First+Law+of+Thermodynamics

The first law of thermodynamics is a version of the law of conservation of energy, specialized for thermodynamical systems. It is usually formulated by stating that the change in the internal energy of a closed system is equal to the amount of heat supplied to the system, minus the amount of work performed by the system on its surroundings. The law of conservation of energy can be stated: The energy of an isolated system is constant. In other words energy can neither be created nor destroyed.

Evidence for the first law of thermodynamics
The first law of thermodynamics is induced from empirically observed evidence. The original discovery of the law was gradual over a period of perhaps half a century or more, and was mostly in terms of cyclic processes.The following is an account in terms of changes of state through compound processes that are not necessarily cyclic, but are composed of segments of two special kinds, adiabatic and isothermal diabatic.

Adiabatic processes
It can be observed that, given a system in an initial state, if work is exerted on the system in an adiabatic (i.e. no heat transfer) way, the final state is the same for a given amount of work, irrespective of how this work is performed. For instance, in Joule's experiment, the initial system is a tank of water with a paddle wheel inside. If we isolate thermally the tank and move the paddle wheel with a pulley and a weight we can relate the increase in temperature with the height descended by the mass. Now the system is returned to its initial state, isolated again, and the same amount of work is done on the tank using different devices (an electric motor, a chemical battery, a spring,...). In every case, the amount of work can be measured independently. The evidence shows that the final state of the water (in particular, its temperature) is the same in every case. It's irrelevant if the work is electrical , mechanical, chemical,... or if done suddenly or slowly, as long as it is performed in an adiabatic way. This evidence leads to a statement of one aspect of the first law of thermodynamics //For all adiabatic processes between two specified states of a closed system, the net work done is the same regardless of the nature of the closed system and the details of the process.// This affirmation of path independence is one aspect of the meaning of the state function that is called internal energy ,. In an adiabatic process, adiabatic work takes the system from a reference state with internal energy to an arbitrary eventual one with internal energy : where, following IUPAC convention we take as positive the work done on the system. To go from a state A to a state B we can take a path that goes through the reference state, since the adiabatic work is independent of the path

Isothermal diabatic processes
A complementary observable aspect of the first law is about heat transfer. When the system does not evolve adiabatically, the work exerted on the system is not equal to the increase in its internal energy: The difference is due to the transfer of heat  into the system. Heat transfer can be measured by calorimetry. If the system is at constant temperature during the heat transfer, the transfer is called isothermal diabatic, and we may write.

Combination of adiabatic and isothermal diabatic processes
Putting the two complementary aspects together, adiabatic and isothermal diabatic, the inequality can be transformed into an equality as This combined statement is the expression the first law of thermodynamics for finite processes composed of distinct adiabatic and isothermal diabatic segments. In particular, if no work is exerted on a thermally isolated system we have . This is one aspect of the law of conservation of energy and can be stated: