Posted on Categories:Thermodynamics, 热力学, 物理代写

# 物理代写|热力学代写Thermodynamics代考|EGM-321 Energy at the Molecular Scale

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## 物理代写|热力学代写Thermodynamics代考|Energy at the Molecular Scale

From the laws of physics, we learn that there are two kinds of energy: potential energy and kinetic energy. At the molecular scale, it is convenient to imagine a third kind of energy possessed by individual molecules, which we call the internal state energy. Each type of energy is associated with a different type of molecular coordinate.

When the subscript $i$ is applied to the quantities in the box above, this refers to the specific molecule $i$. Thus, $E_{K, i}$ is the kinetic energy associated with the translational (center-of-mass) motion of molecule $i$. “Potential energy” refers to the intermolecular interactions among molecules. Strictly speaking, $E_P$ depends on the relative (to other molecules) molecular positions, rather than on the absolute (center-of-mass) molecular positions, $\left(x_i, y_i, z_i\right)$. However, this distinction does not matter for our purposes. Finally, $E_I$ represents the energy associated with intramolecular interactions. For most thermodynamics applications, $E_{I, i}$ can be regarded as the chemical energy stored in molecule $i$ – if indeed, it need even be regarded at all.

## 物理代写|热力学代写Thermodynamics代考|Internal Energy

Because $E$ can change over time even when the thermodynamic state does not, this means that $E$ itself cannot be a thermodynamic quantity, according to Definition $4.1$ (p. 26). So how can we define a meaningful thermodynamic energy quantity? This is where the idea of statistical averaging comes in-specifically, time averaging (Section 2.3).

A plot of the molecular state energy, $E$, as a function of time, $t$, for a macroscopic system in thermodynamic equilibrium, would look something like Figure 5.1. Note that $E(t)$ oscillates very quickly, but never gets far from the time-averaged value. These oscillations are called fluctuations. According to statistical mechanics, fluctuations quickly cancel out when averaged over macroscopic time scales, and can therefore be ignored-provided that the system is both large $(N \rightarrow \infty)$, and in equilibrium.
In practice, the time-averaged quantity,
$$\langle E(t)\rangle=\frac{1}{t_{\text {final }}} \int_0^{t_{\text {final }}} E(t) d t,$$
is much more useful than $E$ itself. Moreover, for a system in equilibrium, $\langle E(t)\rangle$ is constant over time, and may therefore be regarded as a true thermodynamic quantity. We take this quantity to be the internal energy (after subtracting the internal state energy).

## 物理代写|热力学代写Thermodynamics代考|Internal Energy

$$\langle E(t)\rangle=\frac{1}{t_{\text {final }}} \int_0^{t_{\text {fimal }}} E(t) d t$$

## MATLAB代写

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