Proof of Independence of U from V & P with Respect to T

In summary, the conversation discusses how to show that the internal energy of a material is independent of volume and pressure, using the equation of state and the fundamental relation of thermodynamics. The use of partial derivatives and Maxwell's relation is suggested, but the exact method is not clear. The possibility of using the chain rule is also mentioned.
  • #1
mahdert
15
0

Homework Statement


Show that the internal energy of a material whose equation of state has the form p = f(V), T is independent of the volume and the pressure. That is

[tex]\left(\frac{\partial U}{\partial V}\right)_{T} = 0 [/tex]

[tex]\left(\frac{\partial U}{\partial p}\right)_{T} = 0 [/tex]

Homework Equations


TdS = dU + pdV


The Attempt at a Solution


I know the answer intiutively, i just don't know how one would go about showing it.
I assume that U = f(p,v) and then take the partial derivatives, but I do not see where T comes into play
 
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  • #2


Hmmm, Divide through by dv to get T ds/dv= dU/dv + p I am guessing you need to use a maxwell relation to get ds/dv to dp/dT I don't see how you get the partials to equal 0 though.
 
  • #3


I am trying to use the fundamental relation: Tds = dU + pdV and equate it with the partial expansion of U = U(V,p), however, I keep getting stuck because I do not know how to get the relationship of the change in U with either a change in V or p for a given temperature.
 
  • #4


not sure this helps but If U=U(V,p) you can use the chain rule to get to dU= (partial U/ partial V) dU + (partial U/partial p) dp
 
  • #5
Simple Thermodynamics Problem

Is this a single variable problem?
 

FAQ: Proof of Independence of U from V & P with Respect to T

What is the meaning of "Proof of Independence of U from V & P with Respect to T"?

The phrase "Proof of Independence of U from V & P with Respect to T" refers to a scientific concept in thermodynamics where the internal energy (U) of a system is shown to be independent of both volume (V) and pressure (P) with respect to temperature (T). This proof demonstrates that the internal energy of a system is solely dependent on the temperature and not affected by changes in volume or pressure.

Why is it important to prove the independence of U from V & P with Respect to T?

Proving the independence of U from V & P with Respect to T is important because it confirms the validity of the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. This proof also allows scientists to accurately calculate the internal energy of a system and make predictions about its behavior.

What evidence is used to support the independence of U from V & P with Respect to T?

The evidence used to support the independence of U from V & P with Respect to T comes from experimental data and mathematical equations. Scientists use measurements of temperature, volume, and pressure to calculate the internal energy of a system and then compare it to the theoretical values predicted by thermodynamic equations. If the calculated and theoretical values match, it provides evidence for the independence of U from V & P with Respect to T.

Can the independence of U from V & P with Respect to T be applied to all systems?

The independence of U from V & P with Respect to T can be applied to most systems, but there are certain cases where it may not hold true. For example, in systems with very high pressures or temperatures, there may be slight deviations from this principle. Additionally, this proof is based on the ideal gas law and may not accurately apply to real gases or non-ideal systems.

How does the independence of U from V & P with Respect to T relate to other thermodynamic properties?

The independence of U from V & P with Respect to T is closely related to other thermodynamic properties, such as enthalpy (H) and entropy (S). Enthalpy is also shown to be independent of volume and pressure with respect to temperature, while entropy is independent of volume and temperature with respect to pressure. These relationships are important in understanding the behavior of thermodynamic systems and can be used to make predictions and calculations.

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