**1.1. System, Surroundings, and State**

Imagine you have a container that holds something you're interested in studying. This container keeps what you're studying separate from everything else around it. Let's say it's like a refrigerator keeping food separate from the rest of the house.

Now, think about what you want to focus on. That thing you're interested in studying is called the "system." Everything else around it is called the "surroundings." These definitions are important because they help us know what part of the universe we're looking at—the system. They also help us understand how the system interacts with what's around it.

Now, let's talk about the system itself. How do we describe it? Well, it depends on what the system is. For example, describing a glass of milk is different from describing the inside of a star. But let's keep it simple for now and talk about a system made of just one kind of gas.

So, what can we say about this gas system? We can describe it by looking at things like its volume (how much space it takes up), its pressure (how much force it's pushing out with), its temperature (how hot or cold it is), how much gas is there, and other things like how it reacts chemically. If we know all these things, we know everything we need to know about the system. We call this knowing the "state" of the system.

When the system's state doesn't seem to be changing, we say it's at "equilibrium" with the surroundings. This equilibrium idea is really important in thermodynamics, a branch of science dealing with energy and its transformations. Even though not all systems are at equilibrium, we often use it as a starting point to understand how systems work.

Now, let's talk about energy. Energy is another important thing to know about a system. It's related to all the other stuff we can measure about the system, like its temperature and pressure. Understanding how energy relates to everything else is what thermodynamics is all about.

So, how do we figure out the state of our system? We look at its physical properties, like pressure, temperature, volume, and how much stuff is there. These are easy to measure and have clear units (like liters for volume, atmospheres for pressure, and so on). Temperature wasn't always easy to measure, but now we have ways to do it accurately.

So, that's the basic idea: we focus on describing a system by looking at its physical properties, and we use those descriptions to understand how the system behaves and interacts with its surroundings.

**1.2. The Zeroth Law of Thermodynamics**

The Zeroth Law of Thermodynamics is a basic principle that helps us understand how temperature works in different systems. It's called the Zeroth Law because it's so fundamental that it's considered even before the First Law of Thermodynamics.

So, what is temperature? It's a measure of how much energy the particles in a system have. The higher the temperature, the more energy the particles have. Temperature is really important because it's related to the energy of a system.

Now, imagine two closed systems, A and B, meaning no matter can go in or out, but energy can move between them. If we bring these systems together, something interesting happens. Energy starts moving from one system to the other until their temperatures become the same. When this happens, we say they're at "thermal equilibrium." Even if the systems are big or small, this rule still applies.

The transfer of energy because of temperature differences is called heat. So, when heat moves from system A to system B, we say that heat has flowed from A to B.

Now, if there's a third system, C, and it's at the same temperature as system A, it means it's also at the same temperature as system B. This idea can apply to any number of systems. This is the Zeroth Law of Thermodynamics in action: If two systems are at the same temperature and one of them is at the same temperature as a third system, then the two original systems are at the same temperature too.

This rule might seem obvious, but it's super important in thermodynamics. It tells us how temperature works in different systems.

Another important thing to note is that a system doesn't remember its past states. So, even if a system started at a different temperature and then reached the same temperature as another system, they're considered to be in the same state. This shows us that what matters is the current values of the properties of the system, not how they got there.

**1.3. Equations of State**

Phenomenological thermodynamics is all about understanding how things work by doing experiments and making measurements in places like labs, garages, or kitchens. It's like learning how cooking works by actually trying out recipes.

For instance, when we're dealing with a certain amount of gas, we can control two important things: pressure (p) and volume (V). We can change pressure while keeping volume the same, or vice versa. And then there's temperature (T), another important factor. Even though we can change temperature independently from pressure and volume, experiments have shown that if we set a particular pressure, volume, and temperature for a gas sample, all the other properties of that gas sample have specific values. So, these three things—pressure, volume, and temperature—determine everything about our gas sample. And, by the way, we also have to keep track of how much gas there is, which we call the amount (n), usually measured in moles.

But here's the thing: we can't just set any values we want for pressure, volume, amount, and temperature all at once. Experiments have shown that only two out of these four variables are truly independent for a given amount of gas. Once we set two values, the third one has to be a certain value. So, there's a mathematical equation that lets us figure out what the third variable must be if we know the other two. This equation is called the equation of state.

The earliest gas laws, like Boyle's, Charles's, and Avogadro's, were based on these kinds of observations and experiments. They help us understand how gases behave under different conditions. For example, Boyle's law shows us how pressure and volume are related when temperature stays the same. Charles's law tells us how volume changes with temperature when pressure is constant. And Avogadro's law helps us understand how volume and amount are related when temperature and pressure stay the same.

These laws can be rewritten in terms of proportionality, like saying volume is proportional to 1/pressure or volume is proportional to temperature. And we can combine these into one equation, called the ideal gas law: pV = nRT. This equation tells us how pressure, volume, amount, and temperature are related in an ideal gas.

But to use this equation properly, we need to use the right temperature scale. In thermodynamics, we use the Kelvin scale, where zero means there's absolutely no heat energy. And there's a simple conversion between Kelvin and Celsius: K = °C + 273.15.

So, when we talk about gases and temperature in thermodynamics, we always use the Kelvin scale. And there's a set of standard conditions called standard temperature and pressure (STP) that we often use in experiments and calculations.

**1.4. Partial Derivatives and Gas Laws**

Partial derivatives are a tool from calculus that help us understand how one variable changes when another variable changes. Imagine you're on a roller coaster—the slope of the track changes as you move along, right? Well, in math, we use derivatives to describe how slopes change.

When we're dealing with equations that have many variables, like in thermodynamics, we use partial derivatives. These are like regular derivatives, but we're only looking at how one variable changes while keeping the others constant.

Let's take our ideal gas equation as an example. Say we want to know how pressure changes when temperature changes, but we're keeping volume and the number of moles constant. We can write this as a partial derivative, like ∂p/∂T at constant V and n.

Using calculus, we can figure out this derivative. First, we rewrite the ideal gas law so that pressure is alone on one side of the equation. Then, we take the derivative of both sides with respect to temperature while treating everything else as constants.

The result is a mathematical expression that tells us how pressure changes with temperature. If we plot pressure against temperature for an ideal gas, we'd get a straight line, and the slope of that line would be equal to nR/V, where n is the number of moles, R is a constant, and V is volume.

This slope can help us find the value of the ideal gas law constant, R, experimentally. So, partial derivatives like these are really useful because they can help us measure things we might not be able to measure directly.

Now, it's worth noting that this derivative suggests that at 0 Kelvin, a true ideal gas would have zero volume. But in reality, atoms and molecules do have volume, and gases don't behave perfectly at such low temperatures anyway. So, while the math suggests this, it's not entirely accurate in the real world.

**1.5. Nonideal Gases**

Real gases don't always behave like ideal gases. They deviate from the ideal gas law, and their behavior can be described using equations of state, but it's more complicated. Let's break down some key concepts:

** 1. Compressibility Factor (Z):** For 1 mole of gas, the ideal gas law can be rewritten to include a compressibility factor (Z). Z indicates how much a real gas deviates from ideal behavior. If Z is close to 1, the gas behaves more ideally.

** 2. Virial Equations:** These are equations that describe real gas behavior using power series in terms of pressure or volume. One common form is the virial equation in terms of volume, which includes virial coefficients like B, C, D, etc.

** 3. van der Waals Equation:** Proposed by Johannes van der Waals, this is a corrected version of the ideal gas law. It includes constants (a and b) related to particle interactions and size, respectively.

__4. Boyle Temperature (TB):__** **This is the temperature at which a real gas behaves more like an ideal gas. At TB, the second virial coefficient (B) is zero.

__5. Correspondence between Equations:____ __Comparing the van der Waals equation with the virial equation allows us to establish relationships between constants like a, b, and B. These relationships help us understand gas behavior.

__6. Derivatives of Equations of State:__** **We can use derivatives to determine how one state variable changes when another changes. For example, we can find expressions for how pressure changes with temperature using the van der Waals equation.

__7. Complex Equations of State:____ __Some equations of state are quite complex and involve numerous parameters. While they can accurately describe real gas behavior, they are challenging to use and often require computer assistance.

__8. Diagrammatic Representations:__** **State variables of gases can be represented graphically. These diagrams provide insights into gas behavior based on the equation of state.

**1.5. More on Derivatives**

Partial derivatives play a crucial role in thermodynamics, allowing us to understand how one state variable changes with respect to another. Two fundamental rules for partial derivatives are the chain rule and the cyclic rule.

__Chain Rule for Partial Derivatives:__

The chain rule allows us to find the partial derivative of a function with respect to one variable while considering that both variables are themselves functions of other variables. For example, if A depends on B and C, and both B and C depend on D and E, then the chain rule states:

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**partial A\partial D = partial A\partial B*partial B\partial D + partial A\partial C*partial C\partial D**

This rule essentially extends the concept of derivatives to functions of multiple variables, akin to how we would handle fractions algebraically.

__Cyclic Rule for Partial Derivatives:__

The cyclic rule is particularly useful in thermodynamics, especially when dealing with pressure, volume, and temperature. It states that for any state variable F, its total derivative with respect to temperature at constant pressure satisfies:

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**(partial F\partial T)p (partial T\partial V)F (partial V\partial p)T = -1**

This rule is independent of the equation of state and allows us to determine one derivative if we know the other two, providing a systematic way of understanding the relationships between pressure, volume, and temperature.

The cyclic rule can be rearranged into a more memorable form, often expressed as:

**(partial T\partial p)V ( partial p\partial V)T (partial V\partial T)p = -1**

**1.6. A Few Partial Derivatives Defined**

The expansion coefficient and the isothermal compressibility are two important properties of gaseous systems that help us understand their behavior under changing conditions.

__Expansion Coefficient (____β____):__

The expansion coefficient, denoted by **β**, measures how the volume of a gas changes when its temperature is varied at constant pressure. It is defined as:

**β= (1\V)(partial V\partial T)p**

This coefficient helps us understand how a gas expands or contracts when heated or cooled under constant pressure conditions. For an ideal gas, β equals **(R\pV)**.

__Isothermal Compressibility (____κ____):__

The isothermal compressibility, denoted by **κ**, measures how the volume of a gas changes when its pressure is varied at constant temperature. It is defined as:

**κ = -1\V (partial V\partial p)T**

This coefficient quantifies the gas's response to changes in pressure while maintaining a constant temperature. The negative sign ensures that **κ** is positive for gases, and for an ideal gas, κ equals **RT/p^2V**.

Both **β** and **κ** include a **(1/V)** term to make them intensive properties, meaning they are independent of the amount of gas.

These coefficients are related through the cyclic rule, as shown by the equation:

** **

**( partial V\partial T)p (partial T\partial p)V = -βκ**

This relationship allows us to express changes in volume under different conditions, even when it's challenging to maintain a constant volume in experimental settings.

**1.7. Summary**

Gases are introduced first in a detailed study of thermodynamics because their behavior is simple. Boyle enunciated his gas law about the relationship between pressure and volume in 1662, making it one of the oldest of modern chemical principles. Although it is certain that not all of the “simple” ideas have been discovered, in the history of science the more straightforward ideas were developed first. Because the behavior of gases was so easy to understand, even with more complicated equations of state, they became the systems of choice for studying other state variables. Also, the calculus tool of partial derivatives is easy to apply to the behavior of gases. As such, a discussion of the properties of gases is a fitting introductory topic for the subject of thermodynamics. A desire to understand the state of a system of interest, which includes state variables not yet introduced and uses some of the tools of calculus, is at the heart of thermodynamics.

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