Phase Rule & Duhem Teorem
Phase Rule
The number of variables that are set in independent form in a system in equilibrium is the difference between the total number of variables of the intensive system's state and the number of independent equations related to the variables.
The intensive state of a PVT system wih N chemical speies and P phases in equilibrium is characterized by the intensive variables, temperature T, pressure P and N-1 mol fraction for any phase, The number of varables given by phase rule is 2+(N-1)P. The number of equations of the independent phase equilibrium is (P-1)(N).
The difference between the number of variables of phase rule and the independent equations that relate them is the number of variables that can be fixed independently. They are known as degrees of freedom (F).
F = 2 + (N - 1) (P) - (P - 1) (N).
Simplyfing.
F = 2 - P + N.
Duhem Theorem.
It is simlar to the phase rule. It is applied to closed systems in equilibrium, in which intensive and extensive states are kept as constant. The system state is completely determined and characterized not only by the 2 + (N - 1) P intensive variables, obtained in the phase rule, if not also by the P extensive variables represented by the masses (or number of moles) of the phases. The total number of variables is.
2 + N P - P N = 2
The number of variables that are set in independent form in a system in equilibrium is the difference between the total number of variables of the intensive system's state and the number of independent equations related to the variables.
The intensive state of a PVT system wih N chemical speies and P phases in equilibrium is characterized by the intensive variables, temperature T, pressure P and N-1 mol fraction for any phase, The number of varables given by phase rule is 2+(N-1)P. The number of equations of the independent phase equilibrium is (P-1)(N).
The difference between the number of variables of phase rule and the independent equations that relate them is the number of variables that can be fixed independently. They are known as degrees of freedom (F).
F = 2 + (N - 1) (P) - (P - 1) (N).
Simplyfing.
F = 2 - P + N.
Duhem Theorem.
It is simlar to the phase rule. It is applied to closed systems in equilibrium, in which intensive and extensive states are kept as constant. The system state is completely determined and characterized not only by the 2 + (N - 1) P intensive variables, obtained in the phase rule, if not also by the P extensive variables represented by the masses (or number of moles) of the phases. The total number of variables is.
2 + N P - P N = 2
For any closed system formed by the knowm masses of the chemical species prescribed, the equlibrium state is determined fully when two independent variables are set.
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These two intependent variables are intensive or extensive. However, the number of independet intensive variables are known by the phase rule. So, when F=1, one of the variables must be extensive, and when F=0, both must be it.
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Gibbs Dühem Relation.
It shows that three intensive variables are not independent - if we know two of them, the value of the third can be determined from the Gibbs-Duhem equation.
If we know how the chemical potential of one species chages, then we can solve to obtain the chemical potential of the second species,
It shows that three intensive variables are not independent - if we know two of them, the value of the third can be determined from the Gibbs-Duhem equation.
If we know how the chemical potential of one species chages, then we can solve to obtain the chemical potential of the second species,
References.
-Smit, J., Van Ness, H. and Abboth, M. (2007). Introducción a la termodinámica en Ingeniería Química. Mexico City: McGraw Hill.
-LearnChemE (2011). Gibbs phase rule fot material science. Retrieved from: http://youtu.be/L0-3WEokrWA
-Addison, R. (2003). The Euler Equation and the Gibbs-Duhem equation. Retrieved from: https://www.tug.org/texshowcase/EulerGibbsDuhem.html
-TMP Chem (2014). Gibbs-Duhem Equation. Retrieved from: http://youtu.be/_fIy8cf5Edk
-Smit, J., Van Ness, H. and Abboth, M. (2007). Introducción a la termodinámica en Ingeniería Química. Mexico City: McGraw Hill.
-LearnChemE (2011). Gibbs phase rule fot material science. Retrieved from: http://youtu.be/L0-3WEokrWA
-Addison, R. (2003). The Euler Equation and the Gibbs-Duhem equation. Retrieved from: https://www.tug.org/texshowcase/EulerGibbsDuhem.html
-TMP Chem (2014). Gibbs-Duhem Equation. Retrieved from: http://youtu.be/_fIy8cf5Edk