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Prove that internal energy is a property of the system.

need answer for the above question which will helpfull for me in my preperation for thermodynamics

Let us assume that we have one system which is undergoing a change of state from initial state 1 to another state 2 via following the path A as shown in following figure. System is returning to initial state i.e. state 1 from state 2 via following the path B. Here, we can say that system is undergoing in a cycle 1-2-1 as displayed in figure.! [enter image description here] Let us recall the “first law of thermodynamics for a system undergoing a change of state” and apply for path A, where system is changing its state from state 1 to state 2. We will have following equation QA- WA= ∆EA Similarly, we will have following equation when system is changing its state from state 2 to state 1 via following the path B. QB- WB= ∆EB We have already seen that system is undergoing in a cycle 1-2-1 as displayed in above figure. Hence, we will use the concept of “first law of thermodynamics for a system undergoing a cycle”. Let us see the equation for system which constitutes a cycle 1-2-1 and we will have following equation. ![enter image description here]

WA+WB= QA+QB WB- QB = QA- WA - (QB- WB) = QA- WA - (∆EB) = ∆EA Let us assume that system is returning to initial state 1 from state 2 via following the path C, in that case we will go ahead similarly as we have gone above and finally we will have following equation - (∆EC) = ∆EA

Now if we will look the end result for first case where system is returning to initial state by following the path B and of second case where system is returning to initial state by following the path C, what we will secure here that change in system energy is same in both cases and it will not depend over the path followed by the system to return to its initial state.

Therefore we can conclude that system energy will have some definite magnitude for each state of the system and it will not depend over the path followed by the system and hence energy will be considered as a point function and also a property of the system.