bridge derivation: latent heat from atomic scale interactions

[bhpayne]

prompt for Gemini 2.0 Flash Thinking Experimental on https://aistudio.google.com/prompts/new_chat

Latent heat is the energy absorbed or released during a phase transition at constant temperature. For example, the latent heat of fusion is the energy required to melt a solid into a liquid.

At the atomic level, this energy is associated with changes in the interatomic bonding and potential energy. In a solid, atoms are tightly bound in a relatively ordered structure. Melting involves breaking some of these bonds, allowing atoms to move more freely in the liquid phase.

In mathematical Physics a derivation is comprised of steps and equations. Is there a derivation that bridges the macroscopic concept of latent heat with the atomic-scale interactions? If yes, provide the steps of the derivation. Format the answer using Latex as a single file.

Latex is a software system for typesetting documents. LaTeX is widely used for the communication and publication of scientific documents and technical note-taking in many fields, owing partially to its support for complex mathematical notation.

Do not have equations inline with the text. Equations should be separate and inside a math environment for Latex:

\begin{equation}
\label{eq:}
\end{equation}

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thoughts

The user is asking for a derivation that connects the macroscopic concept of latent heat with the atomic-scale interactions, using Latex format.
The derivation should bridge the macroscopic observation of energy absorption/release during phase transition with the microscopic changes in interatomic potential energy and bonding.

Let's consider the latent heat of fusion as an example, which is the energy required to melt a solid.

At the macroscopic level, we can define latent heat of fusion $L_f$ as the heat $Q$ required to change the phase of a mass $m$ of a substance from solid to liquid at constant temperature $T_m$ (melting point).
$Q = m L_f$

At the microscopic level, the energy is related to the potential energy of the interatomic interactions. In a solid, atoms are in a potential well due to interatomic forces. To melt the solid, we need to provide energy to overcome these interatomic forces and allow atoms to move more freely as in a liquid.

Let's think about the internal energy $U$ of a system. The change in internal energy during a phase transition is related to the latent heat.
For a process at constant pressure, the heat absorbed is equal to the change in enthalpy $H$.
$Q = \Delta H = \Delta U + P \Delta V$
where $P$ is pressure and $V$ is volume.

The internal energy $U$ of a system can be considered as the sum of kinetic energy $K$ and potential energy $V_{pot}$ of the atoms/molecules.
$U = K + V_{pot}$

During a phase transition at constant temperature, the kinetic energy of the molecules (related to temperature) remains approximately constant. The main change in internal energy comes from the change in potential energy due to the change in interatomic distances and arrangements.

In a solid, atoms are in a relatively ordered structure with strong interatomic bonds, leading to a lower potential energy. In a liquid, some of these bonds are broken, and atoms have more freedom to move, resulting in a higher potential energy.

Let's consider the potential energy per atom. In a solid, each atom is in a potential well due to its neighbors. To melt, we need to provide enough energy to overcome this potential well.

We can relate the latent heat to the change in potential energy per atom/molecule and the number of atoms/molecules.

Let $N$ be the number of atoms or molecules in the system.
Let $u_{solid}$ be the average potential energy per atom/molecule in the solid phase.
Let $u_{liquid}$ be the average potential energy per atom/molecule in the liquid phase.
Generally, $u_{liquid} > u_{solid}$ because bonds are broken in the liquid phase, leading to a less negative (or more positive) potential energy.

The change in potential energy for the entire system during melting is approximately $N (u_{liquid} - u_{solid})$.
This change in potential energy is related to the latent heat of fusion.

We can consider the enthalpy $H = U + PV$. The change in enthalpy during fusion is $\Delta H = \Delta U + P \Delta V$.
$\Delta U \approx N (u_{liquid} - u_{solid})$
$\Delta V = V_{liquid} - V_{solid}$

Latent heat of fusion per unit mass $L_f = \frac{Q}{m} = \frac{\Delta H}{m} = \frac{\Delta U + P \Delta V}{m} \approx \frac{N (u_{liquid} - u_{solid}) + P (V_{liquid} - V_{solid})}{m}$

We can express the number of atoms $N$ and mass $m$ in terms of molar mass $M$ and Avogadro's number $N_A$.
If we have $n$ moles, then $m = n M$ and $N = n N_A$.
$L_f = \frac{n N_A (u_{liquid} - u_{solid}) + P (V_{liquid} - V_{solid})}{n M} = \frac{N_A (u_{liquid} - u_{solid})}{M} + \frac{P (V_{liquid} - V_{solid})}{m}$

Let's consider latent heat per mole $L_{f,mol} = M L_f = N_A (u_{liquid} - u_{solid}) + P (V_{liquid,mol} - V_{solid,mol})$
where $V_{liquid,mol}$ and $V_{solid,mol}$ are molar volumes.

If we neglect the $P \Delta V$ term (especially for solid-liquid transitions at moderate pressures), then
$L_{f,mol} \approx N_A (u_{liquid} - u_{solid})$.
Latent heat of fusion per atom/molecule is approximately $\Delta u = u_{liquid} - u_{solid}$.
$L_f \approx \frac{N}{m} (u_{liquid} - u_{solid}) = \frac{1}{\bar{m}} (u_{liquid} - u_{solid})$ where $\bar{m}$ is the average mass per atom/molecule.

Let's think about the interatomic potential. A simple model is the Lennard-Jones potential.
$V_{LJ}(r) = 4 \epsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^{6} \right]$
where $\epsilon$ is the depth of the potential well and $\sigma$ is the distance at which the potential is zero.

In a solid, atoms are arranged in a lattice structure and are near the minimum of the potential well. In a liquid, the average interatomic distance might change, and the structure is less ordered.
The difference in potential energy per atom between liquid and solid phase comes from the change in average interatomic distances and coordination number.

Steps for Derivation:

1. Define latent heat macroscopically as heat required for phase transition.
2. Relate macroscopic heat to change in enthalpy and internal energy.
3. Express internal energy at atomic level as sum of kinetic and potential energy.
4. Argue that at constant temperature, change in kinetic energy is negligible, so change in internal energy is mainly due to change in potential energy.
5. Relate change in potential energy to the difference in average potential energy per atom/molecule in different phases.
6. Express latent heat in terms of change in potential energy per atom/molecule and number of atoms/molecules, and possibly the work done due to volume change.
7. Simplify the expression by neglecting pressure-volume work or focusing on the dominant term from potential energy change.
8. Discuss the interatomic potential and how the change in phase affects the potential energy.

More Concrete Derivation Steps:

Step 1: Macroscopic definition of latent heat of fusion.
Step 2: Thermodynamic relation to enthalpy change.
Step 3: Internal energy as sum of kinetic and potential energy.
Step 4: Constant temperature implies constant average kinetic energy per particle.
Step 5: Change in internal energy is dominated by change in potential energy.
Step 6: Latent heat related to change in potential energy and pressure-volume work.
Step 7: Approximate latent heat by change in potential energy per particle times number of particles, neglecting P-V term for simplicity initially and then including it for completeness.
Step 8: Discuss potential energy difference in terms of interatomic interactions and bonding in solid vs liquid.

Let's write out the derivation using Latex now.

response

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}

\title{Derivation Bridging Macroscopic Latent Heat with Atomic-Scale Interactions}
\date{}

\begin{document}
\maketitle

\section{Introduction}
Latent heat is a macroscopic thermodynamic property that quantifies the energy absorbed or released during a phase transition at constant temperature. This energy is required to change the state of matter, for instance, from solid to liquid (latent heat of fusion) or from liquid to gas (latent heat of vaporization). To understand the origin of latent heat, we need to delve into the atomic-scale interactions and potential energy changes that occur during phase transitions. This derivation aims to bridge the macroscopic concept of latent heat with these atomic-scale phenomena.

\section{Macroscopic Definition of Latent Heat}
Macroscopically, the latent heat of fusion, $L_f$, is defined as the amount of heat energy, $Q$, required to completely melt a mass, $m$, of a solid substance at its melting temperature, $T_m$, under constant pressure. Mathematically, this is expressed as:
\begin{equation}
\label{eq:1}
Q = m L_f
\end{equation}
Similarly, for latent heat of vaporization, $L_v$, the heat required to vaporize a mass $m$ of liquid at its boiling temperature $T_b$ is $Q = m L_v$. In general, for a phase transition, the heat exchanged $Q$ is related to the latent heat $L$ and mass $m$ by:
\begin{equation}
\label{eq:2}
Q = m L
\end{equation}

\section{Thermodynamic Relation to Enthalpy Change}
For a process occurring at constant pressure, the heat absorbed by the system is equal to the change in enthalpy, $\Delta H$. Thus, the latent heat can be related to the change in enthalpy during the phase transition.
\begin{equation}
\label{eq:3}
Q = \Delta H = H_{final} - H_{initial}
\end{equation}
Where $H_{initial}$ and $H_{final}$ are the enthalpies of the initial and final phases, respectively. Enthalpy $H$ is defined as:
\begin{equation}}
\label{eq:4}
H = U + PV
\end{equation}
where $U$ is the internal energy, $P$ is the pressure, and $V$ is the volume of the system. Therefore, the change in enthalpy during a phase transition is:
\begin{equation}
\label{eq:5}
\Delta H = \Delta U + P \Delta V
\end{equation}
where $\Delta U = U_{final} - U_{initial}$ and $\Delta V = V_{final} - V_{initial}$ are the changes in internal energy and volume, respectively.

\section{Internal Energy and Atomic-Scale Interactions}
At the atomic level, the internal energy $U$ of a substance can be considered as the sum of the kinetic energy ($K$) and potential energy ($V_{pot}$) of its constituent atoms or molecules.
\begin{equation}
\label{eq:6}
U = K + V_{pot}
\end{equation}
The kinetic energy $K$ is associated with the motion of atoms or molecules (translational, rotational, vibrational), and is directly related to the temperature of the system. The potential energy $V_{pot}$ arises from the interatomic or intermolecular forces and depends on the relative positions of the atoms or molecules.

During a phase transition at constant temperature, the average kinetic energy per particle remains approximately constant. The primary change in internal energy comes from the change in potential energy due to the rearrangement of atoms or molecules and changes in interatomic distances and bonding.

For example, in melting a solid to a liquid, energy is required to break some of the interatomic bonds that hold the atoms in a fixed lattice structure in the solid phase. This process increases the potential energy as atoms become less constrained and move to a state with different average interatomic distances and coordination.

\section{Change in Potential Energy during Phase Transition}
Let $N$ be the number of atoms or molecules in the system. Let $u_{pot, initial}$ be the average potential energy per atom/molecule in the initial phase (e.g., solid) and $u_{pot, final}$ be the average potential energy per atom/molecule in the final phase (e.g., liquid). The change in potential energy for the entire system is approximately:
\begin{equation}
\label{eq:7}
\Delta V_{pot} \approx N (u_{pot, final} - u_{pot, initial})
\end{equation}
Assuming that the change in kinetic energy $\Delta K$ is negligible during a phase transition at constant temperature, the change in internal energy $\Delta U$ is dominated by the change in potential energy:
\begin{equation}
\label{eq:8}
\Delta U \approx \Delta V_{pot} \approx N (u_{pot, final} - u_{pot, initial})
\end{equation}

\section{Latent Heat and Atomic Potential Energy Change}
Substituting equation \eqref{eq:8} into equation \eqref{eq:5}, we get:
\begin{equation}
\label{eq:9}
\Delta H \approx N (u_{pot, final} - u_{pot, initial}) + P \Delta V
\end{equation}
From equation \eqref{eq:3} and \eqref{eq:9}, the latent heat $Q = \Delta H$ can be expressed as:
\begin{equation}
\label{eq:10}
Q \approx N (u_{pot, final} - u_{pot, initial}) + P \Delta V
\end{equation}
The latent heat per unit mass, $L = Q/m$, can be written as:
\begin{equation}
\label{eq:11}
L \approx \frac{N}{m} (u_{pot, final} - u_{pot, initial}) + \frac{P \Delta V}{m}
\end{equation}
We know that $N/m = N_A / M$, where $N_A$ is Avogadro's number and $M$ is the molar mass. Therefore,
\begin{equation}
\label{eq:12}
L \approx \frac{N_A}{M} (u_{pot, final} - u_{pot, initial}) + \frac{P \Delta V}{m}
\end{equation}
Or, in terms of molar latent heat $L_{mol} = M L$:
\begin{equation}
\label{eq:13}
L_{mol} \approx N_A (u_{pot, final} - u_{pot, initial}) + P \Delta V_{mol}
\end{equation}
where $\Delta V_{mol} = M \Delta V / m = V_{mol, final} - V_{mol, initial}$ is the change in molar volume during the phase transition.

For many solid-liquid transitions, the volume change $\Delta V$ is relatively small, and the $P \Delta V$ term is significantly smaller than the change in potential energy, especially at moderate pressures. In such cases, the latent heat is primarily attributed to the change in potential energy due to the change in interatomic interactions.
\begin{equation}
\label{eq:14}
L_{mol} \approx N_A (u_{pot, final} - u_{pot, initial})
\end{equation}
This equation provides a direct link between the macroscopic latent heat and the microscopic change in potential energy per atom/molecule during a phase transition. The term $(u_{pot, final} - u_{pot, initial})$ represents the average energy required per atom/molecule to break bonds and transition from the initial to the final phase in terms of potential energy.

\section{Conclusion}
This derivation shows how the macroscopic concept of latent heat is rooted in the atomic-scale changes in potential energy. During a phase transition, the energy supplied as latent heat is primarily used to overcome interatomic or intermolecular forces, leading to an increase in the potential energy of the system. While the pressure-volume work term ($P \Delta V$) contributes to the enthalpy change, particularly for vaporization, the dominant contribution to latent heat often comes from the change in potential energy associated with changes in atomic or molecular arrangements and bonding. This microscopic perspective provides a fundamental understanding of why energy is required or released during phase transitions at constant temperature.

\end{document}

[bhpayne]

Minor manual reformatting yields

\documentclass{article}
\usepackage{amsmath}

% margins of 1 inch:
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{9in}
\setlength{\oddsidemargin}{0in}
\setlength{\textwidth}{6.5in}

\usepackage[pdftex]{hyperref} % hyperlink equation and bibliographic citations



\title{Derivation Bridging Macroscopic Latent Heat with Atomic-Scale Interactions}
\date{\today}
\author{Ben Payne with Gemini 2.0}



\begin{document}
\maketitle

\section{Introduction}
Latent heat is a macroscopic thermodynamic property that quantifies the energy absorbed or released during a phase transition at constant temperature. This energy is required to change the state of matter, for instance, from solid to liquid (latent heat of fusion) or from liquid to gas (latent heat of vaporization). To understand the origin of latent heat, we need to delve into the atomic-scale interactions and potential energy changes that occur during phase transitions. This derivation aims to bridge the macroscopic concept of latent heat with these atomic-scale phenomena.

\section{Macroscopic Definition of Latent Heat}
Macroscopically, the latent heat of fusion, $L_{\rm fusion}$, is defined as the amount of heat energy, $Q$, required to completely melt a mass, $m$, of a solid substance at its melting temperature, $T_{\rm melt}$, under constant pressure. This is expressed as:
\begin{equation}
\label{eq:1}
Q = m L_{\rm fusion}
\end{equation}
Similarly, for latent heat of vaporization, $L_{\rm vaporization}$, the heat required to vaporize a mass $m$ of liquid at its boiling temperature $T_b$ is 
\begin{equation}
    Q = m L_{\rm vaporization}
\end{equation}
In general, for a phase transition, the heat exchanged $Q$ is related to the latent heat $L$ and mass $m$ by:
\begin{equation}
\label{eq:2}
Q = m L
\end{equation}

\section{Thermodynamic Relation to Enthalpy Change}
For a process occurring at constant pressure, the heat absorbed by the system is equal to the change in enthalpy, $\Delta H$. Thus, the latent heat can be related to the change in enthalpy during the phase transition.
\begin{equation}
\label{eq:3}
Q = \Delta H = H_{\rm final} - H_{\rm initial}
\end{equation}
Where $H_{\rm initial}$ and $H_{\rm final}$ are the enthalpies of the initial and final phases, respectively. Enthalpy $H$ is defined as:
\begin{equation}
\label{eq:4}
H = U + PV
\end{equation}
where $U$ is the internal energy, $P$ is the pressure, and $V$ is the volume of the system. Therefore, the change in enthalpy during a phase transition is:
\begin{equation}
\label{eq:5}
\Delta H = \Delta U + P \Delta V
\end{equation}
where $\Delta U = U_{\rm final} - U_{\rm initial}$ and $\Delta V = V_{\rm final} - V_{\rm initial}$ are the changes in internal energy and volume, respectively.

\section{Internal Energy and Atomic-Scale Interactions}
At the atomic level, the internal energy $U$ of a substance can be considered as the sum of the kinetic energy ($K$) and potential energy ($V_{\rm potential}$) of its constituent atoms or molecules.
\begin{equation}
\label{eq:6}
U = K + V_{\rm potential}
\end{equation}
The kinetic energy $K$ is associated with the motion of atoms or molecules (translational, rotational, vibrational), and is directly related to the temperature of the system. The potential energy $V_{\rm potential}$ arises from the interatomic or intermolecular forces and depends on the relative positions of the atoms or molecules.

During a phase transition at constant temperature, the average kinetic energy per particle remains approximately constant. The primary change in internal energy comes from the change in potential energy due to the rearrangement of atoms or molecules and changes in interatomic distances and bonding.

For example, in melting a solid to a liquid, energy is required to break some of the interatomic bonds that hold the atoms in a fixed lattice structure in the solid phase. This process increases the potential energy as atoms become less constrained and move to a state with different average interatomic distances and coordination.

\section{Change in Potential Energy during Phase Transition}
Let $N$ be the number of atoms or molecules in the system. Let $u_{\rm pot, initial}$ be the average potential energy per atom/molecule in the initial phase (e.g., solid) and $u_{\rm pot, final}$ be the average potential energy per atom/molecule in the final phase (e.g., liquid). The change in potential energy for the entire system is approximately:
\begin{equation}
\label{eq:7}
\Delta V_{\rm potential} \approx N (u_{\rm pot, final} - u_{\rm pot, initial})
\end{equation}
Assuming that the change in kinetic energy $\Delta K$ is negligible during a phase transition at constant temperature, the change in internal energy $\Delta U$ is dominated by the change in potential energy:
\begin{equation}
\label{eq:8}
\Delta U \approx \Delta V_{\rm potential} \approx N (u_{\rm pot, final} - u_{\rm pot, initial})
\end{equation}

\section{Latent Heat and Atomic Potential Energy Change}
Substituting equation \eqref{eq:8} into equation \eqref{eq:5}, we get:
\begin{equation}
\label{eq:9}
\Delta H \approx N (u_{\rm pot, final} - u_{\rm pot, initial}) + P \Delta V
\end{equation}
From equation \eqref{eq:3} and \eqref{eq:9}, the latent heat $Q = \Delta H$ can be expressed as:
\begin{equation}
\label{eq:10}
Q \approx N (u_{\rm pot, final} - u_{\rm pot, initial}) + P \Delta V
\end{equation}
The latent heat per unit mass, $L = Q/m$, can be written as:
\begin{equation}
\label{eq:11}
L \approx \frac{N}{m} (u_{\rm pot, final} - u_{\rm pot, initial}) + \frac{P \Delta V}{m}
\end{equation}
We know that $N/m = N_A / M$, where $N_A$ is Avogadro's number and $M$ is the molar mass. Therefore,
\begin{equation}
\label{eq:12}
L \approx \frac{N_A}{M} (u_{\rm pot, final} - u_{\rm pot, initial}) + \frac{P \Delta V}{m}
\end{equation}
Or, in terms of molar latent heat $L_{mol} = M L$:
\begin{equation}
\label{eq:13}
L_{mol} \approx N_A (u_{\rm pot, final} - u_{\rm pot, initial}) + P \Delta V_{mol}
\end{equation}
where $\Delta V_{mol} = M \Delta V / m = V_{mol, final} - V_{mol, initial}$ is the change in molar volume during the phase transition.

For many solid-liquid transitions, the volume change $\Delta V$ is relatively small, and the $P \Delta V$ term is significantly smaller than the change in potential energy, especially at moderate pressures. In such cases, the latent heat is primarily attributed to the change in potential energy due to the change in interatomic interactions.
\begin{equation}
\label{eq:14}
L_{mol} \approx N_A (u_{\rm pot, final} - u_{\rm pot, initial})
\end{equation}
This equation provides a direct link between the macroscopic latent heat and the microscopic change in potential energy per atom/molecule during a phase transition. The term $(u_{\rm pot, final} - u_{\rm pot, initial})$ represents the average energy required per atom/molecule to break bonds and transition from the initial to the final phase in terms of potential energy.

\section{Conclusion}
This derivation shows how the macroscopic concept of latent heat is rooted in the atomic-scale changes in potential energy. During a phase transition, the energy supplied as latent heat is primarily used to overcome interatomic or intermolecular forces, leading to an increase in the potential energy of the system. While the pressure-volume work term ($P \Delta V$) contributes to the enthalpy change, particularly for vaporization, the dominant contribution to latent heat often comes from the change in potential energy associated with changes in atomic or molecular arrangements and bonding. This microscopic perspective provides a fundamental understanding of why energy is required or released during phase transitions at constant temperature.

\end{document}