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65-596: Under these conditions, essentially all molecules which arrive at the hole continue and pass through the hole, since collisions between molecules in the region of the hole are negligible. Conversely, when the diameter is larger than the mean free path of the gas, flow obeys the Sampson flow law. In medical terminology, an effusion refers to accumulation of fluid in an anatomic space , usually without loculation . Specific examples include subdural , mastoid , pericardial and pleural effusions . The word effusion derives from

130-421: A v g ≈ 1.085   v a v g {\textstyle v_{\rm {rms}}={\sqrt {3\pi /8}}\ v_{\rm {avg}}\approx 1.085\ v_{\rm {avg}}} ). The rate Φ N {\displaystyle \Phi _{N}} at which a gas of molar mass M {\displaystyle M} effuses (typically expressed as the number of molecules passing through

195-533: A Lennard-Jones potential . One way to deal with such "soft" molecules is to use the Lennard-Jones σ parameter as the diameter. Another way is to assume a hard-sphere gas that has the same viscosity as the actual gas being considered. This leads to a mean free path where m {\displaystyle m} is the molecular mass, ρ = m p / ( k B T ) {\displaystyle \rho =mp/(k_{\text{B}}T)}

260-419: A molality , the proportionality constant is known as the cryoscopic constant ( K f ) and is characteristic for each solvent. If w represents the mass fraction of the solute in solution, and assuming no dissociation of the solute, the molar mass is given by The boiling point of a solution of an involatile solute is higher than that of the pure solvent , and the boiling-point elevation ( Δ T )

325-409: A compound in g/mol thus is equal to the mass of this number of molecules of the compound in grams. The molar mass of atoms of an element is given by the relative atomic mass of the element multiplied by the molar mass constant , M u  ≈  1.000 000 × 10  kg/mol = 1 g/mol. For normal samples from earth with typical isotope composition, the atomic weight can be approximated by

390-416: A gas is inversely proportional to the square root of the mass of its particles. In other words, the ratio of the rates of effusion of two gases at the same temperature and pressure is given by the inverse ratio of the square roots of the masses of the gas particles. where M 1 {\displaystyle M_{1}} and M 2 {\displaystyle M_{2}} represent

455-533: A medium with dimensions smaller than the mean free path of electrons occurs through ballistic conduction or ballistic transport. In such scenarios electrons alter their motion only in collisions with conductor walls. If one takes a suspension of non-light-absorbing particles of diameter d with a volume fraction Φ , the mean free path of the photons is: where Q s is the scattering efficiency factor. Q s can be evaluated numerically for spherical particles using Mie theory . In an otherwise empty cavity,

520-420: A precision of a few parts per million . This is accurate enough to directly determine the chemical formula of a molecule. The term formula weight has a specific meaning when used in the context of DNA synthesis: whereas an individual phosphoramidite nucleobase to be added to a DNA polymer has protecting groups and has its molecular weight quoted including these groups, the amount of molecular weight that

585-414: A sample which has been distilled will be enriched in the lighter isotopes of all the elements present. This complicates the calculation of the standard uncertainty in the molar mass. A useful convention for normal laboratory work is to quote molar masses to two decimal places for all calculations. This is more accurate than is usually required, but avoids rounding errors during calculations. When

650-523: A separate dimension of measurement . Until 2019, the mole was defined as the amount of substance that has as many constituent particles as there are atoms in 12 grams of carbon-12 . During that period, the molar mass of carbon-12 was thus exactly 12 g/mol, by definition. Since 2019, a mole of any substance has been redefined in the SI as the amount of that substance containing an exactly defined number of particles, 6.022 140 76 × 10 . The molar mass of

715-461: A vapor at low pressure by sublimation . The vapor slowly effuses through a pinhole, and the loss of mass is proportional to the vapor pressure and can be used to determine this pressure. The heat of sublimation can also be determined by measuring the vapor pressure as a function of temperature, using the Clausius–Clapeyron relation . Mean free path In physics , mean free path is

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780-414: A volumetric flow rate as follows: or where Φ V {\displaystyle \Phi _{V}} is the volumetric flow rate of the gas, P a v g {\displaystyle P_{\rm {avg}}} is the average pressure on either side of the orifice, and d {\displaystyle d} is the hole diameter. At constant pressure and temperature,

845-409: Is The fraction of particles that are not stopped ( attenuated ) by the slab is called transmission T = I / I 0 = e − x / ℓ {\displaystyle T=I/I_{0}=e^{-x/\ell }} , where x is equal to the thickness of the slab. In the kinetic theory of gases , the mean free path of a particle, such as a molecule ,

910-451: Is distinct but related to the molar mass, which is a measure of the average molecular mass of all the molecules in a sample and is usually the more appropriate measure when dealing with macroscopic (weigh-able) quantities of a substance. Molecular masses are calculated from the atomic masses of each nuclide , while molar masses are calculated from the standard atomic weights of each element . The standard atomic weight takes into account

975-405: Is given by Combining these two equations gives an expression for the molar mass in terms of the vapour density for conditions of known pressure and temperature : The freezing point of a solution is lower than that of the pure solvent , and the freezing-point depression ( Δ T ) is directly proportional to the amount concentration for dilute solutions. When the composition is expressed as

1040-447: Is kg/mol. However, for historical reasons, molar masses are almost always expressed in g/mol. The mole was defined in such a way that the molar mass of a compound, in g/mol, is numerically equal to the average mass of one molecule or formula unit, in daltons. It was exactly equal before the redefinition of the mole in 2019 , and is now only approximately equal, but the difference is negligible for all practical purposes. Thus, for example,

1105-436: Is limited by the knowledge of the isotopic distribution of the element. If a more accurate value of the molar mass is required, it is necessary to determine the isotopic distribution of the sample in question, which may be different from the standard distribution used to calculate the standard atomic mass. The isotopic distributions of the different elements in a sample are not necessarily independent of one another: for example,

1170-784: Is much greater than pinhole diameter and the gas can be treated as an ideal gas . If a small area A {\displaystyle A} on the container is punched to become a small hole, the effusive flow rate will be Q effusion = J impingement × A = P A 2 π m k B T = P A N A 2 π M R T {\displaystyle {\begin{aligned}Q_{\text{effusion}}&=J_{\text{impingement}}\times A\\&={\frac {PA}{\sqrt {2\pi mk_{\text{B}}T}}}\\&={\frac {PAN_{\text{A}}}{\sqrt {2\pi MRT}}}\end{aligned}}} where M {\displaystyle M}

1235-461: Is particularly important in polymer science , where there is usually a molar mass distribution of non-uniform polymers so that different polymer molecules contain different numbers of monomer units. The average molar mass of mixtures M ¯ {\displaystyle {\overline {M}}} can be calculated from the mole fractions x i of the components and their molar masses M i : It can also be calculated from

1300-447: Is sometimes called a number of mean free paths image. In macroscopic charge transport, the mean free path of a charge carrier in a metal ℓ {\displaystyle \ell } is proportional to the electrical mobility μ {\displaystyle \mu } , a value directly related to electrical conductivity , that is: where q is the charge , τ {\displaystyle \tau }

1365-630: Is the Boltzmann constant . The average molecular speed can be calculated from the Maxwell speed distribution as v a v g = 8 / 3 π   v r m s ≈ 0.921   v r m s {\textstyle v_{\rm {avg}}={\sqrt {8/3\pi }}\ v_{\rm {rms}}\approx 0.921\ v_{\rm {rms}}} (or, equivalently, v r m s = 3 π / 8   v

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1430-432: Is the absolute temperature . Assuming the pressure difference between the two sides of the barrier is much smaller than P a v g {\displaystyle P_{\rm {avg}}} , the average absolute pressure in the system ( i.e. Δ P ≪ P a v g {\displaystyle \Delta P\ll P_{\rm {avg}}} ), it is possible to express effusion flow as

1495-489: Is the mean free time , m is the effective mass , and v F is the Fermi velocity of the charge carrier. The Fermi velocity can easily be derived from the Fermi energy via the non-relativistic kinetic energy equation. In thin films , however, the film thickness can be smaller than the predicted mean free path, making surface scattering much more noticeable, effectively increasing the resistivity . Electron mobility through

1560-806: Is the molar mass , N A {\displaystyle N_{\text{A}}} is the Avogadro constant , and R = N A k B {\displaystyle R=N_{\text{A}}k_{\text{B}}} is the molar gas constant . The average velocity of effused particles is v x ¯ = v y ¯ = 0 v z ¯ = π k B T 2 m . {\displaystyle {\begin{aligned}{\overline {v_{x}}}&={\overline {v_{y}}}=0\\{\overline {v_{z}}}&={\sqrt {\frac {\pi k_{\text{B}}T}{2m}}}.\end{aligned}}} Combined with

1625-427: Is the average distance the particle travels between collisions with other moving particles. The derivation above assumed the target particles to be at rest; therefore, in reality, the formula ℓ = ( n σ ) − 1 {\displaystyle \ell =(n\sigma )^{-1}} holds for a beam particle with a high speed v {\displaystyle v} relative to

1690-507: Is the density of ideal gas, and μ is the dynamic viscosity. This expression can be put into the following convenient form with R s p e c i f i c = k B / m {\displaystyle R_{\rm {specific}}=k_{\text{B}}/m} being the specific gas constant , equal to 287 J/(kg*K) for air. The following table lists some typical values for air at different pressures at room temperature. Note that different definitions of

1755-418: Is the distance traveled by the beam through the target, and I 0 is the beam intensity before it entered the target; ℓ is called the mean free path because it equals the mean distance traveled by a beam particle before being stopped. To see this, note that the probability that a particle is absorbed between x and x + dx is given by Thus the expectation value (or average, or simply mean) of x

1820-426: Is the effective cross-sectional area for collision. The area of the slab is L , and its volume is L   dx . The typical number of stopping atoms in the slab is the concentration n times the volume, i.e., n L   dx . The probability that a beam particle will be stopped in that slab is the net area of the stopping atoms divided by the total area of the slab: where σ is the area (or, more formally,

1885-538: Is the mass of one molecule (of any single isotopic composition), and to the atomic mass , which is the mass of one atom (of any single isotope). The dalton , symbol Da, is also sometimes used as a unit of molar mass, especially in biochemistry , with the definition 1 Da = 1 g/mol, despite the fact that it is strictly a unit of mass (1 Da = 1 u = 1.660 539 068 92 (52) × 10  kg , as of 2022 CODATA recommended values). Obsolete terms for molar mass include gram atomic mass for

1950-397: Is the molar mass of the atoms multiplied by the number of atoms in each molecule: The molar mass of a compound is given by the sum of the relative atomic mass A r of the atoms which form the compound multiplied by the molar mass constant M u ≈ 1  g/mol {\displaystyle M_{u}\approx 1{\text{ g/mol}}} : Here, M r

2015-1312: Is the relative molar mass, also called formula weight. For normal samples from earth with typical isotope composition, the standard atomic weight or the conventional atomic weight can be used as an approximation of the relative atomic mass of the sample. Examples are: M ( NaCl ) = [ 22.98976928 ( 2 ) + 35.453 ( 2 ) ] × 1  g/mol = 58.443 ( 2 )  g/mol M ( C 12 H 22 O 11 ) = [ 12 × 12.0107 ( 8 ) + 22 × 1.00794 ( 7 ) + 11 × 15.9994 ( 3 ) ] × 1  g/mol = 342.297 ( 14 )  g/mol {\displaystyle {\begin{array}{ll}M({\ce {NaCl}})&={\bigl [}22.98976928(2)+35.453(2){\bigr ]}\times 1{\text{ g/mol}}\\&=58.443(2){\text{ g/mol}}\\[4pt]M({\ce {C12H22O11}})&={\bigl [}12\times 12.0107(8)+22\times 1.00794(7)+11\times 15.9994(3){\bigr ]}\times 1{\text{ g/mol}}\\&=342.297(14){\text{ g/mol}}\end{array}}} An average molar mass may be defined for mixtures of compounds. This

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2080-454: Is ultimately added by this nucleobase to a DNA polymer is referred to as the nucleobase's formula weight (i.e., the molecular weight of this nucleobase within the DNA polymer, minus protecting groups). The precision to which a molar mass is known depends on the precision of the atomic masses from which it was calculated (and very slightly on the value of the molar mass constant , which depends on

2145-617: The Latin word, effundo, which means "shed", "pour forth", "pour out", "utter", "lavish", "waste". Effusion from an equilibrated container into outside vacuum can be calculated based on kinetic theory . The number of atomic or molecular collisions with a wall of a container per unit area per unit time ( impingement rate ) is given by: J impingement = P 2 π m k B T . {\displaystyle J_{\text{impingement}}={\frac {P}{\sqrt {2\pi mk_{\text{B}}T}}}.} assuming mean free path

2210-506: The X-ray spectrum changes with distance. Sometimes one measures the thickness of a material in the number of mean free paths . Material with the thickness of one mean free path will attenuate to 37% (1/ e ) of photons. This concept is closely related to half-value layer (HVL): a material with a thickness of one HVL will attenuate 50% of photons. A standard x-ray image is a transmission image, an image with negative logarithm of its intensities

2275-487: The isotopic distribution of the element in a given sample (usually assumed to be "normal"). For example, water has a molar mass of 18.0153(3) g/mol , but individual water molecules have molecular masses which range between 18.010 564 6863 (15) Da ( H 2 O ) and 22.027 7364 (9) Da ( H 2 O ). The distinction between molar mass and molecular mass is important because relative molecular masses can be measured directly by mass spectrometry , often to

2340-414: The kinetic theory of gases , the kinetic energy for a gas at a temperature T {\displaystyle T} is where m {\displaystyle m} is the mass of one molecule, v r m s {\displaystyle v_{\rm {rms}}} is the root-mean-square speed of the molecules, and k B {\displaystyle k_{\rm {B}}}

2405-411: The mass fractions w i of the components: As an example, the average molar mass of dry air is 28.96 g/mol. Molar mass is closely related to the relative molar mass ( M r ) of a compound and to the standard atomic weights of its constituent elements. However, it should be distinguished from the molecular mass (which is confusingly also sometimes known as molecular weight), which

2470-440: The molar mass ( M ) (sometimes called molecular weight or formula weight , but see related quantities for usage) of a chemical compound is defined as the ratio between the mass and the amount of substance (measured in moles ) of any sample of the compound. The molar mass is a bulk, not molecular, property of a substance. The molar mass is an average of many instances of the compound, which often vary in mass due to

2535-469: The nucleus before they interact with other nucleons. The effective mean free path of a nucleon in nuclear matter must be somewhat larger than the nuclear dimensions in order to allow the use of the independent particle model. This requirement seems to be in contradiction to the assumptions made in the theory ... We are facing here one of the fundamental problems of nuclear structure physics which has yet to be solved. Molar mass In chemistry ,

2600-422: The relative molar mass ( M r ). This is a dimensionless quantity (i.e., a pure number, without units) equal to the molar mass divided by the molar mass constant . The molecular mass ( m ) is the mass of a given molecule: it is usually measured in daltons (Da or u). Different molecules of the same compound may have different molecular masses because they contain different isotopes of an element. This

2665-449: The " scattering cross-section ") of one atom. The drop in beam intensity equals the incoming beam intensity multiplied by the probability of the particle being stopped within the slab: This is an ordinary differential equation : whose solution is known as Beer–Lambert law and has the form I = I 0 e − x / ℓ {\displaystyle I=I_{0}e^{-x/\ell }} , where x

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2730-412: The average distance over which a moving particle (such as an atom , a molecule , or a photon ) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a result of one or more successive collisions with other particles. Imagine a beam of particles being shot through a target, and consider an infinitesimally thin slab of the target (see

2795-413: The average mass of a molecule of water is about 18.0153 daltons, and the molar mass of water is about 18.0153 g/mol. For chemical elements without isolated molecules, such as carbon and metals, the molar mass is computed dividing by the number of moles of atoms instead. Thus, for example, the molar mass of iron is about 55.845 g/mol. Since 1971, SI defined the "amount of substance" as

2860-407: The calculation of the mean free path is more complicated, because photons are not mono-energetic, but have some distribution of energies called a spectrum . As photons move through the target material, they are attenuated with probabilities depending on their energy, as a result their distribution changes in process called spectrum hardening. Because of spectrum hardening, the mean free path of

2925-400: The concept of the mean free path is not commonly used, being replaced by the similar concept of attenuation length . In particular, for high-energy photons, which mostly interact by electron–positron pair production , the radiation length is used much like the mean free path in radiography. Independent-particle models in nuclear physics require the undisturbed orbiting of nucleons within

2990-407: The effusive flow rate, the recoil/thrust force on the system itself is F = m v z ¯ × Q effusion = P A 2 . {\displaystyle F=m{\overline {v_{z}}}{\times }Q_{\text{effusion}}={\frac {PA}{2}}.} An example is the recoil force on a balloon with a small hole flying in vacuum. According to

3055-411: The figure). The atoms (or particles) that might stop a beam particle are shown in red. The magnitude of the mean free path depends on the characteristics of the system. Assuming that all the target particles are at rest but only the beam particle is moving, that gives an expression for the mean free path: where ℓ is the mean free path, n is the number of target particles per unit volume, and σ

3120-431: The hole per second) is then Here Δ P {\displaystyle \Delta P} is the gas pressure difference across the barrier, A {\displaystyle A} is the area of the hole, N A {\displaystyle N_{\text{A}}} is the Avogadro constant , R {\displaystyle R} is the gas constant and T {\displaystyle T}

3185-412: The mass, in grams, of one mole of atoms of an element, and gram molecular mass for the mass, in grams, of one mole of molecules of a compound. The gram-atom is a former term for a mole of atoms, and gram-molecule for a mole of molecules. Molecular weight (M.W.) (for molecular compounds) and formula weight (F.W.) (for non-molecular compounds), are older terms for what is now more correctly called

3250-551: The mean free path is where k B is the Boltzmann constant , p {\displaystyle p} is the pressure of the gas and T {\displaystyle T} is the absolute temperature. In practice, the diameter of gas molecules is not well defined. In fact, the kinetic diameter of a molecule is defined in terms of the mean free path. Typically, gas molecules do not behave like hard spheres, but rather attract each other at larger distances and repel each other at shorter distances, as can be described with

3315-481: The mean free path of a single particle bouncing off the walls is: where V is the volume of the cavity, S is the total inside surface area of the cavity, and F is a constant related to the shape of the cavity. For most simple cavity shapes, F is approximately 4. This relation is used in the derivation of the Sabine equation in acoustics, using a geometrical approximation of sound propagation. In particle physics

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3380-428: The measured value of the dalton ). Most atomic masses are known to a precision of at least one part in ten-thousand, often much better (the atomic mass of lithium is a notable, and serious, exception). This is adequate for almost all normal uses in chemistry: it is more precise than most chemical analyses , and exceeds the purity of most laboratory reagents. The precision of atomic masses, and hence of molar masses,

3445-703: The molar mass is greater than 1000 g/mol, it is rarely appropriate to use more than one decimal place. These conventions are followed in most tabulated values of molar masses. Molar masses are almost never measured directly. They may be calculated from standard atomic masses, and are often listed in chemical catalogues and on safety data sheets (SDS). Molar masses typically vary between: While molar masses are almost always, in practice, calculated from atomic weights, they can also be measured in certain cases. Such measurements are much less precise than modern mass spectrometric measurements of atomic weights and molecular masses, and are of mostly historical interest. All of

3510-406: The molar masses of the gases. This equation is known as Graham's law of effusion . The effusion rate for a gas depends directly on the average velocity of its particles. Thus, the faster the gas particles are moving, the more likely they are to pass through the effusion orifice. The Knudsen cell is used to measure the vapor pressures of a solid with very low vapor pressure. Such a solid forms

3575-407: The molecular diameter, as well as different assumptions about the value of atmospheric pressure (100 vs 101.3 kPa) and room temperature (293.17 K vs 296.15 K or even 300 K) can lead to slightly different values of the mean free path. In gamma-ray radiography the mean free path of a pencil beam of mono-energetic photons is the average distance a photon travels between collisions with atoms of

3640-461: The most authoritative sources define it differently. The difference is that molecular mass is the mass of one specific particle or molecule, while the molar mass is an average over many particles or molecules. The molar mass is an intensive property of the substance, that does not depend on the size of the sample. In the International System of Units (SI), the coherent unit of molar mass

3705-604: The number of collisions is 2 {\displaystyle {\sqrt {2}}} times the number with stationary targets. Therefore, the following relationship applies: and using n = N / V = p / ( k B T ) {\displaystyle n=N/V=p/(k_{\text{B}}T)} ( ideal gas law ) and σ = π d 2 {\displaystyle \sigma =\pi d^{2}} (effective cross-sectional area for spherical particles with diameter d {\displaystyle d} ), it may be shown that

3770-579: The presence of isotopes . Most commonly, the molar mass is computed from the standard atomic weights and is thus a terrestrial average and a function of the relative abundance of the isotopes of the constituent atoms on Earth. The molar mass is appropriate for converting between the mass of a substance and the amount of a substance for bulk quantities. The molecular mass (for molecular compounds) and formula mass (for non-molecular compounds, such as ionic salts ) are commonly used as synonyms of molar mass, differing only in units ( daltons vs g/mol); however,

3835-443: The procedures rely on colligative properties , and any dissociation of the compound must be taken into account. The measurement of molar mass by vapour density relies on the principle, first enunciated by Amedeo Avogadro , that equal volumes of gases under identical conditions contain equal numbers of particles. This principle is included in the ideal gas equation : where n is the amount of substance . The vapour density ( ρ )

3900-489: The relative speed is v r e l = v r e l a t i v e 2 ¯ = v 1 2 + v 2 2 ¯ = 2 v . {\displaystyle v_{\rm {rel}}={\sqrt {\overline {\mathbf {v} _{\rm {relative}}^{2}}}}={\sqrt {\overline {\mathbf {v} _{1}^{2}+\mathbf {v} _{2}^{2}}}}={\sqrt {2}}v.} This means that

3965-418: The root-mean-square speed and therefore the effusion rate are inversely proportional to the square root of the molecular weight. Gases with a lower molecular weight effuse more rapidly than gases with a higher molecular weight, so that the number of lighter molecules passing through the hole per unit time is greater. Scottish chemist Thomas Graham (1805–1869) found experimentally that the rate of effusion of

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4030-1061: The square of relative velocity is: v r e l a t i v e 2 ¯ = ( v 1 − v 2 ) 2 ¯ = v 1 2 + v 2 2 − 2 v 1 ⋅ v 2 ¯ . {\displaystyle {\overline {\mathbf {v} _{\rm {relative}}^{2}}}={\overline {(\mathbf {v} _{1}-\mathbf {v} _{2})^{2}}}={\overline {\mathbf {v} _{1}^{2}+\mathbf {v} _{2}^{2}-2\mathbf {v} _{1}\cdot \mathbf {v} _{2}}}.} In equilibrium, v 1 {\displaystyle \mathbf {v} _{1}} and v 2 {\displaystyle \mathbf {v} _{2}} are random and uncorrelated, therefore v 1 ⋅ v 2 ¯ = 0 {\displaystyle {\overline {\mathbf {v} _{1}\cdot \mathbf {v} _{2}}}=0} , and

4095-486: The standard atomic weight or the conventional atomic weight. Multiplying by the molar mass constant ensures that the calculation is dimensionally correct: standard relative atomic masses are dimensionless quantities (i.e., pure numbers) whereas molar masses have units (in this case, grams per mole). Some elements are usually encountered as molecules , e.g. hydrogen ( H 2 ), sulfur ( S 8 ), chlorine ( Cl 2 ). The molar mass of molecules of these elements

4160-488: The target material. It depends on the material and the energy of the photons: where μ is the linear attenuation coefficient , μ/ρ is the mass attenuation coefficient and ρ is the density of the material. The mass attenuation coefficient can be looked up or calculated for any material and energy combination using the National Institute of Standards and Technology (NIST) databases. In X-ray radiography

4225-404: The velocities of an ensemble of identical particles with random locations. In that case, the motions of target particles are comparatively negligible, hence the relative velocity v r e l ≈ v {\displaystyle v_{\rm {rel}}\approx v} . If, on the other hand, the beam particle is part of an established equilibrium with identical particles, then

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