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).