A system could be transferred from an initial to a final state by means of a quasi – static process and how the work done during the process could be calculated. There are, however, other means of changing the state of a system that do not necessarily involve the performance of work. Consider the four processes shown in figure 4-1, which involve closed systems, where a closed system is a system in which no matter passes between the system and surroundings. In figure 1 (a), the system is a composite one consisting of water and a paddle wheel, which is caused to rotate and churn the water by means of a falling weight. As a result, the temperature of the water rises from room temperature to a slightly higher temperature. In figure 1 (b), the water and the resistor constitute the composite system, the electric current in the resistor being maintained by a generator turned by means of a falling weight. Again, the temperature of the water rises. In both cases, the state of the system is caused to change; and since the agency for changing the state of the system is a falling weight, both processes involve the performance of work.
In figure 1 (c) and 1 (d), however, the situation is quite different. The system in both these cases is water in a diathermic container. In figure 1 (c), the system is in contact with the burning gases at high temperature; whereas, in figure 1 (d), the system is near but not in contact with a lamp whose temperature is much higher than that of the water. In both cases, the system is caused to change, but in neither case can the agency of change be described by mechanical means.
Figure 1 Distinction between work and heat: (a)and (b) show work being done on the system by means falling body, whereas (c) and (d) show heat entering the system from a hotter substance.
When a closed system is completely surrounded by an adiabatic boundary, the system may still be coupled to the surroundings so that work may be done. Four examples of different systems experiencing the process of working in an adiabatic container, so – called adiabatic work, are shown in figure 2. It was a series of experiments using a paddle wheel, like the one in figure 1 (a), that established the important fact that the state of a system may be caused to change from a given initial state to the same final state by the performance of adiabatic work only.
Mechanical systems are not easily controlled in changing the state of a system, so let us consider a composite electrical system composed of a resistor immersed in water. The initial state i is characterized by the thermodynamic coordinates Pi= 1 atm and Ti= 287,7 K (145°C) and the final state f is characterized by the coordinates Pf = 1 atm and Tf= 288,7 K (15,5°C), as shown in figure 3. To cause the system to proceed from i to f along path by the performance of adiabatic work only, it would be necessary to surround the water with an adiabatic wall, keep the water at atmospheric pressure, and maintain a current in the resistor for a suitable interval of time.
But, in path I is not the only path by wich the system may be changed from i to f by the performance of adiabatic work only. We might compress the water adiabatically from i to a, then use a current in a resistor from a to b, and then expand from b to f, the whole series of processes being designated by path II.
Figure 2 Adiabatic work for different types of systems.
Or we might make use of a similiar adiabatic path III. There are an infinite number of paths by which a system may be transferred from an initial state to a final state by the performance of adiabatic work only. Although further measurements of adiabatic work along different paths between the same two states were not made after Joule’s pioneering work, indirect experiments and the validity of subsequent results indicate that the adiabatic work is a restricted statement of the first law of thermodynamics:
If a closed system is caused to change from an initial state by adiabatic means only, then the work done on the system is the same for all adiabatic paths connectingthe two states.
Whenever a quantity is known to depend only on the initial and final states, and not on the path connecting them, an important conclusion can be drawn. Recall from mechanics that, in moving an object from one point in a gravitational field to another point, in the absence of friction, the work done depends only on the position of the two points and not on the path through which body was moved.
Figure 3 Changing the state of a system from the initial state i to the final state f along three different adiabatic paths.
It was concluded that , for a conservative force, there exists a function of the space coordinates of the body whose final value minus its initial value is equal to the work done. This function was called the potential – energy function. Similiarly, the work done in moving an electric charge from one point in a conservative electric field to another point is also independent of the path and, therefore, is also expressible as the value of the electric potential function at the final state minus its value at the initial state. Therefore, it follows from the restricted statement of the first law of thermodynamics that there exists a function of the coordinates of a thermodynamic system whose value at the final state minus its value at the initial state is equal to the adiabatic work in going from one state to the other. This function is known as the internal – energy function.
Denoting the internal – energy function by U, we have
Where the signs are such that, if positive work is done on the system, Uf will be greater than Ui. Its found by experiment that it is not always possible to take a system from an initial atate i to any final state f by the performance of adiabatic only. It will be shown later, when entropy is discussed , that if f cannot be reached in this way, then it is always possible to go from f to i by adiabatic means, in which case the change in internal energy from i to f, instead of being +Wià f , is –W fà i . The importance of equation is that thermodynamics work, which is generally path- dependent, becomes path- independent for an adiabatic process.
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