Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat being transferred. As a result, adding up all the changes in heat divided by temperature (dq/T) for the whole cycle, the total comes out to zero. If the cycle is broken down into smaller and smaller steps, this summation sign turns into an integral. If the overall process is performed along two different paths, labeled as I and II, the integral also separates into two parts. Because the process can be run in reverse, the limits of the integral for path II are reversed. The integral of dq/T is equivalent to the change in entropy, which is a measure of disorder or randomness in a system. Because entropy, like internal energy, is a state function (meaning it depends only on the current state of the system, not the path taken to get there), the change in entropy between points A and B is the same no matter which path (I or II) is taken.
Consider an arbitrary, reversible cyclic process operating between two states, A and B, broken into small Carnot cycles.
Each cycle maintains a constant ratio of heat exchanged during the two reversible isothermal processes to their respective temperatures.
The remaining two processes are reversible and adiabatic, resulting in no heat exchange. As a result, the summation of dq/T terms for the complete cycle - composed of many steps - equals zero.
In infinitesimal steps, this summation sign becomes an integral.
Given that the overall process is performed along two distinct reversible paths, I and II, the integral separates into two parts. Simplifying the equation shows that the integrals over both paths evaluate to the same quantity.
Since the integral of dq/T defines the entropy change, the entropy difference between states A and B is identical along either path.
This means that entropy, like internal energy, is a state function.
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