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# 物理代写|热力学代写Thermodynamics代考|ENME485 Temperature Contribution to Entropy

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

Thus far, we have considered only changes in $V$ at constant $T$. To round out the full thermodynamic story, we must also consider changes in $T$ at constant $V$. Because $V$ is fixed, there is no change in our information about the molecular positions; these therefore play no role in the origin of $\Delta S$ in this case. The entropy does change, however-owing to a change in the available velocity states, induced by the change in temperature.
$\triangleright \triangleright \triangleright$ Helpful Hint: Remember: $V$ corresponds to the available molecular positions; $T$ corresponds to the available molecular velocities.

Recall from Section $6.2$ that for a system in thermal equilibrium, the distribution of particle velocities is not uniform, but adheres to the MaxwellBoltzmann distribution [Equation (6.2) and Figure 6.1]. Recall, also, that this same distribution describes all three velocity components of all system particles – although for simplicity, we refer to a single, ‘ $v_x$ ‘ component only.

## 物理代写|热力学代写Thermodynamics代考|Combined Entropy Expression

Comparing the rightmost parts of Equations (11.3) and (11.7), we find that $A(T) V=T^{3 / 2} B(V)$. This can only be true if $A(T)=C T^{3 / 2}$ and $B(V)=$ $C V$, where $C$ is a true thermodynamic constant. This means that $C$ depends neither on $T$ nor $V$, and is therefore independent of the thermodynamic state. We thus obtain
$$\left.S(T, V)=N k \ln \left(C V T^{3 / 2}\right), \quad \text { [ideal gas }\right] \quad \text { (11.9) }$$
i.e., the desired explicit $S(T, V)$ state function for the ideal gas.
Note that in Equation (11.9) above, the value of the constant $C$ is still unspecified. This means that the above analysis is insufficient to provide us with absolute entropy values – i.e., specific numerical values for $S(T, V)$ itself-though it does specify $S(T, V)$ to within an additive constant. Nevertheless, $C$ is indeed found to have a specific value in nature, which can be determined using quantum mechanics (a discipline that lies outside the scope of this book).

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

DDD有用的提示: 记住: $V$ 对应于可用的分子位置; $T$ 对应于可用的分子速度。 组件。

## 物理代写|热力学代写Thermodynamics代考|Combined Entropy Expression

$$\left.S(T, V)=N k \ln \left(C V T^{3 / 2}\right), \quad \text { [ideal gas }\right]$$

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