# Internal Energy as a function of U(S,V,A,Ni)

• mcoth420
In summary, the conversation discusses the function describing internal energy and how the total differential is derived for a one component system. The variables T, P, γ, and μ are defined and their relation to the partial derivatives of internal energy is explained. This technique is known as the Gibbs Formulation and can be found in various physical chemistry texts.
mcoth420
A general thermo question...
for the function describing internal energy U(S,V,A,N)

U=TS-PV+γA+μN

please explain how the total differential is

dU=TdS-PdV+γdA+μdN (for a one component system)

Basically why is dT=dP=dγ=dμ=0? Is it because they are intensive or potentials?

Thank you,

M

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Does this help?

$$\begin{array}{l} U = U(S,V,.N,A...) \\ dU = {\left( {\frac{{\partial U}}{{\partial S}}} \right)_{V,N,A...}}dS + {\left( {\frac{{\partial U}}{{\partial V}}} \right)_{S,N,A}}dV... \\ T = {\left( {\frac{{\partial U}}{{\partial S}}} \right)_{V,N,A...}} \\ P = {\left( {\frac{{\partial U}}{{\partial V}}} \right)_{S,N,A}} \\ {\rm{etc}} \\ \end{array}$$

I will leave you to fill in the bits for moles and area or other quantities.

Thank you for your reply...what is this technique called? I am working with Legendre transforms and this is similar...

Another thing...how can you derive T=partialU/partialS or others without knowledge of the internal energy equation?

This is the Gibbs Formulation.

Carington : Basic Thermodynamics P 187ff : Oxford University Press

Also in many Physical Chemistry texts.

aggie

I am happy to provide an explanation for the total differential of internal energy U(S,V,A,N). First, let's break down the function into its individual components: U stands for internal energy, T is temperature, S is entropy, P is pressure, V is volume, γ is surface tension, A is area, μ is chemical potential, and N is the number of particles.

The total differential of a function is a mathematical concept used to describe how the function changes when one or more of its variables change. In this case, the total differential of U(S,V,A,N) is represented as dU, and it is equal to the sum of the partial differentials of each variable multiplied by its corresponding differential. In other words, dU is the sum of all the infinitesimal changes in U due to infinitesimal changes in each of its variables.

Now, let's take a closer look at the total differential of internal energy, dU. We see that it is composed of four terms: TdS, -PdV, γdA, and μdN. Each of these terms represents the change in internal energy due to a change in one of its variables while holding the others constant.

For a one-component system, the temperature, pressure, surface tension, and chemical potential are all intensive properties, meaning they do not depend on the size or amount of the system. This is why their differentials, dT, dP, dγ, and dμ, are all equal to zero. In other words, a small change in these properties does not affect the internal energy of the system. On the other hand, the entropy, volume, area, and number of particles are extensive properties, meaning they depend on the size or amount of the system. Thus, their differentials, dS, dV, dA, and dN, are not equal to zero and must be considered in the total differential of internal energy.

In summary, the reason why dT, dP, dγ, and dμ are equal to zero in the total differential of internal energy is because they are intensive properties, while dS, dV, dA, and dN are extensive properties and must be included in the total differential. I hope this explanation helps to clarify the concept of total differentials in thermodynamics.

## 1. What is Internal Energy?

Internal Energy is the total energy of a system, including the kinetic energy of its particles and the potential energy of its interactions. It is a state function, meaning its value depends only on the current state of the system and not on how it reached that state.

## 2. How is Internal Energy related to Entropy?

Internal Energy and Entropy are both thermodynamic state functions, meaning they depend only on the current state of the system. However, they are not directly related to each other. Entropy measures the disorder or randomness of a system, while Internal Energy measures the total energy of the system.

## 3. What factors affect the Internal Energy of a system?

The Internal Energy of a system is affected by its temperature, pressure, volume, and the number of particles present. Changes in these factors can result in a change in the Internal Energy of the system.

## 4. How is Internal Energy calculated?

In a closed system, Internal Energy can be calculated using the equation U = Q - W, where U is the Internal Energy, Q is the heat added to the system, and W is the work done by the system. In an open system, the equation is modified to include the change in the number of particles present.

## 5. What is the significance of Internal Energy in thermodynamics?

Internal Energy is a fundamental concept in thermodynamics and is used to understand and predict the behavior of systems. It is a crucial factor in determining the stability and equilibrium of a system, as well as its ability to do work and transfer heat.

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