Why do we study chemistry? There are three reasons. First, chemistry is fun. That means there are four reasons why we do chemistry. First, chemistry is fun, second, we want to understand why a chemical reaction happens, third we want to harness the energy released in a chemical process, and fourth, we want to exploit the properties of the products formed in a chemical reaction. To understand the second item, we need to understand what the Gibbs energy is and why a system will evolve until its Gibbs energy is a minimum. To understand the third reason we need to understand what makes a reaction exothermic, that is, we need to define enthalpy and what controls it. To understand the fourth reason we need to understand phases, intermolecular interactions and the quantum mechanics that is behind them. Our study of thermodynamics will allow us to address items two and three. As we shall see, we will need to define enthalpy (first law), entropy (second and third laws) and fix the relationship between them (solved in the third law). The next three chapters will define these laws of thermodynamics. The subsequent two will develop these fundamental themes and apply them specifically to the understanding of reactivity, thermochemistry and equilibrium phenomena.

Why do you need to understand thermodynamics? The good academic reason is that you cannot understand chemistry fundamentally without some knowledge of thermodynamics. The good practical reason is that you might injure or kill yourself or others if you do not. The following is an example that is related to an actual laboratory accident.

Consider the reaction of 15.0 g of MgN with H₂O to produce MgO, N₂ and H₂ in a glass vessel with a volume of 1.00 l. What difference does it make if you use: (a) 10 g, (b) 150 g or (c) 1000 g of H₂O? What about an open versus a closed vessel?

To arrive at approximate answers, let us apply the rules for problem solving that were outlined in Chapter 1.

1. start with a balanced reaction (include phases):
   
   
   
   
2. Evaluate the physical processes that occur, draw a diagram, or chart the number of moles. Is the reaction exothermic or endothermic? It is exothermic, so the temperature is going to increase. We need to develop methods of calculating the correct answers for both energy release and temperature increase because you cannot look up this reaction in a table.
   
3. Draw a straight line or otherwise present results appropriately. In this case we are not going to present data in a graph. Instead, a tabular representation of the moles of reactants and products and the reaction conditions (i.e., a reaction chart) is more appropriate. What is the initial composition? What is the final composition? What is the final temperature and pressure? How should the reaction be run?
   
4. Answer with appropriate units, always use SI units unless specifically asked for other units. Answers: Closed system = grave danger in all three cases. Open system: (a) utter destruction; (b) geyser; (c) toasty.

7.1 Some definitions and fundamental concepts in thermodynamics

• System – the material in the process under study.
  
• Surroundings – the rest of the universe.
  
• Open system – system can exchange material with surroundings.
  
• Closed system – no exchange of material.
  
• Isolated system – no exchange of material or energy.
  
• Thermodynamic equilibrium – p, T, composition are constant and the system is in its most stable state. Also called chemical equilibrium.
  
• Thermal equilibrium – two systems have same temperature.
  
• Metastable state – a system in which p, T and composition are constant; however, the system could relax to a lower energy state if it were sufficiently perturbed. A system in a metastable state is in a state of metastable equilibrium, and so can be described consistently (but with caution) by thermodynamic methods.
  
• Steady state – p, T and composition are constant but are maintained at these values in an open system by a balance of inward and outward fluxes of matter and energy.
  
• Isothermal: ΔT = 0; isobaric Δp = 0; isochoric ΔV = 0.
  
• Adiabatic in a thermodynamic sense means that the system exchanges no heat, q = 0.
  
• Diathermal barrier – allows for the transport of heat but not matter between systems.
  
• Temperature T in kelvin (absolute temperature) is related to temperature t in degrees Celsius by

(7.1)

7.2 Laws of thermodynamics

To allay the suspense, we will state up front the most important rules in thermodynamics. Combining the laws of thermodynamics with the understanding that all systems evolve to lower their Gibbs energy (or Helmholtz energy, as appropriate), and that the rates of the reactions that lead to equilibration (or a steady state) are governed by concentrations, temperature and activation energies, we can essentially explain all of chemistry. Thus, is it very important that we state the laws of thermodynamics and fully understand their implications and application, as we will over the course of the next few chapters.

• Zeroth Law of Thermodynamics

• First Law of Thermodynamics

• Second Law of Thermodynamics

• Third Law of Thermodynamics

We have already encountered the Zeroth Law in Chapter 2. Essentially, we require it and a definition of temperature before we can make any other statements about thermodynamics. With it in hand, we can proceed to a discussion of the First Law.

7.3 Internal energy and the first law

The total energy of a system is its internal energy, U. The internal energy is the sum of the kinetic and potential energies of the molecules composing the system as well as rotational and vibrational energy and energy involved in chemical and physical interactions. Changes in internal energy are denoted

(7.2)

Note the sign convention. Whenever we calculate changes – that is, for all delta variables such as ΔU above – we will always take the final state minus the initial state. Equation (7.2) can be used to state the first law of thermodynamics in an alternate fashion:

The internal energy, U, of an isolated system is constant, ΔU = 0.

For a system in contact with its surroundings

(7.3)

Therefore,

(7.4)

that is, the internal energy change in the system must be equal and opposite to that of the surroundings with which it has thermal contact.

Heat and work, which we shall define precisely shortly, act equivalently with respect to changing a system’s internal energy.

(7.5)

Equation (7.5) is central to thermodynamics, much of which revolves around determining how to calculate or measure q, the amount of heat transfer to or from a system, and w, the work done on or by a system. The first law is an expression of the conservation of energy. Energy can be converted from one form to another, for example between work and heat, but cannot be created or destroyed. Equation (7.5) means that the units of work, heat and energy are the same.

Equation (7.5) also sets the sign convention for q and w. A positive value of q or w leads to an increase in the internal energy (a positive value of ΔU). A negative value of q or w leads to a decrease in internal energy (a negative value of ΔU). In other words, when the system does work on the surroundings the work is done at the expense of the internal energy. This corresponds to a negative value. When work is done by the surroundings on the system, the internal energy increases and a positive value of work is performed.

7.3.1 State versus path functions and reversibility

The internal energy is a state function. It depends only on the state of the system and not how it came to that state, just like the variable z in Fig. 7.1, the value of which depends on p and T but not how the system arrived at this point. Examples of state functions are temperature, volume, pressure, enthalpy, entropy and Gibbs energy. State functions are interrelated and if we know the composition of a system and any two state functions, we can work out the values for all other state functions. Note that since U is a state function, so too is ΔU. The work and heat evolved in going from the initial state to the final state depend on the way the system transforms from initial to final state. Heat and work are path functions. For instance, along an isochoric path for a closed system (and no non-pV work), w = 0; thus, ΔU = q. On the other hand, in an adiabatic process there is no exchange of heat. Thus, q = 0 and ΔU = w for an adiabatic process. When a process is non-adiabatic (q ≠ 0), the amount of work done is path-dependent and will have to change accordingly if the change in internal energy along this non-adiabatic path is to be the same as along some other path that connects the same initial and final states.

In a reversible process, at each step of the process the direction of change can go either forward or backward. In an irreversible process, the direction of change is always only in one direction. For example, at equilibrium, room-temperature liquid water and gaseous water are connected by a reversible chemical process. Water molecules are able to evaporate from the surface of the liquid water to enter the gas phase. They do so at exactly the same rate as water molecules condense from the gas phase to enter the liquid phase. However, a piece of iron irreversibly oxidizes to form rust when a polished iron surface is exposed to air. Oxygen and water molecules decompose on the surface of the iron to form various iron oxides. The reverse process (decomposition of the oxides) does not occur spontaneously at room temperature.

How can a system change state along a reversible path? Think of the role of fluctuations. Collisions lead to the establishment of the Maxwell–Boltzmann distribution of populations among energy levels. On average, these distributions and all thermodynamic parameters such as p and T are constant. However, there also exist fluctuations around these mean values. The probability of any given fluctuation in energy decreases exponentially with the magnitude of the fluctuation away from the mean value. Nonetheless, these fluctuations are a part of the equilibrium distribution. As long as a system is evolving slowly enough such that its fluctuations lie within a standard deviation of the mean value, there is no way to tell that the system has departed from equilibrium. Thus, a slowly changing system can maintain a reversible path that is described by equilibrium conditions.

Heat and work are path-dependent. Thus, their values differ for reversible and irreversible paths connecting the same two states. The internal energy change does not depend on the path. Consider infinitesimal changes in heat and work (, respectively) along these two paths that lead to an infinitesimal change in internal energy dU.¹ Denoting the reversible and irreversible paths rev and irrev, respectively, we can write

From the First Law, this implies that

That is, by changing the path what is taken from heat is given to work to affect the same net change in internal energy. For those of you who are true sticklers for detail, once the path is defined an inexact differential becomes exact. For instance, along an adiabatic path, the change in work is exact. Thus, one is justified in writing dw instead of in this case.

Summarizing we state:

Irreversible process

• Need not take place at an infinitesimal rate.
  
• Passes through nonequilibrium states.
  
• Intensive variables need not be continuous across boundaries.
  
• ΔU_forward = −ΔU_reverse.

7.3.1.1 Example

Calculate the internal energy change and the molar internal energy change when 100 J of heat are transferred into 2 mol of ideal gas while simultaneously 100 J of work are performed by the surroundings on the ideal gas.

In this example q = +100 J and w = +100 J. Therefore, the internal energy change is

and the molar internal energy change is

7.4 Work

Work is the transfer of energy that makes use of the uniform motion of atoms. As a reflection of the fact that the work done depends on the path taken, there are numerous expressions for the amount of work done. The correct expression is path-dependent – that is, it depends on how the system is changing and how it is interacting with its surroundings.

First let us define mechanical work. The infinitesimal amount of work dw done when an object is displaced by ds under the aegis of a force F is given by

(7.6)

where F_s is the component of the vector F along the direction of ds. Displacing the object from an initial point i to a final point f, the total work performed on it is given by the integral

(7.7)

Work (regardless of whether it is mechanical or thermodynamic) is path-dependent. It depends on the path taken and how the force changes along that path. Work is a process variable, it is not a property of the object. The object does not contain a certain amount of work. Instead, work is a quantity that is exchanged between systems. Work is only defined with respect to motion.

Remember that if a parameter is constant it can be pulled out of the integral. Therefore, for a constant force moving an object a total distance Δs, the integration is trivial and the work is

Constant force:

(7.8)

Note that for this example of mechanical work, we have taken dw to be an exact differential. As long as the force acting on the object is derivable from a potential, dw is an exact differential. This is the case whenever the system is a conservative system. When some or all of the forces acting in the system are not derivable from potentials, the system is dissipative. In mechanical systems this arises, for example, when friction is present. When we speak of thermodynamic systems in general, we find that typically the forces are not derivable from potentials. Therefore, we must discuss the general case of thermodynamic work treating it as an inexact differential.

Now we turn to thermodynamic work. One of the most frequently encountered forms of work for chemical systems is the expansion of a gas against an external pressure p_ex, also known as pV work. In calculating thermodynamic work, we need to take account of the sign convention imposed by the First Law of Thermodynamics: when the system performs work on the surroundings, its internal energy must go down and the sign of work is negative. Let us assume that p_ex is applied uniformly over our system (say an ideal gas contained in a box with one movable wall). The infinitesimal work done when the volume changes is

(7.9)

The negative sign must be inserted to enforce compliance with the First Law: expansion must lead to negative work. We confirm this by performing the integration. If the volume changes from its initial value V_i to a final value V_f, then the total work is

(7.10)

For a constant external pressure, we again move the constant out of the integral and evaluate explicitly,

Isobaric work:

(7.11)

Consistent with the First Law, expansion corresponding to ΔV > 0 leads to w < 0.

Similar to pV work, surface expansion work is given by the product of the surface tension γ and the surface area σ

(7.12)

Work can be done by moving electrons. This is one of the most common forms of non-pV work. The electrical work of transporting electrical charge Q through an electrical potential φ is

(7.13)

This is equivalent to the work performed to charge a capacitor at the applied potential difference φ. If the current I and potential are constant, then the electrical work is the product of electrical power P_el (power, electrical or otherwise, has SI units of watts W = J s⁻¹) and time t

(7.14)

At this point it is good to recall Ohm’s law,

(7.15)

which allows us to relate electrical power to current, resistance R as well as the charge passed,

(7.16)

7.4.1 Example

Calculate the work done by a gas when expansion is performed isothermally and reversibly.

It is important to define the path to calculate the work appropriately. Defining the path often involves deciphering the vocabulary of thermodynamics and chemical change. If the expansion is reversible, then the internal and external pressures must be equal at every step along the path of expansion or compression. This means p_int = p_ex = p. The temperature does not change in an isothermal process (ΔT = 0). We know from the ideal gas law that p = nRT/V. Substituting this result into Eq. (7.10) we find

Take a look at the form of this integral. It is a frequently encountered form, which will have a familiar solution ∫^b_adx/x = ln b − ln a.

Reversible, isothermal work:

(7.17)

7.4.2 Example

One mole of an ideal gas expands from 5 bar to 1 bar at 298 K. Calculate the work done for (a) a reversible expansion and (b) an expansion against a constant pressure of 1 bar.

1. From Eq. (7.17)
   
   
   
   

   For an ideal gas, p_iV_i = p_fV_f, thus V_f/V_i = p_i/p_f and

   
   
   
   
   
   
   
   
   
   
2. In this case of irreversible expansion, the external pressure is constant. The internal pressure changes from the initial value and decreases until expansion stops when the internal and external pressures equalize.

This example indicates what is a general result: the maximum work is done on the surroundings when the process is carried out reversibly. Note that the negative sign means that the internal energy is decreasing as a result of the system performing work on the surroundings.

7.5 Intensive and extensive variables

Some variables in thermodynamics are independent of the mass of the system. These variables, for example pressure and temperature, are known as intensive variables. Other variables are dependent upon the mass when the intensive variables are held constant. These so-called extensive variables include volume and internal energy. Another way to express the difference is to state that intensive variables can be measured at any point in the system (i.e., the pressure or temperature of a system at thermodynamic equilibrium is everywhere the same), whereas extensive variable are properties of the system as a whole.

Each intensive variable α has a related conjugate variable β. Conjugate variables are related by a particular work process. We can determine what variables are conjugate variables by noting first that the product of a variable and an infinitesimal change in its conjugate variable is equal to work. Work and energy have the same units. Therefore, any combination of variables, the product of which has units of energy, is a candidate pair. Finally, suppose that α has the same value in the system and surroundings. The conjugate extensive variable β is the variable that satisfies the equation

(7.18)

Conjugate extensive and intensive variables correspond, respectively, to the generalized coordinates and generalized forces of mechanics. In analogy to expressions for the potential energy of mechanical systems involving generalized coordinates and forces, conjugate variables are related by the negative of the partial derivative of the internal energy, holding all other independent extensive variables constant. That is,

(7.19)

When an extensive quantity is divided by the amount of substance, it becomes an intensive quantity. Note that ‘amount of substance’ should be used in place of the colloquial ‘number of moles’ just as ‘mass’ is referred to and never ‘number of kilograms.’ The result is called a molar quantity. Some examples are given in Table 7.1. Always check to make sure that you are using consistent units. Molar quantities will often be highlighted with the subscript m. When an extensive quantity is divided by the mass, the result is called a specific quantity. Similarly, if an extensive quantity is divided by volume, area or length, the results are called volumic, areic and lineic, respectively.

7.6 Heat

If the energy of a system changes such that the temperature changes, we say that energy has been exchanged in the form of heat. Heat is the exchange of energy that makes use of random motions of atoms. If a process occurs such that it cannot exchange heat with its surroundings, the process is adiabatic.

The most fundamental definition of heat is

(7.20)

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