# Solving E Extensive Parameter Relation: cE = E(cS, cV, cN)

• ehrenfest
In summary, the conversation is discussing the relationship between extensive parameters and how they are affected by a constant c. It is stated that an extensive parameter is proportional to N and that replacing a system with one twice as big can either result in recalculating the values of S, V, and N or just doubling E. It is also mentioned that the proper definition of an extensive quantity is that it is additive and independent of the existence of a function. An example of this is given with the internal energy of systems consisting of mono-atomic ideal gases. Finally, it is concluded that the energy E is an extensive quantity independent of the suggested function.
ehrenfest
[SOLVED] extensive quantity

## Homework Statement

My teacher said that E=E(S,V,N) implies that c E = E(cS,cV,cN) where c is some constant. The justification was that E is an extensive parameter. I know what an extensive parameter is (proportional to N), but I do not see how that relation follows.

## The Attempt at a Solution

1. E, S, V, and N are extensive quantities.
2. You have a system with a certain S, V, and N, and you calculate the energy E.
3. I replace the system with one twice as big.
4. You can either double S, V, and N and recalculate or just double E, right? This statement is identical to your equation.

the proper definition of an extensive quantity is: if there is a system with extensive quanty X' and another system with X" then the two systems together has property X, with X=X'+X". (the property is additive). This is independent of the existence of a function X(a,b,c) where a b and c are also extensive quantities. For example take the internal energy of systems consisting of mono-atomic ideal gases with for helium U'=n'(3/2)RT' and for neon U"=n"(3/2)RT". (Hence U=U(n,T)). Now if we take the systems together U=U'+U" and n=n'+n", independent of the actual values of T'and T". (and we do not need the constant c)
U (=E) is and extensive quantity independent from the suggested function, that is actually non-existent for most systems.

## 1. What is the purpose of solving E Extensive Parameter Relation?

The purpose of solving E Extensive Parameter Relation is to understand the relationship between extensive parameters, such as energy (E), entropy (S), volume (V), and number of particles (N). This can help in predicting the behavior of a system and making calculations in thermodynamics and statistical mechanics.

## 2. What does the equation cE = E(cS, cV, cN) represent?

The equation cE = E(cS, cV, cN) represents the extensive parameter relation, where cE is the extensive parameter of energy and cS, cV, cN are the intensive parameters of entropy, volume, and number of particles, respectively. This equation shows how the extensive parameter of energy is dependent on the intensive parameters.

## 3. How do you solve for cE in the equation cE = E(cS, cV, cN)?

To solve for cE in the equation cE = E(cS, cV, cN), you need to manipulate the equation by algebraic methods to isolate cE on one side. This can involve using mathematical operations such as addition, subtraction, multiplication, and division. The resulting solution will give you the value of cE.

## 4. Are there any assumptions involved in solving E Extensive Parameter Relation?

Yes, there are certain assumptions involved in solving E Extensive Parameter Relation. One of the main assumptions is that the system is in thermal equilibrium, meaning the temperature is constant throughout the system. Additionally, the system is assumed to be closed, meaning there is no exchange of matter or energy with the surroundings.

## 5. How is E Extensive Parameter Relation used in practical applications?

E Extensive Parameter Relation has many practical applications, especially in thermodynamics and statistical mechanics. It is used to calculate and predict the behavior of various systems, such as gases, liquids, and solids. This can help in designing and optimizing various industrial processes, such as refrigeration, power generation, and chemical reactions.

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