Internal energy of an ideal gas -- confusion

[misko]

We know that internal energy of ideal gas depends only on temperature.
Let's say we have 1 mole of ideal gas with pressure P₁, volume V₁ and temperature T. Let's call this the state 1. Equation of state for ideal gas applies: PV=RT.

Now if we expand (or compress gas) isothermally, gas will then have different pressure P₂, volume V₂ while temperature will remain the same. Let's call this the state 2.

Since internal energy is dependent only on temperature it follows that it remained the same after this process, so U₍₁₎ = U₍₂₎.

My question is now this: is system in the same state or different state after this process? Is state(1) same as state(2)? If internal energy remained the same it means that state also remained the same, right? But how then two states that are same (equivalent, identical) have different state variables like pressure and volume?

This confuses me... Are state functions in thermodynamic a "One-to-one" functions mathematically? Meaning that for one state there is only one value for internal energy.
Or can thermodynamic systems be degenerate, in terms of multiple different states having the same internal energy? (eg. degenerate as in quantum mechanical systems).

 

[Eucliddo]

In the state 1 that you have mentioned, it is in the ideal gas state.In the formula PV is equal to nRT, we are using the universal gas constant. In the state 2, temperature is held constant. So the formula has been revised into P1V1 equals P2V2, this means that the gas changes in pressure and volume but not in temperature. And since the temperature doesn't change, the internal energy would remain as is.

 

[DrClaude]

This confuses me... Are state functions in thermodynamic a "One-to-one" functions mathematically? Meaning that for one state there is only one value for internal energy.
Or can thermodynamic systems be degenerate, in terms of multiple different states having the same internal energy? (eg. degenerate as in quantum mechanical systems).

As you have figured out, it is the latter that is true. For a given energy, there are actually an infinity of different states (PV products) that will correspond to it. Knowing the energy of an ideal gas is not sufficient information to completely describe its state.

 

[misko]

Ok thanks for the clarification.
For some reason I thought that value of the state function uniquely corresponds to one and only one state of the system.

Based on the definition from wikipedia: "state function is a property of a system that depends only on the current equilibrium state of the system" I incorrectly concluded that two different states of the system will always give two different values for some state function (such as internal energy).

So is there some other "state function" that has this one-to-one property? Like, is there some state function X that once we know it's value we can uniquely identify the state of the system for that value?

PS. Sorry for my naive questions, I am struggling to grasp these basic concepts before I go more deeply into thermodynamics.

 

[Chestermiller]

It takes specification of two intensive properties to determine the state of a single phase system of constant composition.