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Introductory Physics Homework Help
Specific heat in the curve of equilibrium
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[QUOTE="Dario SLC, post: 5804208"] [h2]Homework Statement [/h2] Consider a system formed by two phases of a substance that consists of a single class of molecules. Determine the specific heat ##c## of a vapor pressure and temperature ##p## ##T## on the curve of liquid-vapor equilibrium. Consider the steam as an ideal gas. Data: ##c_p##, ##c_v##, ##l## heat of the phase transition of liquid-vapor. [h2]Homework Equations[/h2] $$c_X=T\frac{\partial S}{\partial T}_X$$ Clausius-Clapeyron equation: $$\frac{dp}{dT}=\frac{l_{l-v}}{T(v_l-v_v)}$$ when ##v_l## is the specific volume of the liquid and ##v_v## is the specific volume of the vapor [h2]The Attempt at a Solution[/h2] Well I think that like ##S=S(T,p(T))## then: $$c_CC=T\frac{\partial S}{\partial T}_p+T\frac{\partial S}{\partial p}_T\frac{dp}{dT}$$ for the rule of the chain, and ##c_{CC}## it is the specific heat on the curve Clausius-Clapeyron. For the Maxwell relation: $$\frac{\partial S}{\partial p}_T=-\frac{\partial V}{\partial T}_p$$ then gathering all $$c_{CC}=c_p-\cancel{T}\frac{\partial V}{\partial T}_p\frac{l_{l-v}}{\cancel{T}(v_l-v_v)}$$ when ##v_l\ll v_v## and using that it is a ideal gas for your equation of state: $$c_{CC}=c_p+\frac{R}{p}\frac{l_{l-v}}{v_v}=c_p+\frac{c_p-c_v}{p}\frac{l_{l-v}}{RT/p}$$ like the presure ##p## it is equal for all curve the expression for specific heat it will (I hope): $$c_{CC}=c_p+\frac{l_{l-v}}{T}$$ [/QUOTE]
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Introductory Physics Homework Help
Specific heat in the curve of equilibrium
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