Introduction

Abstract

This chapter introduces the macroscopic approach to matter and defines thermodynamic systems and their exchange with the surroundings. The state variables temperature and pressure are presented and linked via the ideal gas law (an equation of state for gas).

Thermodynamics is a scientific discipline firmly rooted in physics, chemistry and engineering with implications for the universe and thus biology, earth sciences and astronomy. It describes the behavior of matter on a macroscale, sets the rules for conversions of energy from one form into another, and governs the direction of processes. Engineers and physicists applying thermodynamics focus on energy conversions and conservation to develop efficient engines, refrigerators, or heat pumps. Chemists use thermodynamics to study chemical reactions and phase equilibria. Thermodynamics is key to understand many processes on Earth including adiabatic compression of air (Fohn winds), the lapse rate due to expansion of air, geothermal gradients in the inner Earth, the effect of pressure on freezing and boiling points, the strength of atmospheric circulation, the stability of minerals as a function of pressure, temperature, and composition, redox zonation in marine sediments and the evolution of biogeochemical cycles.

Thermodynamics is a science about the macroscopic world, i.e., the human scale and larger. Most of the thermodynamic concepts have been developed, and can be understood, without knowledge of the molecular structure. However, sometimes it is easier understood if we use our knowledge on the properties of individual molecules.

Thermodynamics is based upon a few statements, i.e., thermodynamic laws, that are based on years of observations and multiple clever experiments connected through theory. Most chemical thermodynamics textbooks present four laws, while engineering treatments are often limited to the first and second laws. The first law is the conservation of energy: energy can be converted from one form into another, but overall, no energy is lost or gained. The second law is concerned with spontaneous change and states that energy is, overall and over time, increasingly being dispersed. Alternatively formulated, the second law reads that spontaneous processes are those which increase the entropy of the universe. The term entropy will be introduced and explained later.

1 Macroscopic Approach, Yet Everything is Made of Atoms

Chemists deal with atoms, ions, and molecules and aim to explain the properties of matter and predict the synthesis of compounds using atom-level knowledge. Although thermodynamics is a core subject in academic chemistry curricula, it deals with macroscopic systems, i.e., large amounts of matter rather than a few molecules. Matter exists in three phases: solid, liquid or gas that differ in the number of particles (atoms, molecules, ions) per volume, hence the distance and interactions among particles, and their spatial orientations and distribution (Fig. 1.1). In a gas molecules are far apart and move randomly through a volume that is primarily empty space. A gas is thus homogenous, fluid, and compressible. Moreover, for an ideal gas the identity and interaction of the particles can be ignored at the macroscopic level. In a liquid the particles are close to each other but are free to move relative to each other, thus a liquid is fluid but less compressible. In a solid constituent, atoms, ions, and molecules are generally close to each other and in fixed positions. Moreover, their identity governs the physical and chemical properties of the solid. Pure substances, matter with a homogenous and definite chemical composition, may exist as a solid, liquid or gas depending on the external conditions (temperature, pressure). For instance, at the Earth surface, H₂O, water, may exist as solid (ice), liquid (water) or gas (water vapor). A single phase may contain multiple substances, e.g., air consists of nitrogen, oxygen, carbon dioxide, etc.

Fig. 1.1

The most common states of matter: solid, liquid and gas

Thermodynamic theory is general and applies to solid, liquid, and gas phases. However, most thermodynamics has been developed based on the behavior of gas. Consider a container filled with a gas. Gas exerts a pressure on the container walls, and this can be understood in terms of randomly moving particles that collide with the wall. The result of these collisions is a force perpendicular to the wall. Gas pressure (P) is thus defined as:

$$ P = \frac{F}{A} , $$

(1.1)

where F = force (N) and A = area (m²). Pressure acts equally in all directions in a gas and is thus a scalar quantity. Pressure has the unit Nm⁻² or Pa (the SI unit), but multiple other units are commonly used (Table 1.1).

Table 1.1 Pressure units and their relation to SI unit Pa

Because of the large amount of space between the molecules in a gas (Fig. 1.1), and thus limited interactions, gases with different compositions will mix well. If they do not react, the total pressure of the gas mixture is the sum of the partial pressure of each gas contributing to the mixture. The partial pressure of gas A (P_A) is the pressure it would have if it were alone. Dalton’s law states that the total pressure (P_tot) for a mixture of gases A, B, C, is given by:

$$ P_{tot} = P_{A} + P_{B} + P_{C} $$

(1.2)

For instance, for air (78% N₂, 21% O₂) with a total pressure of 1 atm, the partial pressure of nitrogen and oxygen gases would be 0.78 and 0.21 atm, respectively.

The particles moving freely in a gas have kinetic energy. Temperature is a measure of how much kinetic energy each particle has on average. The higher the temperature, the more energy a system has, all other factors being the same. Temperature is not a form of energy, but a measurable parameter to compare amounts of energy of different systems. Most solids, liquids, and gases expand roughly linearly with increasing temperature because temperature governs the kinetic energy of particles. The expansion of liquids such as mercury and alcohol are commonly used to quantify temperature using the freezing (0) and boiling (100) points of water for calibration. This centigrade scale has been superseded by the closely related Celsius scale anchored to the triple point of water (0.01 °C, see Sect. 4.3) at which ice, water and gaseous water are at equilibrium. The Celsius scale is widely used but the scale is arbitrary: it is based on water and its macroscopic phases. The thermodynamic, or absolute, temperature scale is independent of the substance used in the thermometer and goes down to the minimum possible temperature of −273.15 °C. The absolute scale, or Kelvin scale, is expressed in K (without a degree sign). Temperature in degrees Celsius and Kelvins are related:

$$ K\, = \,^\circ {\text{C}} + 273.15 $$

(1.3)

This Kelvin scale should be used in thermodynamics.

Having defined pressure (P) and temperature (T). We can define two reference states: standard temperature and pressure (STP) refers to P = 1 bar and T = 273.15 K = 0.0 °C, while standard ambient temperature and pressure (SATP) refers to T = 298.15 K = 25 °C and 1 bar for P (1 bar = 10⁵ Pa).

2 Ideal Gas Law

Experimental studies on the physical properties of gases have resulted in a few empirical laws that were eventually combined into what we call today the ideal gas law. Boyle (1627–1691) studied the relationship between the pressure and volume of a fixed amount of gas at constant temperature and observed that the product of pressure and volume is constant. Boyle’s law states that the volume (V) of a given amount (e.g., mass) of a gas is inversely proportional to pressure (P) when the temperature is constant:

$$ V \propto \frac{1}{P}, $$

(1.4a)

where $\propto$ represents the proportionality symbol, or alternatively:

$$ P \cdot V = constant. $$

(1.4b)

Subsequent work by Charles (1746–1823) showed that at constant pressure (P) the volume (V) of a given amount of a gas is directly proportional to its absolute temperature (T). Charles’s law states:

$$ V \propto T. $$

(1.5a)

Or alternatively,

$$ \frac{V}{T} = constant. $$

(1.5b)

Next, Avogrado (1776–1856) related volume (V) and the amount (n in moles), but at fixed temperature and pressure. Avogrado’s law states:

$$ V \propto n , $$

(1.6a)

Or alternatively,

$$ \frac{V}{n} = constant. $$

(1.6b)

These three gas laws can be combined because volume (V) appears in all three:

$$ V \propto \frac{nT}{P}. $$

(1.7)

The proportionality symbol ($\propto )$ can be turned into an equality through the introduction of a proportionality constant (R):

$$ V = R \cdot \frac{nT}{P}, $$

(1.8a)

This equation is usually written as:

$$ PV = nRT, $$

(1.8b)

which is the well-known ideal gas law. R is the universal gas law constant, which has the value of 8.314 J mol⁻¹ K⁻¹ in SI units, when n is expressed in moles. Note that thermodynamic data are often provided in various units (e.g., pressure in bar, atm or tor rather than Pa; volume in liters rather than m³); the units and values of R then need modification to maintain consistency of units. For instance, the value of R is 0.0820568 when expressed in L atm mol⁻¹ K⁻¹.

3 System, Surrounding and Equations of State

Systems are a central concept in thermodynamics. The object of interest is defined as the system while everything else, the rest of the universe, is defined as surroundings. A system can be separated from the surroundings via imaginary boundaries or physically real boundaries such as the wall of a container. The system can be a chemical reaction taking place in a solution (e.g., calcite dissolution), it can be your cup of hot coffee cooling while you are enthusiastically reading this text, or the entire System Earth receiving short-wave radiation from the sun and emitting long-wave radiation to the surrounding cosmos.

A system interacts with its environment via mass transfer, or energy transfer via heat or work exchange (Fig. 1.2). An isolated system does not exchange heat, work, or matter with the surroundings. A closed system does not exchange matter but can exchange energy with the environment. An open system exchanges matter with the surroundings and may exchange energy (as heat or work) as well. Changes to the system are called isothermal if the temperature is kept constant, isobaric if pressure is kept constant, isochoric or isovolumetric if the volume is constant, and adiabatic if no heat is exchanged, i.e., if it occurs in thermal isolation (Fig. 1.3).

Fig. 1.2

Open systems exchange matter and energy; Closed systems exchange energy but no matter; Isolated system can exchange energy nor matter

Fig. 1.3

Isothermal, adiabatic, isochoric and isobaric changes to a thermodynamic system (e.g., a gas)

The state of a macroscopic system is described using observable quantities, i.e. state variables, such as pressure, temperature, volume, and number of moles. Equations of state relate the various state variables of a system. The ideal gas law (Eqs. 1.8a and 1.8b) is an equation of state for an ideal gas. For one mol of a gas (n = 1), the system is fully determined with two out of the three variables (V, T or P). For instance, for a standard temperature and pressure (STP, 0 °C and 1 bar) the gas would then have a volume of 22.4 L. For SATP conditions (25 °C and 1 bar) the molar volume (V_m) of an ideal gas would be about 24.5 L mol⁻¹.

Thermodynamic variables are referred to as intensive if they are independent of the amount, while extensive variables are additive, i.e., they depend on the sample size. Of the four variables in the gas law (Eqs. 1.8a and 1.8b), P and T are independent of the amount of gas and are referred to as intensive variables, while V and n are proportional to the amount of gas and are extensive variables. In thermodynamics, changes in extensive quantities are associated with changes in the respective specific intensive quantities and their product has the dimension of energy: e.g., PV in the ideal gas law. The ratio of two extensive variables can be an intensive variable, e.g., density (mass divided by volume).

The energy of a system is related to all the other measurables of the system via equations of state. Thermodynamics literally means ‘heat movement’ because it describes how the energy of a system relates to measurable variables. If the state of a system shows no tendency to change, i.e., the system is at equilibrium, the state functions have values which are independent of the history of the system. Consequently, the changes in a function of state do not depend on the route by which one goes from one state to another, i.e., they are path independent. Equilibrium thermodynamics, the topic of this course, focuses on differences in system states and we use the symbol Δ to indicate changes in state variables and functions.

Box 1 Math intermezzo: Partial derivatives and state functions

Equations of state usually involve multiple variables. The total differential of a function F (x,y,z) is defined as:

$$ dF = \left( {\frac{\partial F}{{\partial x}}} \right)_{y,z} dx + \left( {\frac{\partial F}{{\partial y}}} \right)_{x,z} dy + \left( {\frac{\partial F}{{\partial z}}} \right)_{x,y} dz $$

(1.9)

where the derivative of F is taken with respect to one variable at a time while the others are kept constant. The first term $\left( {\frac{\partial F}{{\partial x}}} \right)_{y,z}$ is the derivative of F with respect to x only and is the partial derivative. These partial derivatives sometimes reveal relationships between state variables and are often used to formally define basic properties such as heat capacity, compressibility, etc.

As an example, suppose we aim to quantify the pressure dependence on temperature of an ideal gas, assuming that the volume and number of molecules remain constant. The relevant partial derivative is

$$ \left( {\frac{\partial P}{{\partial T}}} \right)_{V,n} $$

We start with rewriting the ideal gas law (Eqs. 1.8a and 1.8b) to isolate pressure on one side of the equation:

$$ P = \frac{nRT}{V} $$

Next, we take the derivative of both sides with respect to temperature T, while considering n and V constant, and obtain

$$ \left( {\frac{\partial P}{{\partial T}}} \right)_{V,n} = \frac{\partial }{\partial T}\left( \frac{nRT}{V} \right) = \frac{nR}{V} \frac{\partial }{\partial T}T = \frac{nR}{V}. $$

(1.10)

In this way, we have analytically derived how pressure varies with temperature and have revealed the phenomenological law of Guy-Lussac stating that, for a given mass (n) and volume (V), the pressure is proportional to absolute temperature, i.e., ΔP/ΔT = constant.