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We can think about the internal energy of a system in two different but consistent ways. The first is the atomic and molecular view, which examines the system on the atomic and molecular scale. The internal energy $U$ of a system is the sum of the kinetic and potential energies of its atoms and molecules. Recall that kinetic plus potential energy is called mechanical energy. Thus internal energy is the sum of atomic and molecular mechanical energy. Because it is impossible to keep track of all individual atoms and molecules, we must deal with averages and distributions. A second way to view the internal energy of a system is in terms of its macroscopic characteristics, which are very similar to atomic and molecular average values.
Macroscopically, we define the change in internal energy $\mathrm{\Delta}U$ to be that given by the first law of thermodynamics:
Many detailed experiments have verified that $\mathrm{\Delta}U=Q-W$ , where $\mathrm{\Delta}U$ is the change in total kinetic and potential energy of all atoms and molecules in a system. It has also been determined experimentally that the internal energy $U$ of a system depends only on the state of the system and not how it reached that state . More specifically, $U$ is found to be a function of a few macroscopic quantities (pressure, volume, and temperature, for example), independent of past history such as whether there has been heat transfer or work done. This independence means that if we know the state of a system, we can calculate changes in its internal energy $U$ from a few macroscopic variables.
In thermodynamics, we often use the macroscopic picture when making calculations of how a system behaves, while the atomic and molecular picture gives underlying explanations in terms of averages and distributions. We shall see this again in later sections of this chapter. For example, in the topic of entropy, calculations will be made using the atomic and molecular view.
To get a better idea of how to think about the internal energy of a system, let us examine a system going from State 1 to State 2. The system has internal energy ${U}_{1}$ in State 1, and it has internal energy ${U}_{2}$ in State 2, no matter how it got to either state. So the change in internal energy $\mathrm{\Delta}U={U}_{2}-{U}_{1}$ is independent of what caused the change. In other words, $\mathrm{\Delta}U$ is independent of path . By path, we mean the method of getting from the starting point to the ending point. Why is this independence important? Note that $\mathrm{\Delta}U=Q-W$ . Both $Q$ and $W$ depend on path , but $\mathrm{\Delta}U$ does not. This path independence means that internal energy $U$ is easier to consider than either heat transfer or work done.
(a) Suppose there is heat transfer of 40.00 J to a system, while the system does 10.00 J of work. Later, there is heat transfer of 25.00 J out of the system while 4.00 J of work is done on the system. What is the net change in internal energy of the system?
(b) What is the change in internal energy of a system when a total of 150.00 J of heat transfer occurs out of (from) the system and 159.00 J of work is done on the system? (See [link] ).
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