In a P-V-T diagram, where V (P,T) show that the volume is a state function.

In a P-V-T diagram, where \(P\) is pressure, \(V\) is volume, and \(T\) is temperature, the volume \(V\) can be demonstrated to be a state function by showing that it depends only on the state of the system (i.e., the specific values of \(P\), \(V\), and \(T\)) and not on how the system reached that state. **Definition of a State Function:** A state function is a property whose value depends only on the state of the system and not on the path taken to reach that state. Examples of state functions include internal energy, enthalpy, entropy, pressure, volume, and temperature. ### How to Show Volume \(V\) is a State Function: 1. **Dependence on State Variables:** In thermodynamics, any change in the system’s state can be fully described by a set of state variables. In the case of ideal gases (for simplicity), the state of the gas is completely defined by \(P\), \(V\), and \(T\) through the ideal gas law: \[ PV = nRT \] where \(n\) is the number of moles, and \(R\) is the ideal gas constant. 2. **Isothermal Process:** Consider an isothermal expansion or compression (constant temperature \(T\)). The law can be expressed as: \[ V = \frac{nRT}{P} \] In this case, for a fixed amount of gas (constant \(n\)), volume \(V\) is solely determined by the state variables \(P\) and \(T\). Regardless of how the process occurs (the specific pathway taken on a graph), the volume as a function of \(P\) and \(T\) is definite and unique. 3. **Path Independence:** To further illustrate that \(V\) is a state function, consider a cyclic process. The system returns to its original state. The total volume change over a complete cycle will be zero, regardless of the pathway taken, confirming that volume is defined by the state, not the path. 4. **Comparison of Different States:** For any two states (A and B) in the P-V-T diagram: - If you move from (A) to (B) along any path, whether it is at constant \(P\), constant \(V\), or any arbitrary path in the P-V-T space, the volume at state A and state B will be defined by their respective \(P\) and \(T\). - Thus, when you move between those two states, you will find that the volume at each state is solely a function of \(P\) and \(T\) at that state and is independent of any previous history or paths taken. ### Conclusion: The volume \(V\) can be proven as a state function because its value is determined entirely by the state of the system defined by the variables \(P\), \(T\), and \(n\) (in the case of ideal gases). Even if the system undergoes various transformations or pathways in the P-V-T space, the volume \(V\) at any given state remains unchanged and defined solely by the properties of that state, thereby verifying its status as a state function.