Abstract
The first law reminds us – see Chap. 3 – of the well known fact that energy is conserved and all of it must be accounted for. Therefore, when heat energy ΔQ′ is added to a system, and none of it escapes, then all of it must still be there. And if there should exist a procedure to convert some of this heat energy to work, say ΔW′, then after such conversion only ΔQ′ − ΔW′ of it will still be present in the system. We call this left over amount the increase, ΔU, in the system’s internal energy. Because the amount of heat energy input surely depends on what the temperature difference between the depositor and the depositee at any given time is, and because the total deposit of heat may have taken finite length of time, its magnitude will also depend on how long any particular instance, with some particular difference in temperature, lasted. etc. Similarly, the work that is done – namely, ΔW′ – will also crucially depend on where, what, how, and when the work was carried out. These facts are generally labeled as path dependencies. Clearly, therefore, the size of both ΔQ′ and ΔW′ must depend on the paths that are taken in carrying out these tasks.
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Notes
- 1.
For instance, as noted by Stephen Hawking with regard to Black Holes.
- 2.
See the chapter on the First Law, especially the comments relating to (3.15).
- 3.
Occasionally, just the conservation of energy is referred to as the first law of thermodynamics. In fact, the conservation of energy has been known since the ancient Greek times and the real value of the first law lies in part (b) which identifies an important state function: the internal energy.
- 4.
Compare (4.60).
- 5.
Recall that u, h, s, refer, respectively, to the internal energy, the enthalpy and the entropy.
- 6.
Namely, that occurring at constant pressure.
- 7.
Namely, one that occurs at constant entropy.
- 8.
The positivity of both C v and χ t is required for thermodynamic stability of states that are in thermal equilibrium.
- 9.
Equivalently, it can also be stated that the negativity of α s implies that normal systems heat up during (quasi-static) adiabatic decrease of volume.
- 10.
Note that this, e.g., is the case for an ideal gas.
- 11.
Note that dΦ is an exact differential because ds, du and dt are exact differentials. Note also that Φ, s, and u are state functions and t is a state variable.
- 12.
See (2.31).
- 13.
The statement prescribed in (5.26) was as follows: p(v, t) = a(v) t + b(v). Clearly, the term b(v) can be transferred to the left hand side and included in the general term p(v, t).
- 14.
This fact was recorded immediately following (5.21).
- 15.
See the chapter titled “Equilibrium, Motive Forces, and Stability” for a discussion of the stability of thermodynamic systems.
- 16.
Note that this, for instance, is the case for an ideal gas.
- 17.
Note: Because ds, dt, and dh are exact differentials, therefore, the same is true for dΨ.
- 18.
This is so because when \(\left ( \frac{{C}_{p}} {pv{\alpha }_{p}}\right )\) is > 0, positive (dp) s implies positive (dt) s .
- 19.
- 20.
Jacobian, Carl Gutav Jacob J., (12/10/1804)–(2/18/1851).
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Tahir-Kheli, R. (2012). First and Second Laws Combined. In: General and Statistical Thermodynamics. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21481-3_5
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DOI: https://doi.org/10.1007/978-3-642-21481-3_5
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