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ehrenfest
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Homework Statement
I am confused about this solution. The question explicitly says that the systems are allowed energy exchange, so why is energy held fixed in the equation after "therefore"?
Yes, that's the combined law for an open system.ehrenfest said:The equation after "since" is always true, correct?
I am just confused about how they went from that to the next equation with the partial derivatives. It seems like they just divided each side by dN_A and then changed the total derivatives to partial derivatives and then added the subscripts V,E which I understand are redundant. I am just confused about how they changed the total derivatives to partial derivatives.
ehrenfest said:Why would the partial [tex]\left(\frac{\partial{E_A}}{\partial{N_A}}\right)_{V,E} [/tex] not be identically 0 because the subscript E means energy is not changing?
siddharth said:It's just a differential, right? For example, if S is a function of N,V,E, ie S=S(N,V,E) then
[tex] dS = \frac{\partial S}{\partial N} dN + \frac{\partial S}{\partial V} dV + \frac{\partial S}{\partial E} dE[/tex]
and at constant V,E
[tex] dS = \frac{\partial S}{\partial N} dN[/tex]
I don't see how that explains the equation after "therefore".
siddharth said:Why not? The notation is to tell you that V and E are held constant. I think it follows directly.
The combined law for an open system is a fundamental principle in thermodynamics that describes the relationship between the pressure, volume, and temperature of an open system. It states that the product of the pressure and volume is directly proportional to the absolute temperature of the system.
The combined law for an open system is a more general form of the ideal gas law, which only applies to closed systems. The combined law takes into account the exchange of matter and energy between the system and its surroundings, while the ideal gas law assumes a constant amount of gas in a closed system.
In thermodynamics, an open system is a system that can exchange energy and matter with its surroundings. This means that both energy and matter can enter or leave the system, allowing it to reach equilibrium with its surroundings.
The combined law for an open system is used in various engineering and scientific fields, such as chemical and mechanical engineering, to analyze and design systems that involve the exchange of matter and energy with the surroundings. It is also used to predict the behavior of gases in different conditions, such as in industrial processes and atmospheric studies.
One limitation of the combined law for an open system is that it assumes the system is in thermal equilibrium, meaning that the temperature is the same throughout the system. It also assumes that the system is in a steady state, meaning that there are no significant changes occurring over time. Additionally, the combined law may not accurately describe the behavior of non-ideal gases or systems with complex interactions between energy and matter.