Breakdown of Energy Possessed by Substance (Entropy and Internal Energy)

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• General

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• Internal energy overview
  • 　Again, let us take the expansion and contraction of volume under constant temperature conditions as an example.

    
    

    　Now consider the entropy change and internal energy change when the volume changes.

    Internal energy is defined as "the sum of the kinetic energies of the molecules composing the system".

    Internal energy is "proportional to temperature."
    

    (I'll add a little more about internal energy later.)

    
    

    　As shown in the picture below, the gas expands or contracts at a constant temperature. Since the temperature is constant, the internal energy does not change before or after the expansion or contraction. As explained in the previous course, if the volume increases tenfold at a constant temperature, entropy increases by only 19.3 (J/K).
    
    
    
    

    　Conversely, if work is applied externally to contract the volume and the temperature is kept constant, entropy is reduced by only 19.3 (J/K) compared to before the contraction, even though the internal energy is maintained the same.

    　In other words, entropy can be reduced by adding the energy required to adjust the aggregate state. Since the internal energy is only proportional to the temperature T₀*, the internal energy of the system does not change before or after expansion or contraction.

    　High entropy before contraction has a large amount of "bound energy" that cannot be extracted by work to the outside world.

      ※ The nature of internal energy will be explained in a later course.
    

• Relationship between internal energy and entropy
  • Therefore,

    we consider the following energies to be inherent in the internal energy (U).

                  【Bound energy that cannot be extracted as work to the outside world】

                                              and

                  【Free energy that can be extracted as work】

     

    Therefore, the change in internal energy ⊿U is expressed as follows.
    

    【Change in internal energy：⊿U】

                                ＝【Entropy change at a given temperature：T⊿S】

                                              ＋【Change in free energy：⊿F】

     

    This free energy inherent in the internal energy is defined as the Helmholtz free energy (F). Thus, the following equation holds
    

    ⊿U = T⊿S + ⊿F.

     

    　We have discussed "internal energy" without defining it. Knowing the definition of internal energy, it is easy to imagine that bound energy is inherent in internal energy, which is briefly mentioned below.
    
  • Bound energy inherent in internal energy (supplement)
    

    　If you have dabbled in physics in high school or first year of college, you have probably heard the term "internal energy. However, you may not have been thoroughly briefed on the definition of internal energy, or you may have forgotten about it. Here we only explain the definition and concept of internal energy.　

    
    

    "The internal energy is the sum of the kinetic energies of the molecules that make up the system"

    　If a molecule of mass m is moving with velocity v, the kinetic energy of a single molecule is 1/2・mv². If n molecules are confined in a box of a certain volume, the sum of their kinetic energies is 1/2∙nmv². Molecules moving at velocity v collide with the wall and give a impulse (2mv). The impulse per unit time and area is equal to the pressure (P). Expressing the sum of kinetic energy (U=1/2・nmv²) in terms of PV and then converting it to RT from the gas equation of state (PV=nRT) yields the interesting result that "internal energy is proportional to temperature". This is expressed in the following equation.

    
    

    U ＝ 1/2Σmv² = ・・・ = 3/2RT

    　The calculation of "・・・" is not so difficult, but I will omit the explanation. (It is described in physics textbooks such as "Fundamental Physics" by KANEHARA Juro, etc.)

    　However, if you believe that "the internal energy is the sum of kinetic energy and proportional to temperature", it is easy to imagine that there is a bound energy inherent in the internal energy that cannot be extracted as work.

    
    

    Explanation for imagining that bound energy is inherent in internal energy

    　If there are 10²³ molecules (100 million x 100 million x 10 million) in a liter (1 dm³) of milk carton space, and each molecule is moving at a breakneck speed of 10 km/s in all directions, it is likely that the molecules will collide with each other and cause chemical reactions and do some work. Compared to a system in which there are only 10²³ molecules in a 10,000-km cubic meter moon-sized box, and each molecule is moving at 10 km/s in all directions. The density of 10²³ molecules in a moon-sized space is equivalent to the density of only 10 molecules flying in a room (10 cubic meters) in your prison-like apartment. Since the size of a molecule is about 0.1 nm (nanometer), no matter how fast the molecules are moving, they cannot be expected to collide with each other and cause chemical reactions in such a scanty space. Even though the total kinetic energy (internal energy) of molecules in a milk carton and a moon-sized space are the same, it is difficult to extract work of chemical change from the latter's internal energy. In other words, the latter's internal energy seems to have a large proportion of "bound energy" that cannot be extracted as work. (This concludes the supplementary explanation of internal energy.)