface radiating out into a half-sphere. This derivation uses spherical coordinates, with φ as the zenith angle and θ as the azimuthal angle; and the small flat blackbody surface lies on the xy-plane, where φ = π/2.
The intensity of the light emitted from the blackbody surface is given by Planck's law :
I(\nu ,T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}}.
where
The quantity I(\nu ,T)~A~d\nu ~d\Omega is the power radiated by a surface of area A through a solid angle dΩ in the frequency range between ν and ν + dν.
The Stefan–Boltzmann law gives the power emitted per unit area of the emitting body,
{\frac {P}{A}}=\int _{0}^{\infty }I(\nu ,T)d\nu \int d\Omega \,
To derive the Stefan–Boltzmann law, we must integrate Ω over the half-sphere and integrate ν from 0 to ∞. Furthermore, because black bodies are Lambertian (i.e. they obey Lambert's cosine law), the intensity observed along the sphere will be the actual intensity times the cosine of the zenith angle φ, and in spherical coordinates, dΩ = sin(φ) dφ dθ.
{\begin{aligned}{\frac {P}{A}}&=\int _{0}^{\infty }I(\nu ,T)\,d\nu \int _{0}^{2\pi }\,d\theta \int _{0}^{\pi /2}\cos \phi \sin \phi \,d\phi \\&=\pi \int _{0}^{\infty }I(\nu ,T)\,d\nu \end{aligned}}
Then we plug in for I:
{\frac {P}{A}}={\frac {2\pi h}{c^{2}}}\int _{0}^{\infty }{\frac {\nu ^{3}}{e^{\frac {h\nu }{kT}}-1}}d\nu \,
To do this integral, do a substitution,
u={\frac {h\nu }{kT}}\,

du={\frac {h}{kT}}\,d\nu
which gives:
{\frac {P}{A}}={\frac {2\pi h}{c^{2}}}\left({\frac {kT}{h}}\right)^{4}\int _{0}^{\infty }{\frac {u^{3}}{e^{u}-1}}\,du.
The integral on the right can be done in a number of ways (one is included in this article's appendix) – its answer is {\frac {\pi ^{4}}{15}}, giving the result that, for a perfect blackbody surface:
j^{\star }=\sigma T^{4}~,~~\sigma ={\frac {2\pi ^{5}k^{4}}{15c^{2}h^{3}}}={\frac {\pi ^{2}k^{4}}{60\hbar ^{3}c^{2}}}.
Finally, this proof started out only considering a small flat surface. However, any differentiable surface can be approximated by a bunch of small flat surfaces. So long as the geometry of the surface does not cause the blackbody to reabsorb its own radiation, the total energy radiated is just the sum of the energies radiated by each surface; and the total surface area is just the sum of the areas of each surface—so this law holds for all convex blackbodies, too, so long as the surface has the same temperature throughout. The law extends to radiation from non-convex bodies by using the fact that the convex hull of a black body radiates as though it were itself a black body.

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