Misplaced Pages

#742257

50-441: Kelvin and his brother James Thomson confirmed the relation experimentally in 1849–50, and it was historically important as a very early successful application of theoretical thermodynamics. Its relevance to meteorology and climatology is the increase of the water-holding capacity of the atmosphere by about 7% for every 1 °C (1.8 °F) rise in temperature. On a pressure – temperature ( P – T ) diagram, for any phase change

100-464: A Phase transition in a closed system composed of two contiguous phases, condensed matter and ideal gas, of a single substance, in mutual thermodynamic equilibrium, at constant temperature and pressure . Therefore, d s = ( ∂ s ∂ v ) T d v . {\displaystyle \mathrm {d} s=\left({\frac {\partial s}{\partial v}}\right)_{T}\,\mathrm {d} v.} Using

150-470: A 500-page collection after his death. The collection is freely available online (see below) and is prefaced by a lengthy (80-page) biography plus a more concise (10-page) biography. It is stated in this book that Thomson is the first to use the words radian , interface and apocentric in English, though he used a number of other neologisms that have not survived. State postulate The state postulate

200-669: A final phase β {\displaystyle \beta } , to obtain d P d T = Δ s Δ v , {\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {\Delta s}{\Delta v}},} where Δ s ≡ s β − s α {\displaystyle \Delta s\equiv s_{\beta }-s_{\alpha }} and Δ v ≡ v β − v α {\displaystyle \Delta v\equiv v_{\beta }-v_{\alpha }} are respectively

250-427: A liquid. d P d T = P L T 2 R , {\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {PL}{T^{2}R}},} v = V n = R T P . {\displaystyle v={\frac {V}{n}}={\frac {RT}{P}}.} The equation expresses this in a more convenient form just in terms of

300-437: A phase change), we obtain Δ s = Δ h T . {\displaystyle \Delta s={\frac {\Delta h}{T}}.} Substituting the definition of molar latent heat L = Δ h {\displaystyle L=\Delta h} gives Δ s = L T . {\displaystyle \Delta s={\frac {L}{T}}.} Substituting this result into

350-551: A professional engineer with special expertise in water transport. In his early thirties, in 1855, he was appointed professor of civil engineering at Queen's University Belfast . He remained there until 1873, when he accepted the Regius professorship of Civil Engineering and Mechanics at the University of Glasgow (in which post he was successor to the influential William Rankine ) until he resigned with failing eyesight in 1889. In 1875 he

400-469: A rough example of how much pressure this is, to melt ice at −7 °C (the temperature many ice skating rinks are set at) would require balancing a small car (mass ~ 1000 kg) on a thimble (area ~ 1 cm). This shows that ice skating cannot be simply explained by pressure-caused melting point depression, and in fact the mechanism is quite complex. While the Clausius–Clapeyron relation gives

450-539: A simple compressible system is completely specified by two independent, intensive properties A more general statement of the state postulate says: the state of a simple system is completely specified by r+1 independent, intensive properties where r is the number of significant work interactions. A system is considered to be a simple compressible one in the absence of certain effects which are uncommon in many engineering applications. These are electromagnetic and gravitational fields, surface tension, and motion. For such

500-545: A simplified form of the Clapeyron equation for the solid-liquid phase boundary . He proposed the term triple point to describe the conditions for which solid, liquid and vapour states are all in equilibrium. He also had contributions in the realm of fluid dynamics of rivers. It is claimed that the term torque was introduced into English scientific literature by Thomson, in 1884. James Thomson's main published research reports in physics and engineering were republished as

550-405: A system, only two independent intensive variables are sufficient to derive all the others by use of an equation of state . In the case of a more complex system, additional variables must be measured in order to solve for the complete state. For example, if gravitation is significant then an elevation may be required. Two properties are considered independent if one can be varied while the other

SECTION 10

#1732844921743

600-469: Is a term used in thermodynamics that defines the given number of properties to a thermodynamic system in a state of equilibrium . It is also sometimes referred to as the state principle. The state postulate allows a finite number of properties to be specified in order to fully describe a state of thermodynamic equilibrium. Once the state postulate is given the other unspecified properties must assume certain values. The state postulate says: The state of

650-432: Is also low, the gas may be approximated by the ideal gas law , so that v g = R T P , {\displaystyle v_{\text{g}}={\frac {RT}{P}},} where P {\displaystyle P} is the pressure, R {\displaystyle R} is the specific gas constant , and T {\displaystyle T} is the temperature. Substituting into

700-484: Is buried on the northern slopes of the Glasgow Necropolis overlooking Glasgow Cathedral . One obituary described Thomson as “a man of singular purity of mind and simplicity of character“, whose “gentle kindness and unfailing courtesy will be long remembered.” James Thomson is known for his work on the improvement of water wheels, water pumps and turbines. Also his innovations in the analysis of regelation , i.e.,

750-559: Is in hPa , and T {\displaystyle T} is in degrees Celsius (whereas everywhere else on this page, T {\displaystyle T} is an absolute temperature, e.g. in kelvins). This is also sometimes called the Magnus or Magnus–Tetens approximation, though this attribution is historically inaccurate. But see also the discussion of the accuracy of different approximating formulae for saturation vapour pressure of water . Under typical atmospheric conditions,

800-657: Is the internal energy of the system. Given constant pressure and temperature (during a phase change) and the definition of molar enthalpy h {\displaystyle h} , we obtain d h = T d s + v d P , {\displaystyle \mathrm {d} h=T\,\mathrm {d} s+v\,\mathrm {d} P,} d h = T d s , {\displaystyle \mathrm {d} h=T\,\mathrm {d} s,} d s = d h T . {\displaystyle \mathrm {d} s={\frac {\mathrm {d} h}{T}}.} Given constant pressure and temperature (during

850-589: Is the isothermal compressibility . James Thomson (engineer) James Thomson FRS FRSE LLD (16 February 1822 – 8 May 1892) was a British engineer and physicist, born in Belfast , and older brother of William Thomson (Lord Kelvin) . Born in Belfast, much of his youth was spent in Glasgow . His father James was professor of mathematics at the University of Glasgow from 1832 onward and his younger brother William

900-434: Is the molar entropy change of the phase transition. Alternatively, the specific values may be used instead of the molar ones. The Clausius–Clapeyron equation applies to vaporization of liquids where vapor follows ideal gas law using the ideal gas constant R {\displaystyle R} and liquid volume is neglected as being much smaller than vapor volume V . It is often used to calculate vapor pressure of

950-452: Is the specific latent heat of the substance. Instead of the specific, corresponding molar values (i.e. L {\displaystyle L} in kJ/mol and R = 8.31 J/(mol⋅K)) may also be used. Let ( P 1 , T 1 ) {\displaystyle (P_{1},T_{1})} and ( P 2 , T 2 ) {\displaystyle (P_{2},T_{2})} be any two points along

1000-792: Is the difference between the volumes of d x {\displaystyle \mathrm {d} x} in the liquid phase and vapor phases. The ratio W / Q {\displaystyle W/Q} is the efficiency of the Carnot engine, d T / T {\displaystyle \mathrm {d} T/T} . Substituting and rearranging gives d p d T = L T ( v ″ − v ′ ) , {\displaystyle {\frac {\mathrm {d} p}{\mathrm {d} T}}={\frac {L}{T(v''-v')}},} where lowercase v ″ − v ′ {\displaystyle v''-v'} denotes

1050-419: Is the molar latent heat of sublimation . If the latent heat is known, then knowledge of one point on the coexistence curve , for instance (1 bar, 373 K) for water, determines the rest of the curve. Conversely, the relationship between ln ⁡ P {\displaystyle \ln P} and 1 / T {\displaystyle 1/T} is linear, and so linear regression

SECTION 20

#1732844921743

1100-466: Is the slope of the tangent to the coexistence curve at any point, L {\displaystyle L} is the molar change in enthalpy ( latent heat , the amount of energy absorbed in the transformation), T {\displaystyle T} is the temperature , Δ v {\displaystyle \Delta v} is the molar volume change of the phase transition, and Δ s {\displaystyle \Delta s}

1150-968: Is the specific entropy , v {\displaystyle v} is the specific volume , and M {\displaystyle M} is the molar mass ) to obtain − ( s β − s α ) d T + ( v β − v α ) d P = 0. {\displaystyle -(s_{\beta }-s_{\alpha })\,\mathrm {d} T+(v_{\beta }-v_{\alpha })\,\mathrm {d} P=0.} Rearrangement gives d P d T = s β − s α v β − v α = Δ s Δ v , {\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {s_{\beta }-s_{\alpha }}{v_{\beta }-v_{\alpha }}}={\frac {\Delta s}{\Delta v}},} from which

1200-642: Is used to estimate the latent heat. Atmospheric water vapor drives many important meteorologic phenomena (notably, precipitation ), motivating interest in its dynamics . The Clausius–Clapeyron equation for water vapor under typical atmospheric conditions (near standard temperature and pressure ) is d e s d T = L v ( T ) e s R v T 2 , {\displaystyle {\frac {\mathrm {d} e_{s}}{\mathrm {d} T}}={\frac {L_{v}(T)e_{s}}{R_{v}T^{2}}},} where The temperature dependence of

1250-430: The Clapeyron equation ) equates the slope d P / d T {\displaystyle \mathrm {d} P/\mathrm {d} T} of the coexistence curve P ( T ) {\displaystyle P(T)} to the function L / ( T Δ v ) {\displaystyle L/(T\,\Delta v)} of the molar latent heat L {\displaystyle L} ,

1300-1842: The coexistence curve between two phases α {\displaystyle \alpha } and β {\displaystyle \beta } . In general, L {\displaystyle L} varies between any two such points, as a function of temperature. But if L {\displaystyle L} is approximated as constant, d P P ≅ L R d T T 2 , {\displaystyle {\frac {\mathrm {d} P}{P}}\cong {\frac {L}{R}}{\frac {\mathrm {d} T}{T^{2}}},} ∫ P 1 P 2 d P P ≅ L R ∫ T 1 T 2 d T T 2 , {\displaystyle \int _{P_{1}}^{P_{2}}{\frac {\mathrm {d} P}{P}}\cong {\frac {L}{R}}\int _{T_{1}}^{T_{2}}{\frac {\mathrm {d} T}{T^{2}}},} ln ⁡ P | P = P 1 P 2 ≅ − L R ⋅ 1 T | T = T 1 T 2 , {\displaystyle \ln P{\Big |}_{P=P_{1}}^{P_{2}}\cong -{\frac {L}{R}}\cdot \left.{\frac {1}{T}}\right|_{T=T_{1}}^{T_{2}},} or ln ⁡ P 2 P 1 ≅ − L R ( 1 T 2 − 1 T 1 ) . {\displaystyle \ln {\frac {P_{2}}{P_{1}}}\cong -{\frac {L}{R}}\left({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}}\right).} These last equations are useful because they relate equilibrium or saturation vapor pressure and temperature to

1350-556: The coexistence curve , d μ α = d μ β . {\displaystyle \mathrm {d} \mu _{\alpha }=\mathrm {d} \mu _{\beta }.} One may therefore use the Gibbs–Duhem relation d μ = M ( − s d T + v d P ) {\displaystyle \mathrm {d} \mu =M(-s\,\mathrm {d} T+v\,\mathrm {d} P)} (where s {\displaystyle s}

1400-510: The denominator of the exponent depends weakly on T {\displaystyle T} (for which the unit is degree Celsius). Therefore, the August–Roche–Magnus equation implies that saturation water vapor pressure changes approximately exponentially with temperature under typical atmospheric conditions, and hence the water-holding capacity of the atmosphere increases by about 7% for every 1 °C rise in temperature. One of

1450-464: The partial derivative of molar entropy may be changed into a total derivative d s = d P d T d v , {\displaystyle \mathrm {d} s={\frac {\mathrm {d} P}{\mathrm {d} T}}\,\mathrm {d} v,} and the total derivative of pressure with respect to temperature may be factored out when integrating from an initial phase α {\displaystyle \alpha } to

1500-657: The Clapeyron equation d P d T = L T Δ v , {\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {L}{T\,\Delta v}},} we can obtain the Clausius–Clapeyron equation d P d T = P L T 2 R {\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {PL}{T^{2}R}}} for low temperatures and pressures, where L {\displaystyle L}

1550-474: The appropriate Maxwell relation gives d s = ( ∂ P ∂ T ) v d v , {\displaystyle \mathrm {d} s=\left({\frac {\partial P}{\partial T}}\right)_{v}\,\mathrm {d} v,} where P {\displaystyle P} is the pressure. Since pressure and temperature are constant, the derivative of pressure with respect to temperature does not change. Therefore,

Clausius–Clapeyron relation - Misplaced Pages Continue

1600-623: The change in specific volume during the transition. For transitions between a gas and a condensed phase with the approximations described above, the expression may be rewritten as ln ⁡ ( P 1 P 0 ) = L R ( 1 T 0 − 1 T 1 ) {\displaystyle \ln \left({\frac {P_{1}}{P_{0}}}\right)={\frac {L}{R}}\left({\frac {1}{T_{0}}}-{\frac {1}{T_{1}}}\right)} where P 0 , P 1 {\displaystyle P_{0},P_{1}} are

1650-456: The change in molar entropy and molar volume. Given that a phase change is an internally reversible process , and that our system is closed, the first law of thermodynamics holds: d u = δ q + δ w = T d s − P d v , {\displaystyle \mathrm {d} u=\delta q+\delta w=T\,\mathrm {d} s-P\,\mathrm {d} v,} where u {\displaystyle u}

1700-440: The condensed phase v c {\displaystyle v_{\text{c}}} . Therefore, one may approximate Δ v = v g ( 1 − v c v g ) ≈ v g {\displaystyle \Delta v=v_{\text{g}}\left(1-{\frac {v_{\text{c}}}{v_{\text{g}}}}\right)\approx v_{\text{g}}} at low temperatures . If pressure

1750-417: The derivation of the Clapeyron equation continues as in the previous section . When the phase transition of a substance is between a gas phase and a condensed phase ( liquid or solid ), and occurs at temperatures much lower than the critical temperature of that substance, the specific volume of the gas phase v g {\displaystyle v_{\text{g}}} greatly exceeds that of

1800-591: The different phases, c p {\displaystyle c_{p}} is the specific heat capacity at constant pressure, α = ( 1 / v ) ( d v / d T ) P {\displaystyle \alpha =(1/v)(\mathrm {d} v/\mathrm {d} T)_{P}} is the thermal expansion coefficient , and κ T = − ( 1 / v ) ( d v / d P ) T {\displaystyle \kappa _{T}=-(1/v)(\mathrm {d} v/\mathrm {d} P)_{T}}

1850-404: The effect of pressure on the freezing point of water, and his studies in glaciology including glacier motion , where he extended the work of James David Forbes . He studied the experimental work of his colleague Thomas Andrews , concerning the continuity of the liquid and gaseous states of matter, and strengthened understanding of it by applying his strong knowledge of thermodynamics. He derived

1900-486: The heat absorbed is Q = L d x , {\displaystyle Q=L\,\mathrm {d} x,} and the corresponding work is W = d p d T d T ( V ″ − V ′ ) , {\displaystyle W={\frac {\mathrm {d} p}{\mathrm {d} T}}\,\mathrm {d} T(V''-V'),} where V ″ − V ′ {\displaystyle V''-V'}

1950-520: The latent heat L v ( T ) {\displaystyle L_{v}(T)} can be neglected in this application. The August – Roche – Magnus formula provides a solution under that approximation: e s ( T ) = 6.1094 exp ⁡ ( 17.625 T T + 243.04 ) , {\displaystyle e_{s}(T)=6.1094\exp \left({\frac {17.625T}{T+243.04}}\right),} where e s {\displaystyle e_{s}}

2000-708: The latent heat of the phase change without requiring specific-volume data. For instance, for water near its normal boiling point , with a molar enthalpy of vaporization of 40.7 kJ/mol and R = 8.31 J/(mol⋅K), P vap ( T ) ≅ 1   bar ⋅ exp ⁡ [ − 40 700   K 8.31 ( 1 T − 1 373   K ) ] . {\displaystyle P_{\text{vap}}(T)\cong 1~{\text{bar}}\cdot \exp \left[-{\frac {40\,700~{\text{K}}}{8.31}}\left({\frac {1}{T}}-{\frac {1}{373~{\text{K}}}}\right)\right].} In

2050-781: The latent heat, for moderate temperatures and pressures. Using the state postulate , take the molar entropy s {\displaystyle s} for a homogeneous substance to be a function of molar volume v {\displaystyle v} and temperature T {\displaystyle T} . d s = ( ∂ s ∂ v ) T d v + ( ∂ s ∂ T ) v d T . {\displaystyle \mathrm {d} s=\left({\frac {\partial s}{\partial v}}\right)_{T}\,\mathrm {d} v+\left({\frac {\partial s}{\partial T}}\right)_{v}\,\mathrm {d} T.} The Clausius–Clapeyron relation describes

Clausius–Clapeyron relation - Misplaced Pages Continue

2100-548: The line separating the two phases is known as the coexistence curve . The Clapeyron relation gives the slope of the tangents to this curve. Mathematically, d P d T = L T Δ v = Δ s Δ v , {\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {L}{T\,\Delta v}}={\frac {\Delta s}{\Delta v}},} where d P / d T {\displaystyle \mathrm {d} P/\mathrm {d} T}

2150-480: The original work by Clapeyron, the following argument is advanced. Clapeyron considered a Carnot process of saturated water vapor with horizontal isobars. As the pressure is a function of temperature alone, the isobars are also isotherms. If the process involves an infinitesimal amount of water, d x {\displaystyle \mathrm {d} x} , and an infinitesimal difference in temperature d T {\displaystyle \mathrm {d} T} ,

2200-448: The pressure derivative given above ( d P / d T = Δ s / Δ v {\displaystyle \mathrm {d} P/\mathrm {d} T=\Delta s/\Delta v} ), we obtain d P d T = L T Δ v . {\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {L}{T\,\Delta v}}.} This result (also known as

2250-422: The pressures at temperatures T 0 , T 1 {\displaystyle T_{0},T_{1}} respectively and R {\displaystyle R} is the ideal gas constant . For a liquid–gas transition, L {\displaystyle L} is the molar latent heat (or molar enthalpy ) of vaporization ; for a solid–gas transition, L {\displaystyle L}

2300-1361: The slope of the coexistence curve, it does not provide any information about its curvature or second derivative . The second derivative of the coexistence curve of phases 1 and 2 is given by d 2 P d T 2 = 1 v 2 − v 1 [ c p 2 − c p 1 T − 2 ( v 2 α 2 − v 1 α 1 ) d P d T ] + 1 v 2 − v 1 [ ( v 2 κ T 2 − v 1 κ T 1 ) ( d P d T ) 2 ] , {\displaystyle {\begin{aligned}{\frac {\mathrm {d} ^{2}P}{\mathrm {d} T^{2}}}&={\frac {1}{v_{2}-v_{1}}}\left[{\frac {c_{p2}-c_{p1}}{T}}-2(v_{2}\alpha _{2}-v_{1}\alpha _{1}){\frac {\mathrm {d} P}{\mathrm {d} T}}\right]\\{}&+{\frac {1}{v_{2}-v_{1}}}\left[(v_{2}\kappa _{T2}-v_{1}\kappa _{T1})\left({\frac {\mathrm {d} P}{\mathrm {d} T}}\right)^{2}\right],\end{aligned}}} where subscripts 1 and 2 denote

2350-642: The temperature T {\displaystyle T} , and the change in molar volume Δ v {\displaystyle \Delta v} . Instead of the molar values, corresponding specific values may also be used. Suppose two phases, α {\displaystyle \alpha } and β {\displaystyle \beta } , are in contact and at equilibrium with each other. Their chemical potentials are related by μ α = μ β . {\displaystyle \mu _{\alpha }=\mu _{\beta }.} Furthermore, along

2400-802: The uses of this equation is to determine if a phase transition will occur in a given situation. Consider the question of how much pressure is needed to melt ice at a temperature Δ T {\displaystyle {\Delta T}} below 0 °C. Note that water is unusual in that its change in volume upon melting is negative. We can assume Δ P = L T Δ v Δ T , {\displaystyle \Delta P={\frac {L}{T\,\Delta v}}\,\Delta T,} and substituting in we obtain Δ P Δ T = − 13.5   MPa / K . {\displaystyle {\frac {\Delta P}{\Delta T}}=-13.5~{\text{MPa}}/{\text{K}}.} To provide

2450-746: Was elected a Fellow of the Royal Society of Edinburgh . His proposers were his younger brother William Thomson , Peter Guthrie Tait , Alexander Crum Brown , and John Hutton Balfour . He was elected a Fellow of the Royal Society of London in June 1877. He served as President of the Institution of Engineers and Shipbuilders in Scotland from 1884 to 1886. In later life he lived at 2 Florentine Gardens, off Hillhead Street. He died of cholera in Glasgow on 8 May 1892. He

2500-464: Was to become Baron Kelvin . James attended Glasgow University from a young age and graduated (1839) with high honours in his late teens. After graduation, he served brief apprenticeships with practical engineers in several domains; and then gave a considerable amount of his time to theoretical and mathematical engineering studies, often in collaboration with his brother, during his twenties in Glasgow. In his late twenties he entered into private practice as

#742257