# 2.1 Energy and Thermochemistry
## What is energy?
Energy is the capacity to do work. It can be in many forms: thermal energy (associated with random motion of atoms and molecules), chemical energy (associated with bonds in chemical substances), potential energy (associated with position of an object), kinetic energy (associated with motion), etc. Each of these forms of energy can be converted to another form (e.g., dropping an object converts potential energy to kinetic energy), but cannot be created or destroyed. This principle is called ***law of conservation of energy***. SI unit of energy is joule or $\pu{J}$.
```{dropdown} Example: Kinetic Energy
The amount of kinetic energy possessed by $\pu{2 kg}$ mass moving at a speed of $\pu{1 m s−1}$, calculated as:
>$$
E_k = \frac{1}{2}m\nu^2 = \frac{1}{2}(\pu{ 2 kg})(\pu{1 m s-2}) = \pu{ 1 J}
$$
```
Matter can undergo physical and chemical changes. For example, freezing of water does not involve breaking down of water molecules and only involves changing the state of matter, i.e., physical change. This can be represented as follows:
$$
\ce{H2O(l) -> H2O(g)}
$$
Chemical change involves making or breaking chemical bonds. Formation of water from constituent elements is an example of chemical change:
$$\ce{2 H2(g) + O2(g) -> 2 H2O(l)} $$
In each case there was a specific type of energy involved in the change. This energy must be released or supplied for changes (reactions) to occur.
Energy changes associated with physical processes and chemical reactions must be discussed in the context of a ***system*** and its ***surroundings***. A system is a specific part of the universe that interests us and the surroundings are rest of the universe. Practically speaking, surroundings are immediately close to the system. In the above examples, ice or water is a system and all other parts beyond this system (container, etc) are the surroundings.
## How is energy transferred?
Physical and chemical changes within systems can result in exchange of the energy from within the system to the surroundings or vice versa.
Mass (e.g., coffee, gasoline) contained in a closed system (e.g., a Thermos flask, gas tank in a car) is a conserved quantity, but if the system is not closed, the amount of mass that goes in or out of the system can be measured. Likewise, we often have a system that is not closed, and would like to know how much energy comes in or out.
Energy is not a physical substance like water or coffee, so energy transfer cannot be measured with a meter or an instrument. So, how can we tell how much useful energy can a car "put out" on one tank of gas?
The ***law of conservation of energy*** guarantees that all the chemical energy in the gasoline will reappear in some form, but not necessarily in a form that is useful for doing farm work. Tractors, like cars, are extremely inefficient, and typically $\pu{90 \%}$ of the energy they consume is converted directly into heat, which is carried away by the exhaust and the air flowing over the radiator. We distinguish the energy that comes out directly as heat from the energy that serves to accelerate a trailer or to plow a field, so we define a technical meaning of the ordinary word "work" to express the distinction.
### What is work?
**Work** ($w$) is the energy that crosses a system boundary in response to a force moving through a distance (e.g., when a system changes volume) and is expresses mathematically as follows:
```{math}
:label: work
\text{Work} = \text{force} \times \text{distance}
```
### What is heat?
Heat represents a type of energy transfer and reflects the part of energy transfer that is not accounted for by mechanical work.
**Heat** ($q$) is the energy that crosses a system boundary in response to a temperature gradient.
### A Pond Analogy
Heat and work are forms of energy that is transferred in different ways. If we consider the water in a deep pond. The amount of water in the pond is quite large, but finite, and could be measured, it corresponds to the *internal energy*, $U$ of a system.
```{figure} /assets/PondAnalogy.png
---
name: PondAnalogy
figclass: margin-caption
---
A pond analogy to relate the concepts of work and heat to the concept of energy.
```
Water may be added and subtracted from the pond either in the form of stream water (heat, $q$) or precipitation/evaporation (work, $w$). All the inflows and outflows out of the lake can be monitored using appropriate instruments. If the only gains and losses of water are as follows: stream inflow and outflow is $q_i$ and $q_o$, respectively, the precipitation is $w_r$, and the evaporation is $w_e$, then, the change in pond volume, $\Delta U$, can be calculated as:
$$
\Delta U = (q_i-q_o)+(w_r-w_e) = q + w
$$
Once water enters the pond, it loses its identity as stream or rain water. The pond does not contain any identifiable stream water or rain water, simply water. Similarly systems do not contain so much heat or work, just energy. The water level in the pond can be raised either by stream water alone *or* by rain water alone, the temperature rise in the water bath can be caused either by heating (transferring energy due to a temperature difference) or by thrashing a paddle wheel about in it (transferring energy by doing work on the system).
Another implication or assumption in our pond analogy is that water is conserved; that is, it cannot simply appear or disappear. The same proposition regarding energy is known as the first law of thermodynamics (an explored in aa following sections). For example, the energy expended in lifting a book off a floor to the table was not lost, but transferred to the book.
## What is thermochemistry?
*Heat* and *thermal energy* are not to be used interchangeably. Heat is the transfer of thermal energy between two bodies that are at different temperatures. Matter does not contain heat - it contains thermal energy. ***Thermochemistry*** is the study of heat (transfer of thermal energy) associated with chemical reactions.
The above chemical reaction (formation of water) results in release of energy and can be written as follows:
$$
\ce{2 H2(g) + O2(g) -> 2 H2O(l) + \text{Energy}}
$$
In this case, the system contains a mixture of all reactants and products. Because, energy cannot be created or destroyed, the energy is transferred between the system and the surroundings. In the above chemical reaction, energy is created in the system and is transferred to its surroundings. This is an example of an ***exothermic reaction***.
Consider the decomposition of mercury(II) oxide ($\ce{HgO}$) at high temperatures:
$$
\ce{\text{Energy} + 2 HgO(s) -> 2 Hg(l) + O2 (g)}
$$
For this reaction to occur, energy has to be supplied to the system from the surroundings. This is an example of an ***endothermic reaction***.
The above video shows how energy is consumed or produced during forming or breaking chemical bonds. In an exothermic reaction, the energy of the products in a reaction is lower than the energy of the reactants causing the excess energy to be released from the system to the surroundings. In an endothermic reaction, the energy of the products is higher than that of the reactants.
```{figure} https://www.fda.gov/files/nutrition-facts-label-download-image2.jpg
---
name: fda-label
figclass: margin
---
An example of a US Food and Drug Administration (FDA)'s nutritional facts label. Note that the Calories amount on this label are actually kilocalories. Image source: [Food Labeling & Nutrition | FDA](https://www.fda.gov/food/food-labeling-nutrition)
```
SI unit for energy is joule ($\pu{J}$), another common unit of energy is calorie ($\pu{cal}$). A calorie is the amount of energy required to raise the temperature of $\pu{1 g}$ of water by $\pu{1 ^\circ C}$. $\pu{1 cal} = \pu{4.184 J}$. Calories listed on nutrition labels (see figure ) are not "calories" but "kilocalories." To make the distinction, the first letter in Calories is capitalized.
$$\pu{1 Cal} = \pu{1000 cal} = \pu{1 kcal} = \pu{4184 J}$$
```{figure} https://esha.com/wp-content/uploads/2015/02/eu_label_1-01.jpg
---
name: eu-label
figclass: margin
---
An example of an European Union's nutritional facts label. Image source: [EU Nutrition Facts Templates - Ingredients List Label | ESHA Research](https://esha.com/products/genesis-rd-food-labeling-software/labels-and-labeling/european-union-nutrition-facts-label/)
```
## What is Thermodynamics?
Thermodynamics is a scientific study that deals with different relative energy levels and transfers of energy between systems and between different states of matter. Chemical thermodynamics specifically deals with energy transfers as related to chemical reactions and is primarily of our interest in geochemistry. We will also focus on the concept of equilibrium and its theoretical underpinnings in thermodynamics. The laws of thermodynamics provide useful guidelines for understanding the exchange of energy between systems and also direction of chemical processes.
### Thermodynamic Systems
It is very important to clearly define the ***system*** in equilibrium thermodynamics. This system can be in the scale of the entire universe (for an astronomer), an aquifer (for a hydrogeologist), or a petri dish (for a microbiologist) The choice depends on the kind of problem of interest. For example, in a shore environment there are different processes occurring and defining the bounds of the system become very important. See {numref}`earth-systems` for an example of how various processes on Earth can be studied using a systems approach.
```{figure} https://mynasadata.larc.nasa.gov/sites/default/files/2019-04/earth_system_diagram_print.jpg
---
name: earth-systems
figclass: margin-caption
---
Systems can be considered at many scales. In this picture we can see how energy and matter (elements, water, etc) is cycled across the entire Earth system at many scales. Image source: [Earth System: Matter and Energy Cycles | MyNASAData](https://mynasadata.larc.nasa.gov/basic-page/earth-system-matter-and-energy-cycles)
```
A system is any portion of the universe that can be isolated from the rest of the universe for the purpose of observing and measuring changes. By saying that a system is any portion of the universe, we mean that the system can be whatever the observer defines it to be. In {numref}`earth-systems` we can see different parts of Earth can be subdivided into subsystems and studied separately for convenience. A system is conceptual be design and needs to be defined more clearly by the observer. It can be large or small, simple or complex. It can be in form of a laboratory experiment in a beaker, or a lake, a small sample of rock, an ocean, a volcano, a mountain range, a continent, or an entire planet. A leaf is a system; it is part of a larger system (a tree), which in turn is part of an even larger system (a forest) and so on.
The fact that a system is isolated from the rest of the universe means that it must have a boundary that sets it apart from its surroundings. The nature of the boundary is one of the most important defining features of a system, allowing us to establish three basic kinds of systems—open, closed, and isolated—with different types of boundaries.
```{admonition} Types of systems
There are three types of systems:
1. Open systems - can exchange mass and energy with surroundings
2. Closed systems - can exchange energy with surroundings
3. Isolated systems - no exchange of mass or energy with surroundings
In {numref}`earth-systems` the Earth system is an open system internally with free exchange of mass and energy. Since there is negligible exchange of mass with Earth and it's surroundings in the solar system, it is a closed system for mass. However, Earth gains and loses energy from within the solar system and is an open system for energy.
```
Since matter and energy is shared between systems and surroundings, it would be useful to quantify the flow of mass and energy. The amount of matter (or energy) that is transferred across the boundary of a system, and the rate at which it is transferred, is called a ***flux***. It is quantified in units of *mass or volume per time*.
The storage and movement of material and energy in a group of interacting systems is often represented as a ***box model***. A box model can be used to show the following essential features of a system:
1. The processes by which matter (or energy) enters and leaves the system, and the rates at which they do so.
2. The processes by which matter (or energy) moves among the various parts of the system internally, and the rates at which this happens.
3. The amount of matter (or energy) in the system at a given time and its distribution within the system.
As shown in {numref}`hydro-box-model`, the “boxes” in a box model represent the places where water (or energy, or whatever might be the material of interest) is stored for a period of time within the system. These storage places are called ***reservoirs***. When the flux of matter into a reservoir matches the flux out of that reservoir, we say that the reservoir is at ***steady state***. The average length of time water spends in any of these reservoirs is called its ***residence time***. The residence time of any material in any particular reservoir is deter- mined by the interaction of many factors, including the physical, chemical, and biologic properties of the material itself, the properties of the reservoir, and any external forces or processes acting on either the material or the reservoir.
```{figure} assets/water-cycle-science.jpeg
---
name: hydro-box-model
figclass: margin-caption
---
The global hydrological cycle is shown as a box model in this figure. Fluxes ($\pu{e3 km3 yr-1}$) and reservoirs ($\pu{e3 km3}$) with natural and anthropogenic cycles are included. Big vertical arrows show total annual precipitation and evapotranspiration over land and ocean ($\pu{e3 km3 yr-1}$), which include annual precipitation and evapotranspiration in major landscapes ($\pu{e3 km3 yr-1}$) presented by small vertical arrows; parentheses indicate area ($\pu{e6 km2}$). The direct groundwater discharge, which is estimated to be about $\pu{10 \%}$ of total river discharge globally, is included in river discharge. Image source: [Global Hydrological Cycles and World Water Resources | Science](https://www.science.org/doi/10.1126/science.1128845)
```
Under steady state conditions, the residence time can be calculated as shown in eq {eq}`residence-time`.
```{math}
:label: residence-time
\text{Residence time} = \dfrac{\text{Reservoir Amount}}{\text{Rate of addition/removal}}
```
### States and State Variables
In thermodynamics, it is important to explicitly define the properties (e.g., composition, energy, temperature, pressure, and volume) of a system -- these properties help us understand changes in the ***state of system*** during a chemical process. When state of the system changes, these properties change. ***State variables*** (or ***state functions***) are properties that are determined by the state of the system, regardless of how they were achieved. That is, when the state of the system changes, the magnitude of the change is only dependent on the initial and final states of the system and not on how the change is accomplished. Energy, pressure, temperature, and volume are examples of state variables. Several important state variables are not measurable in the absolute sense in a particular equilibrium state, though they do have fixed, finite values in these states. However, the changes between the equilibrium states can be measured. Changes in system variables can be denoted as shown below. The $\Delta$ is a standard mathematical symbol for change and refers to the *final* minus *initial* values as shown below.
$$
\begin{eqnarray*}
\Delta E &=& E_{final}-E_{initial}\\
\Delta P &=& P_{final}-P_{initial}\\
\Delta V &=& V_{final}-V_{initial}\\
\Delta T &=& T_{final}-T_{initial}
\end{eqnarray*}
$$
Thermodynamic properties can be extensive properties or intensive properties. ***Extensive properties*** are dependent on the amount of the material in the system (e.g., mass, volume.) ***Intensive properties*** are independent on the amount of material. They are the same for any subset of the material in the system (e.g., concentration, temperature, pressure). {numref}`state-properties` lists a summary of the most commonly-used state variables.
```{table} Commonly-used state extensive and intensive properties and their standard SI units. Note: Molar properties are normalized by moles of material while specific properties are normalized by mass.
:name: state-properties
| **Type** | **Extensive Property** | **Intensive Property** |
| --------- | ----------- | ------------------ |
| Mass | mass ($\pu{ g}$ or $\pu{ mol}$) | concentration ($\pu{ g L-1}$ or $\pu{ mol L-1}$) |
| | | density ($\pu{ kg L-1}$) |
| Volume | volume ($\pu{ L}$) | specific volume ($\pu{ L kg-1}$) |
| | | molar volume ($\pu{ L mol-1}$) |
| Thermal | heat capacity ($\pu{ J K-1}$) | specific heat ($\pu{ J K g-1}$) |
| | energy ($\pu{ kJ}$) | molar energy ($\pu{ kJ mol-1}$) |
| | enthalpy ($\pu{ kJ}$) | molar enthalpy ($\pu{ kJ mol-1}$) |
| | entropy ($\pu{ J K-1}$) | molar entropy ($\pu{ J K mol-1}$) |
| | free energy ($\pu{ kJ}$) | molar free energy ($\pu{ kJ mol-1}$) |
| Other | | pressure ($\pu{ atm}$) |
| | | temperature ($\pu{ K}$) |
```
Some thermodynamic properties are additive even after systems are transformed or when two systems are combined. These properties are *conserved*. Intensive properties are not conserved as they are not additive even within the same system. Only extensive properties are conserved.
```{dropdown} Example: Conservation of Mass
Conservation of mass Let's determine concentration in a solution ($S$) formed when one spoon ($\pu{5 mL}$) of seawater ($sw$) from Folly Beach ($=\pu{10.8 g L^{-1}}$) at $\pu{25 ^\circ C}$ is added to $\pu{1.0 L}$ of freshwater ($fw$) from Goose Creek reservoir ($=\pu{6 mg L^{-1}}$). Since, concentration is an intensive property and is not additive, it needs to be transformed to an extensive property. The extensive property equivalent of concentration is mass ($m = C\times V$, where $m$ is mass, $C$ is concentration and $V$ is volume).
$$
\begin{aligned}
m_s &= m_{sw} + m_{fw} = C_{sw}V_{sw} + C_{fw}V_{fw}\\
&= (\pu{5e-5 L})(\pu{10.8 g L-1}) + (\pu{1 L})(\pu{6e-3 g L-1})\\
&= \pu{6.54e-3 g}\\
V_s &= V_{sw} + V_{fw} \\
&= \pu{5e-5 L} + \pu{1 L}\\
&= \pu{1.00005 L}\\
&\text{Now, convert the mass back to concentration}\\
C_s &= \frac{m_s}{V_s}\\
&= \frac{\pu{6.54e-3 g}}{\pu{1.00005 L}} = \pu{6.5 mg L-1}
\end{aligned}
$$
```
### Phases and Components
Thermodynamic properties are used to define the state of the system. But we need to know how many of these properties are required to actually define a system? You can intuitively guess that this would depend on the complexity of the system. The number of properties required is determined by the number of phases and components present in the system.
A ***phase*** is defined as a homogeneous body of matter that has distinct boundaries with adjacent phases, and hence can be mechanically separable. Shape, orientation, and position of a phase in relation to adjacent phases is irrelevant, so a single phase may occur in several locations within a system. Examples of single phases include, quartz in granite and a salt solution. There are three common phases -- solid, liquid, gas or vapor. A system having only a single phase is *homogeneous* and a system with multiple phases is *heterogeneous*.
***Components*** are used to describe the chemical composition of a system. They are defined as the smallest set of chemical formula required to completely describe the composition of all the phases that make up a system. For example, in a $\ce{NaCl}$ solution (doesn't matter if there is water vapor or calcite precipitate in the solution), the combination of $\ce{NaCl}$ and describes the composition of all phases.
So, how many thermodynamic properties does one need in a system that $P$ phases and $C$ components? According to the ***Gibbs Phase Rule***, $C-P+2$ properties are needed.
*Why do we need to know thermodynamic properties?* This will become clear as we study the laws of thermodynamics.
Unlike what you have learned about the definition of a law according to the scientific method, the thermodynamics laws are concepts held to be true without derivation or proof. The basis of thermodynamics is built on three conservation statements: in any process, mass, energy, and momentum are conserved.
Originally, there were only three laws of thermodynamics that were formulated. However, when the fourth law was formulated, it was decided that this fourth law was even more fundamental than the original three laws. Instead of renumbering the original three laws, the fourth law was named the "Zeroth" law.