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# 物理代写|热力学代写Thermodynamics代考|ENME485 Measuring Our Molecular Ignorance

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## 物理代写|热力学代写Thermodynamics代考|Measuring Our Molecular Ignorance

Our goals in this chapter are two-fold. First, we seek to prove that starting from the statistical, or information definition of $S$ as presented in Equation (10.2), we can derive the thermodynamic form of Equation (10.1), under reversible conditions. As discussed in Chapter 10, the general proof is too advanced for the scope of this book. For the ideal gas, on the other hand, a simple and straightforward derivation is possible. For simplicity, we assume an ideal gas of point particles – although the derivation can easily be generalized to incorporate rotating molecules or those with other internal structure, provided that they are noninteracting.

The second goal is to use Equation (10.2) to derive an expression for the entropy state function, $S$, as an explicit function of the thermodynamic variables, $(T, V)$, for the ideal gas. We verify that the result is equivalent to the famous Sackur-Tetrode equation [Equation (11.11), p. 93] to within an additive constant. This again confirms the agreement of the two entropies, for the ideal gas case. In this fashion, we establish a connection between the statistical and thermodynamic definitions of entropy, using the information theory approach.

## 物理代写|热力学代写Thermodynamics代考|Volume Contribution to Entropy

Consider the change in entropy, $\Delta S$, brought about via isothermal (true) expansion of the ideal gas. Because $T$ is fixed, there is no change in the (statistically-averaged) particle velocities. Therefore, all of $\Delta S$ is due to the increase in volume, which in turn is associated with changes in the particle positions.

Now consider the physical box within which the system resides, as depicted in Figure 11.1. As macroscopic observers, we know that every particle must lie somewhere inside the box, but we have no idea precisely where. Thus, if a measurement of a given particle’s position were to be performed, we know a priori that there would be zero probability of finding the particle outside the box, and equal probability of finding it at any point inside the box.

$\triangleright \triangleright \triangleright$ To Ponder… Although it is convenient to think of entropy as the amount of information gained by the observer as a result of precise particle measurement, it is important to realize that such a measurement is imagined to be hypothetical only. The reality is that modern science actually does enable us to peer at individual molecules, in a manner that would have been inconceivable to the founding fathers of thermodynamics. In that sense, we are no longer true macroscopic observers.

It is natural to think of each $(x, y, z)$ point as constituting a different “position state” for a single particle. The total number of position states available to a given particle is thus equal to the number of points inside the box. Since space is continuous, this number is technically infinite. However, that is not too problematic, since (as we will see) the determination of $\Delta S$ requires only the relative number of available states. In any case, it is evident that the number of points inside the box-and therefore $\Omega_1$ (at constant $T$ ) – is proportional to the box volume, $V$. We thus have
$$\Omega_1=A V,$$

where $A$ remains constant under isothermal expansion. In principle, $A$ can depend on $T$, but it must be independent of $V$.
From Equations (10.2), (11.1), and (11.2), we thus find
$$S=k \ln \left(\Omega_1^N\right)=N k \ln \left(\Omega_1\right)=N k \ln (A V) . \quad[\text { ideal gas }]$$
For macroscopic systems, $\Omega=(A V)^N$ grows enormously quickly with $V$. However, the logarithm converts the exponent into a linear $N$ factor. Thus, entropy is extensive, as it should be. In the ideal gas case, the extensivity of $S$ follows from the additivity of information for independent measurements (Section 10.3).

## 物理代写|热力学代写THERMODYNAMICS代考|MEASURING OUR MOLECULAR IGNORANCE

Equation $(11.11), p .93$

## 物理代寻|热力学代写THERMODYNAMICS代考|VOLUME CONTRIBUTION TO ENTROPY

$\triangleright \triangleright \triangleright$ 思考．…虽然将熵视为观察者通过精确粒子测量获得的信息量很方便，但重要的是要认识到这种测量只是假设的。事实上，现代科学确实使我们能够以热力 学的莤基人无法想象的方式观察单个分子。从这个意义上说，我们不再是真正的宏观观察者。

\Omega_1=A V
$$在哪里 A 在等温膨胀下保持不变。原则上， A 可以依赖 T, 但它必须独立于 V. 从方程式 10.2,11.1 ，和 11.2, 我们因此发现$$
S=k \ln \left(\Omega_1^N\right)=N k \ln \left(\Omega_1\right)=N k \ln (A V) . \quad \text { [ideal gas ] }


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