Clausius–clapeyron relation – wikipedia gas x tablets himalaya

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Using the state postulate, take the specific entropy s {\displaystyle s} for a homogeneous substance to be a function of specific volume v {\displaystyle v} and temperature T {\displaystyle T} . [3] : 508 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.}

where P {\displaystyle P} is the pressure. Since pressure and temperature are constant, by definition the derivative of pressure with respect to temperature does not change. [4] [5] : 57, 62 671 Therefore, the partial derivative gas house edwards of specific 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}}

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 the change in specific entropy and gas and water llc specific volume. Given that a phase change is an internally reversible process, and that our system is closed, the first law of thermodynamics holds

This result (also known as the Clapeyron equation) equates the slope of the tangent to the coexistence curve d P / d T {\displaystyle \mathrm {d} P/\mathrm {d} T} , at any given point on the curve, to the function L / T Δ v {\displaystyle {L}/{T{\Delta v}}} of the specific latent heat L {\displaystyle L} , the temperature T {\displaystyle T} , and the change in specific volume Δ v {\displaystyle \Delta v} .

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_{\mathrm {g} }} greatly exceeds that of the condensed phase v c {\displaystyle v_{\mathrm {c} }} . Therefore, one may approximate

Let ( P 1 , T 1 ) {\displaystyle (P_{1},T_{1})} and 76 gas station locations ( P 2 , T 2 ) {\displaystyle (P_{2},T_{2})} be any two points along 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 constant,

d P P = L R d T T 2 , {\displaystyle {\frac {\mathrm {d} P}{P}}={\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}}={\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 \left.\ln P\right|_{P=P_{1}}^{P_{2}}=-{\frac {L}{R}}\cdot \left.{\frac {1}{T}}\right|_{T=T_{1}}^{T_{2}}}

where c {\displaystyle c} is a constant. For a liquid-gas transition, L {\displaystyle L} is the specific latent heat (or specific enthalpy) of vaporization; for a solid-gas transition, L {\displaystyle L} is the specific latent heat of sublimation a gas has. If the latent heat is known, then knowledge of one point on the coexistence curve 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 is used to estimate the latent heat.

Under typical atmospheric conditions, the denominator of the exponent depends weakly on T {\displaystyle T} (for which the unit is Celsius). Therefore, the August-Roche-Magnus equation implies that saturation water vapor pressure changes approximately exponentially with temperature under typical atmospheric conditions, and gas constant for air hence the water-holding capacity of the atmosphere increases by about 7% for every 1 °C rise in temperature. [10] Example [ edit ]

While the Clausius–Clapeyron relation gives 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 [12] 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 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}} is the isothermal compressibility.

Using the state postulate, take the specific entropy s {\displaystyle s} for a homogeneous substance to be a function of specific volume v {\displaystyle v} and temperature T {\displaystyle T} . [3] : 508 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 gas lighting urban dictionary s}{\partial T}}\right)_{v}\mathrm {d} T.}

where P {\displaystyle P} is the pressure. Since pressure and temperature are constant, by definition the derivative of pressure with respect to temperature does not change. [4] [5] : 57, 62 671 Therefore, the partial derivative of specific 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}}

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 the change in specific entropy and specific volume. Given that electricity trading jobs a phase change is an internally reversible process, and that our system is closed, the first law of thermodynamics holds

This result (also known as the Clapeyron equation) equates the slope of the tangent to the coexistence curve d P / d T {\displaystyle \mathrm {d} P/\mathrm {d} T} , at any given point on the 9gag memes curve, to the function L / T Δ v {\displaystyle {L}/{T{\Delta v}}} of the specific latent heat L {\displaystyle L} , the temperature T {\displaystyle T} , and the change in specific volume Δ v {\displaystyle \Delta v} .

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_{\mathrm {g} }} greatly exceeds that of the condensed phase v c {\displaystyle v_{\mathrm {c} }} . Therefore, one may approximate

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 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 constant,

d P P = L R d T T 2 , {\displaystyle {\frac {\mathrm {d} P}{P}}={\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 gasset y ortega filosofia _{P_{1}}^{P_{2}}{\frac {\mathrm {d} P}{P}}={\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 \left.\ln P\right|_{P=P_{1}}^{P_{2}}=-{\frac {L}{R}}\cdot \left.{\frac {1}{T}}\right|_{T=T_{1}}^{T_{2}}}

where c {\displaystyle c} is a constant. For a liquid-gas transition, L {\displaystyle L} is the specific latent heat (or specific enthalpy) of vaporization; for a solid-gas transition, L {\displaystyle L} is the specific latent heat of sublimation. If the latent heat is known, then knowledge of one point on the coexistence curve 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 is used to estimate the latent heat.

Under typical atmospheric conditions, the denominator of the exponent depends weakly on T {\displaystyle T} (for which the unit is 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. [10] Example static electricity bill nye full episode [ edit ]

While the Clausius–Clapeyron relation gives 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 [12] 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 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}} is the electricity physics formulas isothermal compressibility.