Fundamental Equation Thermodynamic Postulates

In summary, the fundamental equation in thermodynamics is a mathematical representation of the relationship between thermodynamic properties such as temperature, pressure, and entropy. The three postulates of thermodynamics are the first law, which states that energy cannot be created or destroyed but can be converted from one form to another; the second law, which states that the total entropy of an isolated system will always increase over time; and the third law, which states that the entropy of a perfect crystal at absolute zero temperature is zero. The fundamental equation is derived from these postulates and provides a framework for understanding and analyzing the behavior of thermodynamic systems. These equations are applicable to all thermal systems but may not be applicable to extreme conditions or non-thermal systems.
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Homework Statement



Consider the fundamental equation S = AUnVmNr where A is a positive constant. Evaluate the permissible values of the three constants n, m, and r if the fundamental equation is to satisfy the thermodynamic postulates and if, in addition, we wish to have P increase with U/V, at constant N. (This latter condition is an intuitive substitute for stability requirements to be studied in Chapter 8.) For definiteness, the zero of energy is to be taken as the energy of the zero-temperature state.

Homework Equations



The thermodynamic postulates in question are -

1 there exist equilibrium states of simple systems macroscopically characterized completely by internal energy, volume, and the mole numbers of the chemical components

2 there exists some S, the entropy function, of the extensive parameters of any composite system, defined for all states of equilibrium, which has the property that the values assumed by the extensive parameters in the absence of an internal constraint are those that maximize, either positively or negatively, the entropy over the manifold of constrained equilibrium states

3 the entropy of a composite system is additive over the constituent subsystems, which implies that S =ΣS(α), where S(α) is the entropy of a subsystem, and n is the number of subsystems; and it is a continuous, differentiable, monotonically-increasing function of the energy, where the energy, U, is a single-valued, continuous, differentiable function of S, V, N1, …, Nr

4 the entropy of any system vanishes in the state for which temperature is zero, where the partial derivative of U with respect to S is zero.

The Attempt at a Solution



I know that I need to develop mathematical constraints on n, m, and r.
 
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First, I will start by using the first and fourth postulates to determine the values of n and r. Since the entropy, S, is a function of U, V, and N, we can rewrite the fundamental equation as S = f(U, V, N) where f is some function. From the first postulate, we know that S must be a function of U, V, and N, so we can write S = S(U, V, N). From the fourth postulate, we know that S = 0 when T = 0, which means that the partial derivative of U with respect to S is zero. Since U is a function of S, V, and N, we can write this as ∂U/∂S = 0. This implies that n = r.

Next, I will use the second postulate to determine the value of m. The second postulate states that the entropy function, S, has the property of maximizing entropy over the manifold of constrained equilibrium states. In order for P to increase with U/V, at constant N, we can rewrite the fundamental equation as P = g(U/V, N) where g is some function. From the second postulate, we know that P must be a function of U/V and N, so we can write P = P(U/V, N). This means that the entropy function, S, must also be a function of U/V and N in order for the fundamental equation to satisfy the second postulate. Therefore, m = 0.

In summary, the permissible values for n, m, and r are n = r and m = 0. This means that the fundamental equation becomes S = AU^2N^2, which satisfies the thermodynamic postulates and also ensures that P increases with U/V at constant N.
 

1. What is the fundamental equation in thermodynamics?

The fundamental equation in thermodynamics is a mathematical representation of the relationship between thermodynamic properties such as temperature, pressure, and entropy. It is the cornerstone of thermodynamics and serves as the basis for analyzing and predicting the behavior of systems in equilibrium.

2. What are the three postulates of thermodynamics?

The three postulates of thermodynamics are the first law, which states that energy cannot be created or destroyed but can be converted from one form to another; the second law, which states that the total entropy of an isolated system will always increase over time; and the third law, which states that the entropy of a perfect crystal at absolute zero temperature is zero.

3. How do the fundamental equation and postulates relate to each other?

The fundamental equation of thermodynamics is derived from the three postulates. The first law is used to define internal energy, the second law is used to define entropy, and the third law is used to set a reference point for the entropy scale. Together, these postulates form the basis for the fundamental equation.

4. What is the significance of the fundamental equation and postulates?

The fundamental equation and postulates provide a framework for understanding and analyzing the behavior of thermodynamic systems. They allow scientists and engineers to make predictions about the changes in thermodynamic properties of a system under different conditions, and they help to explain the fundamental principles behind energy conversion and the limitations of thermodynamic processes.

5. Can the fundamental equation and postulates be applied to all systems?

The fundamental equation and postulates are applicable to all thermal systems, whether they are gases, liquids, or solids. They can also be applied to non-thermal systems, such as electrical or mechanical systems, as long as the properties of these systems can be related to thermodynamic properties. However, these equations may not be applicable to systems that operate at extreme conditions, such as very high pressures or temperatures, or in the presence of strong gravitational fields.

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