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49-465: In a general sense, an ingredient is a substance which forms part of a mixture . In cooking , recipes specify which ingredients are used to prepare a dish. Many commercial products contain secret ingredients purported to make them better than competing products. In the pharmaceutical industry, an active ingredient is the ingredient in a formulation which invokes biological activity . National laws usually require prepared food products to display

98-427: A region D in three-dimensional space is given by the triple or volume integral of the constant function f ( x , y , z ) = 1 {\displaystyle f(x,y,z)=1} over the region. It is usually written as: ∭ D 1 d x d y d z . {\displaystyle \iiint _{D}1\,dx\,dy\,dz.} In cylindrical coordinates ,

147-437: A reservoir , the container's volume is modeled by shapes and calculated using mathematics. To ease calculations, a unit of volume is equal to the volume occupied by a unit cube (with a side length of one). Because the volume occupies three dimensions, if the metre (m) is chosen as a unit of length, the corresponding unit of volume is the cubic metre (m ). The cubic metre is also a SI derived unit . Therefore, volume has

196-457: A unit dimension of L . The metric units of volume uses metric prefixes , strictly in powers of ten . When applying prefixes to units of volume, which are expressed in units of length cubed, the cube operators are applied to the unit of length including the prefix. An example of converting cubic centimetre to cubic metre is: 2.3 cm = 2.3 (cm) = 2.3 (0.01 m) = 0.0000023 m (five zeros). Commonly used prefixes for cubed length units are

245-417: A blend of them). All mixtures can be characterized as being separable by mechanical means (e.g. purification , distillation , electrolysis , chromatography , heat , filtration , gravitational sorting, centrifugation ). Mixtures differ from chemical compounds in the following ways: In the example of sand and water, neither one of the two substances changed in any way when they are mixed. Although

294-413: A foam, these are a solid and a fluid, or a liquid and a gas. On larger scales both constituents are present in any region of the mixture, and in a well-mixed mixture in the same or only slightly varying concentrations. On a microscopic scale, however, one of the constituents is absent in almost any sufficiently small region. (If such absence is common on macroscopic scales, the combination of the constituents

343-612: A formula exists for the shape's boundary. Zero- , one- and two-dimensional objects have no volume; in four and higher dimensions, an analogous concept to the normal volume is the hypervolume. The precision of volume measurements in the ancient period usually ranges between 10–50 mL (0.3–2 US fl oz; 0.4–2 imp fl oz). The earliest evidence of volume calculation came from ancient Egypt and Mesopotamia as mathematical problems, approximating volume of simple shapes such as cuboids , cylinders , frustum and cones . These math problems have been written in

392-458: A homogeneous mixture, a solution has one phase (solid, liquid, or gas), although the phase of the solute and solvent may initially have been different (e.g., salt water). Gases exhibit by far the greatest space (and, consequently, the weakest intermolecular forces) between their atoms or molecules; since intermolecular interactions are minuscule in comparison to those in liquids and solids, dilute gases very easily form solutions with one another. Air

441-697: A list of ingredients and specifically require that certain additives be listed. Law typically requires that ingredients be listed according to their relative weight within the product. From Middle French ingredient, from Latin ingredientem, present participle of ingredior (“to go or enter into or onto”). An artificial ingredient usually refers to an ingredient which is artificial or human-made, such as: Mixture Mixtures are one product of mechanically blending or mixing chemical substances such as elements and compounds , without chemical bonding or other chemical change, so that each ingredient substance retains its own chemical properties and makeup. Despite

490-441: A negative value, similar to length and area . Like all continuous monotonic (order-preserving) measures, volumes of bodies can be compared against each other and thus can be ordered. Volume can also be added together and be decomposed indefinitely; the latter property is integral to Cavalieri's principle and to the infinitesimal calculus of three-dimensional bodies. A 'unit' of infinitesimally small volume in integral calculus

539-402: A single phase . A solution is a special type of homogeneous mixture where the ratio of solute to solvent remains the same throughout the solution and the particles are not visible with the naked eye, even if homogenized with multiple sources. In solutions, solutes will not settle out after any period of time and they cannot be removed by physical methods, such as a filter or centrifuge . As

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588-457: Is a dispersed medium , not a mixture.) One can distinguish different characteristics of heterogeneous mixtures by the presence or absence of continuum percolation of their constituents. For a foam, a distinction is made between reticulated foam in which one constituent forms a connected network through which the other can freely percolate, or a closed-cell foam in which one constituent is present as trapped in small cells whose walls are formed by

637-401: Is a measure of regions in three-dimensional space . It is often quantified numerically using SI derived units (such as the cubic metre and litre ) or by various imperial or US customary units (such as the gallon , quart , cubic inch ). The definition of length and height (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of

686-403: Is a matter of the scale of sampling. On a coarse enough scale, any mixture can be said to be homogeneous, if the entire article is allowed to count as a "sample" of it. On a fine enough scale, any mixture can be said to be heterogeneous, because a sample could be as small as a single molecule. In practical terms, if the property of interest of the mixture is the same regardless of which sample of it

735-439: Is an example of a solution as well: a homogeneous mixture of gaseous nitrogen solvent, in which oxygen and smaller amounts of other gaseous solutes are dissolved. Mixtures are not limited in either their number of substances or the amounts of those substances, though in a homogeneous mixture the solute-to-solvent proportion can only reach a certain point before the mixture separates and becomes heterogeneous. A homogeneous mixture

784-454: Is called homogeneous, whereas a mixture of non-uniform composition and of which the components can be easily identified, such as sand in water, it is called heterogeneous. In addition, " uniform mixture " is another term for homogeneous mixture and " non-uniform mixture " is another term for heterogeneous mixture . These terms are derived from the idea that a homogeneous mixture has a uniform appearance , or only one visible phase , because

833-579: Is characterized by uniform dispersion of its constituent substances throughout; the substances exist in equal proportion everywhere within the mixture. Differently put, a homogeneous mixture will be the same no matter from where in the mixture it is sampled. For example, if a solid-liquid solution is divided into two halves of equal volume , the halves will contain equal amounts of both the liquid medium and dissolved solid (solvent and solute). In physical chemistry and materials science , "homogeneous" more narrowly describes substances and mixtures which are in

882-431: Is common for measuring small volume of fluids or granular materials , by using a multiple or fraction of the container. For granular materials, the container is shaken or leveled off to form a roughly flat surface. This method is not the most accurate way to measure volume but is often used to measure cooking ingredients . Air displacement pipette is used in biology and biochemistry to measure volume of fluids at

931-412: Is one such example: it can be more specifically described as a gaseous solution of oxygen and other gases dissolved in nitrogen (its major component). The basic properties of solutions are as drafted under: Examples of heterogeneous mixtures are emulsions and foams . In most cases, the mixture consists of two main constituents. For an emulsion, these are immiscible fluids such as water and oil. For

980-526: Is taken for the examination used, the mixture is homogeneous. Gy's sampling theory quantitatively defines the heterogeneity of a particle as: where h i {\displaystyle h_{i}} , c i {\displaystyle c_{i}} , c batch {\displaystyle c_{\text{batch}}} , m i {\displaystyle m_{i}} , and m aver {\displaystyle m_{\text{aver}}} are respectively:

1029-634: Is the volume element ; this formulation is useful when working with different coordinate systems , spaces and manifolds . The oldest way to roughly measure a volume of an object is using the human body, such as using hand size and pinches . However, the human body's variations make it extremely unreliable. A better way to measure volume is to use roughly consistent and durable containers found in nature, such as gourds , sheep or pig stomachs , and bladders . Later on, as metallurgy and glass production improved, small volumes nowadays are usually measured using standardized human-made containers. This method

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1078-503: Is used when integrating by an axis parallel to the axis of rotation. The general equation can be written as: V = π ∫ a b | f ( x ) 2 − g ( x ) 2 | d x {\displaystyle V=\pi \int _{a}^{b}\left|f(x)^{2}-g(x)^{2}\right|\,dx} where f ( x ) {\textstyle f(x)} and g ( x ) {\textstyle g(x)} are

1127-711: The Moscow Mathematical Papyrus (c. 1820 BCE). In the Reisner Papyrus , ancient Egyptians have written concrete units of volume for grain and liquids, as well as a table of length, width, depth, and volume for blocks of material. The Egyptians use their units of length (the cubit , palm , digit ) to devise their units of volume, such as the volume cubit or deny (1 cubit × 1 cubit × 1 cubit), volume palm (1 cubit × 1 cubit × 1 palm), and volume digit (1 cubit × 1 cubit × 1 digit). The last three books of Euclid's Elements , written in around 300 BCE, detailed

1176-410: The cube , cuboid and cylinder , they have an essentially the same volume calculation formula as one for the prism : the base of the shape multiplied by its height . The calculation of volume is a vital part of integral calculus. One of which is calculating the volume of solids of revolution , by rotating a plane curve around a line on the same plane. The washer or disc integration method

1225-574: The imperial gallon was defined to be the volume occupied by ten pounds of water at 17 °C (62 °F). This definition was further refined until the United Kingdom's Weights and Measures Act 1985 , which makes 1 imperial gallon precisely equal to 4.54609 litres with no use of water. The 1960 redefinition of the metre from the International Prototype Metre to the orange-red emission line of krypton-86 atoms unbounded

1274-509: The sester , amber , coomb , and seam . The sheer quantity of such units motivated British kings to standardize them, culminated in the Assize of Bread and Ale statute in 1258 by Henry III of England . The statute standardized weight, length and volume as well as introduced the peny, ounce, pound, gallon and bushel. In 1618, the London Pharmacopoeia (medicine compound catalog) adopted

1323-439: The volume integral is ∭ D r d r d θ d z , {\displaystyle \iiint _{D}r\,dr\,d\theta \,dz,} In spherical coordinates (using the convention for angles with θ {\displaystyle \theta } as the azimuth and φ {\displaystyle \varphi } measured from the polar axis; see more on conventions ),

1372-487: The Roman gallon or congius as a basic unit of volume and gave a conversion table to the apothecaries' units of weight. Around this time, volume measurements are becoming more precise and the uncertainty is narrowed to between 1–5 mL (0.03–0.2 US fl oz; 0.04–0.2 imp fl oz). Around the early 17th century, Bonaventura Cavalieri applied the philosophy of modern integral calculus to calculate

1421-461: The average mass of a particle in the population. During sampling of heterogeneous mixtures of particles, the variance of the sampling error is generally non-zero. Pierre Gy derived, from the Poisson sampling model, the following formula for the variance of the sampling error in the mass concentration in a sample: in which V is the variance of the sampling error, N is the number of particles in

1470-487: The contained volume does not need to fill towards the container's capacity, or vice versa. Containers can only hold a specific amount of physical volume, not weight (excluding practical concerns). For example, a 50,000 bbl (7,900,000 L) tank that can just hold 7,200 t (15,900,000 lb) of fuel oil will not be able to contain the same 7,200 t (15,900,000 lb) of naphtha , due to naphtha's lower density and thus larger volume. For many shapes such as

1519-588: The container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. By metonymy , the term "volume" sometimes is used to refer to the corresponding region (e.g., bounding volume ). In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simple three-dimensional shapes can have their volume easily calculated using arithmetic formulas . Volumes of more complicated shapes can be calculated with integral calculus if

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1568-399: The cubic millimetre (mm ), cubic centimetre (cm ), cubic decimetre (dm ), cubic metre (m ) and the cubic kilometre (km ). The conversion between the prefix units are as follows: 1000 mm = 1 cm , 1000 cm = 1 dm , and 1000 dm = 1 m . The metric system also includes the litre (L) as a unit of volume, where 1 L = 1 dm = 1000 cm = 0.001 m . For

1617-533: The exact formulas for calculating the volume of parallelepipeds , cones, pyramids , cylinders, and spheres . The formula were determined by prior mathematicians by using a primitive form of integration , by breaking the shapes into smaller and simpler pieces. A century later, Archimedes ( c.  287 – 212 BCE ) devised approximate volume formula of several shapes using the method of exhaustion approach, meaning to derive solutions from previous known formulas from similar shapes. Primitive integration of shapes

1666-472: The fact that there are no chemical changes to its constituents, the physical properties of a mixture, such as its melting point , may differ from those of the components. Some mixtures can be separated into their components by using physical (mechanical or thermal) means. Azeotropes are one kind of mixture that usually poses considerable difficulties regarding the separation processes required to obtain their constituents (physical or chemical processes or, even

1715-558: The golden crown to find its volume, and thus its density and purity, due to the extreme precision involved. Instead, he likely have devised a primitive form of a hydrostatic balance . Here, the crown and a chunk of pure gold with a similar weight are put on both ends of a weighing scale submerged underwater, which will tip accordingly due to the Archimedes' principle . In the Middle Ages , many units for measuring volume were made, such as

1764-400: The heterogeneity of the i {\displaystyle i} th particle of the population, the mass concentration of the property of interest in the i {\displaystyle i} th particle of the population, the mass concentration of the property of interest in the population, the mass of the i {\displaystyle i} th particle in the population, and

1813-415: The litre unit, the commonly used prefixes are the millilitre (mL), centilitre (cL), and the litre (L), with 1000 mL = 1 L, 10 mL = 1 cL, 10 cL = 1 dL, and 10 dL = 1 L. Various other imperial or U.S. customary units of volume are also in use, including: Capacity is the maximum amount of material that a container can hold, measured in volume or weight . However,

1862-580: The metre, cubic metre, and litre from physical objects. This also make the metre and metre-derived units of volume resilient to changes to the International Prototype Metre. The definition of the metre was redefined again in 1983 to use the speed of light and second (which is derived from the caesium standard ) and reworded for clarity in 2019 . As a measure of the Euclidean three-dimensional space , volume cannot be physically measured as

1911-598: The microscopic scale. Calibrated measuring cups and spoons are adequate for cooking and daily life applications, however, they are not precise enough for laboratories . There, volume of liquids is measured using graduated cylinders , pipettes and volumetric flasks . The largest of such calibrated containers are petroleum storage tanks , some can hold up to 1,000,000  bbl (160,000,000 L) of fluids. Even at this scale, by knowing petroleum's density and temperature, very precise volume measurement in these tanks can still be made. For even larger volumes such as in

1960-505: The modern integral calculus, which remains in use in the 21st century. On 7 April 1795, the metric system was formally defined in French law using six units. Three of these are related to volume: the stère  (1 m ) for volume of firewood; the litre  (1 dm ) for volumes of liquid; and the gramme , for mass—defined as the mass of one cubic centimetre of water at the temperature of melting ice. Thirty years later in 1824,

2009-422: The other constituents. A similar distinction is possible for emulsions. In many emulsions, one constituent is present in the form of isolated regions of typically a globular shape, dispersed throughout the other constituent. However, it is also possible each constituent forms a large, connected network. Such a mixture is then called bicontinuous . Making a distinction between homogeneous and heterogeneous mixtures

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2058-471: The particles are evenly distributed. However, a heterogeneous mixture has non-uniform composition , and its constituent substances are easily distinguishable from one another (often, but not always, in different phases). Several solid substances, such as salt and sugar , dissolve in water to form a special type of homogeneous mixture called a solution , in which there is both a solute (dissolved substance) and solvent (dissolving medium) present. Air

2107-409: The plane curve boundaries. The shell integration method is used when integrating by an axis perpendicular to the axis of rotation. The equation can be written as: V = 2 π ∫ a b x | f ( x ) − g ( x ) | d x {\displaystyle V=2\pi \int _{a}^{b}x|f(x)-g(x)|\,dx} The volume of

2156-425: The population (before the sample was taken), q   i is the probability of including the i th particle of the population in the sample (i.e. the first-order inclusion probability of the i th particle), m   i is the mass of the i th particle of the population and a   i is the mass concentration of the property of interest in the i th particle of the population. The above equation for

2205-408: The sand is in the water it still keeps the same properties that it had when it was outside the water. The following table shows the main properties and examples for all possible phase combinations of the three "families" of mixtures : Mixtures can be either homogeneous or heterogeneous : a mixture of uniform composition and in which all components are in the same phase, such as salt in water,

2254-445: The variance of the sampling error becomes: where a batch is that concentration of the property of interest in the population from which the sample is to be drawn and M batch is the mass of the population from which the sample is to be drawn. Air pollution research show biological and health effects after exposure to mixtures are more potent than effects from exposures of individual components. Volume Volume

2303-416: The variance of the sampling error is an approximation based on a linearization of the mass concentration in a sample. In the theory of Gy, correct sampling is defined as a sampling scenario in which all particles have the same probability of being included in the sample. This implies that q   i no longer depends on  i , and can therefore be replaced by the symbol  q . Gy's equation for

2352-399: The volume of any object. He devised Cavalieri's principle , which said that using thinner and thinner slices of the shape would make the resulting volume more and more accurate. This idea would then be later expanded by Pierre de Fermat , John Wallis , Isaac Barrow , James Gregory , Isaac Newton , Gottfried Wilhelm Leibniz and Maria Gaetana Agnesi in the 17th and 18th centuries to form

2401-495: Was also discovered independently by Liu Hui in the 3rd century CE, Zu Chongzhi in the 5th century CE, the Middle East and India . Archimedes also devised a way to calculate the volume of an irregular object, by submerging it underwater and measure the difference between the initial and final water volume. The water volume difference is the volume of the object. Though highly popularized, Archimedes probably does not submerge

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