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54-739: The Wildcat Hills are an escarpment between the North Platte River and Pumpkin Creek in the western Panhandle , in the state of Nebraska in the Great Plains region of the United States. Located in Banner , Morrill , and Scotts Bluff counties, the high tableland between the streams has been eroded by wind and water into a region of forested buttes , ridges and canyons that rise 150 to 300 m (490 to 980 ft) above

108-399: A l − L i n i t i a l ) L i n i t i a l {\displaystyle \varepsilon _{\mathrm {thermal} }={\frac {(L_{\mathrm {final} }-L_{\mathrm {initial} })}{L_{\mathrm {initial} }}}} where L i n i t i a l {\displaystyle L_{\mathrm {initial} }}

162-534: A l ) {\displaystyle \Delta T=(T_{\mathrm {final} }-T_{\mathrm {initial} })} is the difference of the temperature between the two recorded strains, measured in degrees Fahrenheit , degrees Rankine , degrees Celsius , or kelvin , and α L {\displaystyle \alpha _{L}} is the linear coefficient of thermal expansion in "per degree Fahrenheit", "per degree Rankine", "per degree Celsius", or "per kelvin", denoted by °F , °R , °C , or K , respectively. In

216-418: A broad range of temperatures. Another example is paraffin which in its solid form has a thermal expansion coefficient that is dependent on temperature. Since gases fill the entirety of the container which they occupy, the volumetric thermal expansion coefficient at constant pressure, α V {\displaystyle \alpha _{V}} , is the only one of interest. For an ideal gas ,

270-460: A fault displaces the ground surface so that one side is higher than the other, a fault scarp is created. This can occur in dip-slip faults , or when a strike-slip fault brings a piece of high ground adjacent to an area of lower ground. Earth is not the only planet where escarpments occur. They are believed to occur on other planets when the crust contracts , as a result of cooling. On other Solar System bodies such as Mercury , Mars , and

324-551: A first approximation, the change in length measurements of an object due to thermal expansion is related to temperature change by a coefficient of linear thermal expansion (CLTE). It is the fractional change in length per degree of temperature change. Assuming negligible effect of pressure, one may write: α L = 1 L d L d T {\displaystyle \alpha _{L}={\frac {1}{L}}\,{\frac {\mathrm {d} L}{\mathrm {d} T}}} where L {\displaystyle L}

378-404: A formula can be readily obtained by differentiation of the ideal gas law , p V m = R T {\displaystyle pV_{m}=RT} . This yields p d V m + V m d p = R d T {\displaystyle p\mathrm {d} V_{m}+V_{m}\mathrm {d} p=R\mathrm {d} T} where p {\displaystyle p}

432-467: A negative coefficient of thermal expansion for temperatures between about 18 and 120 kelvins (−255 and −153 °C; −427 and −244 °F). ALLVAR Alloy 30, a titanium alloy, exhibits anisotropic negative thermal expansion across a wide range of temperatures. Unlike gases or liquids, solid materials tend to keep their shape when undergoing thermal expansion. Thermal expansion generally decreases with increasing bond energy, which also has an effect on

486-486: A result, the total volumetric expansion is distributed unequally among the three axes. If the crystal symmetry is monoclinic or triclinic, even the angles between these axes are subject to thermal changes. In such cases it is necessary to treat the coefficient of thermal expansion as a tensor with up to six independent elements. A good way to determine the elements of the tensor is to study the expansion by x-ray powder diffraction . The thermal expansion coefficient tensor for

540-399: A significant length, like rods or cables, an estimate of the amount of thermal expansion can be described by the material strain , given by ε t h e r m a l {\displaystyle \varepsilon _{\mathrm {thermal} }} and defined as: ε t h e r m a l = ( L f i n

594-412: A solid has been reported for a Ti-Nb alloy. (The formula α V ≈ 3 α is usually used for solids.) When calculating thermal expansion it is necessary to consider whether the body is free to expand or is constrained. If the body is free to expand, the expansion or strain resulting from an increase in temperature can be simply calculated by using the applicable coefficient of thermal expansion. If

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648-451: A substance, which changes the volume of the substance while negligibly changing its mass (the negligible amount comes from mass–energy equivalence ), thus changing its density, which has an effect on any buoyant forces acting on it. This plays a crucial role in convection of unevenly heated fluid masses, notably making thermal expansion partly responsible for wind and ocean currents . The coefficient of thermal expansion describes how

702-423: Is a monotonic function of the average molecular kinetic energy of a substance. As energy in particles increases, they start moving faster and faster, weakening the intermolecular forces between them and therefore expanding the substance. When a substance is heated, molecules begin to vibrate and move more, usually creating more distance between themselves. The relative expansion (also called strain ) divided by

756-486: Is a particular length measurement and d L / d T {\displaystyle \mathrm {d} L/\mathrm {d} T} is the rate of change of that linear dimension per unit change in temperature. The change in the linear dimension can be estimated to be: Δ L L = α L Δ T {\displaystyle {\frac {\Delta L}{L}}=\alpha _{L}\Delta T} This estimation works well as long as

810-456: Is a strong function of temperature; doubling the temperature will halve the thermal expansion coefficient. From 1787 to 1802, it was determined by Jacques Charles (unpublished), John Dalton , and Joseph Louis Gay-Lussac that, at constant pressure, ideal gases expanded or contracted their volume linearly ( Charles's law ) by about 1/273 parts per degree Celsius of temperature's change up or down, between 0° and 100 °C. This suggested that

864-993: Is composed of three mutually orthogonal directions. Thus, in an isotropic material, for small differential changes, one-third of the volumetric expansion is in a single axis. As an example, take a cube of steel that has sides of length L . The original volume will be V = L 3 {\displaystyle V=L^{3}} and the new volume, after a temperature increase, will be V + Δ V = ( L + Δ L ) 3 = L 3 + 3 L 2 Δ L + 3 L Δ L 2 + Δ L 3 ≈ L 3 + 3 L 2 Δ L = V + 3 V Δ L L . {\displaystyle V+\Delta V=\left(L+\Delta L\right)^{3}=L^{3}+3L^{2}\Delta L+3L\Delta L^{2}+\Delta L^{3}\approx L^{3}+3L^{2}\Delta L=V+3V{\frac {\Delta L}{L}}.} We can easily ignore

918-1015: Is equal to the gas constant . For an isobaric thermal expansion, d p = 0 {\displaystyle \mathrm {d} p=0} , so that p d V m = R d T {\displaystyle p\mathrm {d} V_{m}=R\mathrm {d} T} and the isobaric thermal expansion coefficient is: α V ≡ 1 V ( ∂ V ∂ T ) p = 1 V m ( ∂ V m ∂ T ) p = 1 V m ( R p ) = R p V m = 1 T {\displaystyle \alpha _{V}\equiv {\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{p}={\frac {1}{V_{m}}}\left({\frac {\partial V_{m}}{\partial T}}\right)_{p}={\frac {1}{V_{m}}}\left({\frac {R}{p}}\right)={\frac {R}{pV_{m}}}={\frac {1}{T}}} which

972-836: Is significant, then the above equation will have to be integrated: ln ⁡ ( V + Δ V V ) = ∫ T i T f α V ( T ) d T {\displaystyle \ln \left({\frac {V+\Delta V}{V}}\right)=\int _{T_{i}}^{T_{f}}\alpha _{V}(T)\,\mathrm {d} T} Δ V V = exp ⁡ ( ∫ T i T f α V ( T ) d T ) − 1 {\displaystyle {\frac {\Delta V}{V}}=\exp \left(\int _{T_{i}}^{T_{f}}\alpha _{V}(T)\,\mathrm {d} T\right)-1} where α V ( T ) {\displaystyle \alpha _{V}(T)}

1026-432: Is some area of interest on the object, and d A / d T {\displaystyle dA/dT} is the rate of change of that area per unit change in temperature. The change in the area can be estimated as: Δ A A = α A Δ T {\displaystyle {\frac {\Delta A}{A}}=\alpha _{A}\Delta T} This equation works well as long as

1080-411: Is the change in temperature (50 °C). The above example assumes that the expansion coefficient did not change as the temperature changed and the increase in volume is small compared to the original volume. This is not always true, but for small changes in temperature, it is a good approximation. If the volumetric expansion coefficient does change appreciably with temperature, or the increase in volume

1134-467: Is the length before the change of temperature and L f i n a l {\displaystyle L_{\mathrm {final} }} is the length after the change of temperature. For most solids, thermal expansion is proportional to the change in temperature: ε t h e r m a l ∝ Δ T {\displaystyle \varepsilon _{\mathrm {thermal} }\propto \Delta T} Thus,

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1188-396: Is the pressure, V m {\displaystyle V_{m}} is the molar volume ( V m = V / n {\displaystyle V_{m}=V/n} , with n {\displaystyle n} the total number of moles of gas), T {\displaystyle T} is the absolute temperature and R {\displaystyle R}

1242-542: Is the rate of change of that volume with temperature. This means that the volume of a material changes by some fixed fractional amount. For example, a steel block with a volume of 1 cubic meter might expand to 1.002 cubic meters when the temperature is raised by 50 K. This is an expansion of 0.2%. If a block of steel has a volume of 2 cubic meters, then under the same conditions, it would expand to 2.004 cubic meters, again an expansion of 0.2%. The volumetric expansion coefficient would be 0.2% for 50 K, or 0.004% K . If

1296-530: Is the volumetric expansion coefficient as a function of temperature T , and T i {\displaystyle T_{i}} and T f {\displaystyle T_{f}} are the initial and final temperatures respectively. For isotropic materials the volumetric thermal expansion coefficient is three times the linear coefficient: α V = 3 α L {\displaystyle \alpha _{V}=3\alpha _{L}} This ratio arises because volume

1350-546: Is usually called negative thermal expansion , rather than "thermal contraction". For example, the coefficient of thermal expansion of water drops to zero as it is cooled to 3.983 °C (39.169 °F) and then becomes negative below this temperature; this means that water has a maximum density at this temperature, and this leads to bodies of water maintaining this temperature at their lower depths during extended periods of sub-zero weather. Other materials are also known to exhibit negative thermal expansion. Fairly pure silicon has

1404-577: The Moon , the Latin term rupes is used for an escarpment. When sedimentary beds are tilted and exposed to the surface, erosion and weathering may occur. Escarpments erode gradually and over geological time . The mélange tendencies of escarpments results in varying contacts between a multitude of rock types. These different rock types weather at different speeds, according to Goldich dissolution series so different stages of deformation can often be seen in

1458-419: The ideal gas law . This section summarizes the coefficients for some common materials. For isotropic materials the coefficients linear thermal expansion α and volumetric thermal expansion α V are related by α V = 3 α . For liquids usually the coefficient of volumetric expansion is listed and linear expansion is calculated here for comparison. For common materials like many metals and compounds,

1512-473: The melting point of solids, so high melting point materials are more likely to have lower thermal expansion. In general, liquids expand slightly more than solids. The thermal expansion of glasses is slightly higher compared to that of crystals. At the glass transition temperature, rearrangements that occur in an amorphous material lead to characteristic discontinuities of coefficient of thermal expansion and specific heat. These discontinuities allow detection of

1566-727: The Wildcat Hills State Recreation Area 41°42′08″N 103°40′02″W  /  41.70222°N 103.66722°W  / 41.70222; -103.66722 in stages between 1929 and 1980; the Wildcat Hills Nature Center, featuring a half-mile boardwalk trail, opened in 1995. Today, the Wildcat Hills are a popular hiking and wildlife viewing destination. Escarpment The terms scarp and scarp face are often used interchangeably with escarpment . Some sources differentiate

1620-576: The area and volumetric thermal expansion coefficient are, respectively, approximately twice and three times larger than the linear thermal expansion coefficient. In the general case of a gas, liquid, or solid, the volumetric coefficient of thermal expansion is given by α = α V = 1 V ( ∂ V ∂ T ) p {\displaystyle \alpha =\alpha _{\text{V}}={\frac {1}{V}}\,\left({\frac {\partial V}{\partial T}}\right)_{p}} The subscript " p " to

1674-406: The area expansion coefficient does not change much over the change in temperature Δ T {\displaystyle \Delta T} , and the fractional change in area is small Δ A / A ≪ 1 {\displaystyle \Delta A/A\ll 1} . If either of these conditions does not hold, the equation must be integrated. For a solid, one can ignore

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1728-757: The area of a face on the cube is just L 2 {\displaystyle L^{2}} . Also, the same considerations must be made when dealing with large values of Δ T {\displaystyle \Delta T} . Put more simply, if the length of a cubic solid expands from 1.00 m to 1.01 m, then the area of one of its sides expands from 1.00 m to 1.02 m and its volume expands from 1.00 m to 1.03 m . Materials with anisotropic structures, such as crystals (with less than cubic symmetry, for example martensitic phases) and many composites , will generally have different linear expansion coefficients α L {\displaystyle \alpha _{L}} in different directions. As

1782-472: The base of the plateau . Scarps are generally formed by one of two processes: either by differential erosion of sedimentary rocks , or by movement of the Earth's crust at a geologic fault . The first process is the more common type: the escarpment is a transition from one series of sedimentary rocks to another series of a different age and composition. Escarpments are also frequently formed by faults. When

1836-480: The body is constrained so that it cannot expand, then internal stress will be caused (or changed) by a change in temperature. This stress can be calculated by considering the strain that would occur if the body were free to expand and the stress required to reduce that strain to zero, through the stress/strain relationship characterised by the elastic or Young's modulus . In the special case of solid materials, external ambient pressure does not usually appreciably affect

1890-416: The change along a length, or over some area. The volumetric thermal expansion coefficient is the most basic thermal expansion coefficient, and the most relevant for fluids. In general, substances expand or contract when their temperature changes, with expansion or contraction occurring in all directions. Substances that expand at the same rate in every direction are called isotropic . For isotropic materials,

1944-402: The change in either the strain or temperature can be estimated by: ε t h e r m a l = α L Δ T {\displaystyle \varepsilon _{\mathrm {thermal} }=\alpha _{L}\Delta T} where Δ T = ( T f i n a l − T i n i t i

1998-412: The change in temperature is called the material's coefficient of linear thermal expansion and generally varies with temperature. If an equation of state is available, it can be used to predict the values of the thermal expansion at all the required temperatures and pressures , along with many other state functions . A number of materials contract on heating within certain temperature ranges; this

2052-413: The derivative indicates that the pressure is held constant during the expansion, and the subscript V stresses that it is the volumetric (not linear) expansion that enters this general definition. In the case of a gas, the fact that the pressure is held constant is important, because the volume of a gas will vary appreciably with pressure as well as temperature. For a gas of low density this can be seen from

2106-502: The effects of pressure on the material, and the volumetric (or cubical) thermal expansion coefficient can be written: α V = 1 V d V d T {\displaystyle \alpha _{V}={\frac {1}{V}}\,{\frac {\mathrm {d} V}{\mathrm {d} T}}} where V {\displaystyle V} is the volume of the material, and d V / d T {\displaystyle \mathrm {d} V/\mathrm {d} T}

2160-437: The expansion coefficient is known, the change in volume can be calculated Δ V V = α V Δ T {\displaystyle {\frac {\Delta V}{V}}=\alpha _{V}\Delta T} where Δ V / V {\displaystyle \Delta V/V} is the fractional change in volume (e.g., 0.002) and Δ T {\displaystyle \Delta T}

2214-583: The field of continuum mechanics , thermal expansion and its effects are treated as eigenstrain and eigenstress. The area thermal expansion coefficient relates the change in a material's area dimensions to a change in temperature. It is the fractional change in area per degree of temperature change. Ignoring pressure, one may write: α A = 1 A d A d T {\displaystyle \alpha _{A}={\frac {1}{A}}\,{\frac {\mathrm {d} A}{\mathrm {d} T}}} where A {\displaystyle A}

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2268-425: The glass transition temperature where a supercooled liquid transforms to a glass. Absorption or desorption of water (or other solvents) can change the size of many common materials; many organic materials change size much more due to this effect than due to thermal expansion. Common plastics exposed to water can, in the long term, expand by many percent. Thermal expansion changes the space between particles of

2322-636: The hills. Cougars (mountain lions), which had been extirpated from the region around 1900, returned to the area in the early 1990s. The Wildcat Hills (along with the Pine Ridge ), are the only areas in Nebraska with a permanent breeding cougar population. The Emigrant Trail passed through the northern Wildcat Hills at Robidoux Pass and after 1851, at Mitchell Pass ; the rock formations were frequently mentioned in emigrant journals and letters. The Nebraska Game and Parks Commission acquired land for

2376-540: The layers where the escarpments have been exposed to the elements. Thermal expansion#Contraction effects (negative thermal expansion) Thermal expansion is the tendency of matter to increase in length , area , or volume , changing its size and density , in response to an increase in temperature (usually excluding phase transitions ). Substances usually contract with decreasing temperature ( thermal contraction ), with rare exceptions within limited temperature ranges ( negative thermal expansion ). Temperature

2430-538: The linear-expansion coefficient does not change much over the change in temperature Δ T {\displaystyle \Delta T} , and the fractional change in length is small Δ L / L ≪ 1 {\displaystyle \Delta L/L\ll 1} . If either of these conditions does not hold, the exact differential equation (using d L / d T {\displaystyle \mathrm {d} L/\mathrm {d} T} ) must be integrated. For solid materials with

2484-405: The materials possessing cubic symmetry (for e.g. FCC, BCC) is isotropic. Thermal expansion coefficients of solids usually show little dependence on temperature (except at very low temperatures) whereas liquids can expand at different rates at different temperatures. There are some exceptions: for example, cubic boron nitride exhibits significant variation of its thermal expansion coefficient over

2538-538: The size of an object and so it is not usually necessary to consider the effect of pressure changes. Common engineering solids usually have coefficients of thermal expansion that do not vary significantly over the range of temperatures where they are designed to be used, so where extremely high accuracy is not required, practical calculations can be based on a constant, average, value of the coefficient of expansion. Linear expansion means change in one dimension (length) as opposed to change in volume (volumetric expansion). To

2592-484: The size of an object changes with a change in temperature. Specifically, it measures the fractional change in size per degree change in temperature at a constant pressure, such that lower coefficients describe lower propensity for change in size. Several types of coefficients have been developed: volumetric, area, and linear. The choice of coefficient depends on the particular application and which dimensions are considered important. For solids, one might only be concerned with

2646-534: The surrounding landscape. Chimney Rock , Scotts Bluff , and Courthouse and Jail Rocks are outcrops along the northern and western edges of the Wildcat Hills. The plant and animal life in the Wildcat Hills is atypical for Nebraska; the ecology more resembles that of the Laramie Mountains , 60 miles to the west. The dominant tree in the region is the ponderosa pine . Bighorn sheep , pronghorn , elk , mule deer , and wild turkeys live in and around

2700-580: The table below, the range for α is from 10 K for hard solids to 10 K for organic liquids. The coefficient α varies with the temperature and some materials have a very high variation; see for example the variation vs. temperature of the volumetric coefficient for a semicrystalline polypropylene (PP) at different pressure, and the variation of the linear coefficient vs. temperature for some steel grades (from bottom to top: ferritic stainless steel, martensitic stainless steel, carbon steel, duplex stainless steel, austenitic steel). The highest linear coefficient in

2754-793: The terms as Δ L is a small quantity which on squaring gets much smaller and on cubing gets smaller still. So Δ V V = 3 Δ L L = 3 α L Δ T . {\displaystyle {\frac {\Delta V}{V}}=3{\Delta L \over L}=3\alpha _{L}\Delta T.} The above approximation holds for small temperature and dimensional changes (that is, when Δ T {\displaystyle \Delta T} and Δ L {\displaystyle \Delta L} are small), but it does not hold if trying to go back and forth between volumetric and linear coefficients using larger values of Δ T {\displaystyle \Delta T} . In this case,

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2808-571: The thermal expansion coefficient is inversely proportional to the melting point . In particular, for metals the relation is: α ≈ 0.020 T m {\displaystyle \alpha \approx {\frac {0.020}{T_{m}}}} for halides and oxides α ≈ 0.038 T m − 7.0 ⋅ 10 − 6 K − 1 {\displaystyle \alpha \approx {\frac {0.038}{T_{m}}}-7.0\cdot 10^{-6}\,\mathrm {K} ^{-1}} In

2862-409: The third term (and sometimes even the fourth term) in the expression above must be taken into account. Similarly, the area thermal expansion coefficient is two times the linear coefficient: α A = 2 α L {\displaystyle \alpha _{A}=2\alpha _{L}} This ratio can be found in a way similar to that in the linear example above, noting that

2916-439: The two terms, with escarpment referring to the margin between two landforms , and scarp referring to a cliff or a steep slope. In this usage an escarpment is a ridge which has a gentle slope on one side and a steep scarp on the other side. More loosely, the term scarp also describes a zone between a coastal lowland and a continental plateau which shows a marked, abrupt change in elevation caused by coastal erosion at

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