Thermodynamics Part 1: Work, Heat, Internal Energy and Enthalpy



Thermodynamics Part 1: Work, Heat, Internal Energy and Enthalpy



James Joule
(1818-1889)



Common Energy Units
Unit SI Equivalent
British thermal unit (BTU) 1,055 J
Calorie (cal) 4.184 J
Electron volt (eV) 1.602 × 10-19 J
Erg (erg) 1.000 × 10-7 J
Foot pound (ft-lb) 1.356 J
Kilowatt-hour (kwh) 3.600 × 106 J
Liter atmosphere (L-atm) 101.3 J

An Open, Diathermic System


A Closed, Diathermic System


An Open, Adiabatic System


A Closed, Adiabatic (Isolated) System



Introduction

Thermochemistry studies the contribution of chemical processes to thermodynamics, the science of energy transfer. Energy is often (unsatisfyingly) defined as the ability to do work, and can be classified as one of two types. Kinetic Energy is energy of motion, such as that possessed by a baseball thrown by a pitcher, a bullet shot from a gun, or a translating H2 gas molecule. Potential Energy is energy of position (technically position in a gradient "field"). The most familiar form of potential energy is gravitational, such as Newton's apple as it hung in the tree. More relevant to chemistry is the potential energy due to position in an electric or magnetic field, such as solvated ions, or atoms transferring charge when forming compounds or molecules. Another important form of potential energy is that found in a coiled spring. Bonded atoms exhibit strikingly similar behavior (vibrational motions) to the action of springs. Potential energy is often thought of as "stored" kinetic energy, meaning that bodies remain stationary in a potential field while held in place by some force, and upon change in this force (such as breaking the twig holding an apple, or breaking the bond between two atoms), potential energy is converted to kinetic form (the apple "falls" or the molecule "dissociates"). The energy unit is a derived physical quantity having dimension energy = mass × length2 × time-2. The SI unit of energy is the Joule (J):

Energy is often measured in other units, specific to particular applications. Some common examples are shown to the left.

Energy manifests itself in many forms:

  1. Nuclear Energy - The potential energy required to bind nucleons in the nucleus
  2. Light Energy - The potential energy possessed by the oscillating electric and magnetic fields that make up electromagnetic radiation
  3. Chemical Energy - The potential energy stored in the electrostatic bonding relationships among atoms in a molecule
  4. Electrical Energy - The potential energy involved in initiating and maintaining electron flow
  5. Mechanical Energy - The energy generated (or stored) by machines which induces (or results from) concerted motion processes in a system
  6. Heat Energy - The kinetic energy associated with random motion of matter including the vibratory and rotatory action of molecules
This lesson will focus on heat and mechanical energy. Subsequent lessons will separately deal with the implications of chemical, electrical, and light energy in chemistry.

The First Law of Thermodynamics

We begin by defining the Universe, or all-encompassing environment. The universe is partitioned into two sections- the system, consisting of the chemical or physical process of interest, and the surroundings which includes everything (and I do mean everything) else. This partitioning at first seems ridiculous, as the system is obviously an infinitessimal component of the universe. However in the realm of thermodynamics, or energy transfer, this definition makes it possible to solve many practical problems. We now categorize systems according to the following definitions.

  • A closed system does not allow matter in or out
  • An open system allows matter in or out
  • An adiabatic system does not allow energy (in this context heat energy) in or out
  • A diathermic system allows energy in or out.
In addition, we define an isolated system to be closed and adiabatic, or one that allows no heat or matter in or out.

We are now positioned to state the First Law of Thermodynamics:


"The universe is an isolated system"

Since no energy is allowed in or out we say that energy is conserved in the universe, or that there is a fixed amount of energy present and that it is forever interconverted among its various forms. It should be noted that energy can also be converted to matter according to Einstein's relationship:
where E is the energy, m the mass, and c the speed of light. Mass-energy interconversion is an important mechanism in the incredibly large potential field required to bind protons in a nucleus. Our study of the first law will focus on heat and mechanical energy transfer between a system and its surroundings.




Specific Heat Values
Substance
S.H.
(J × gram-1 × oC-1)
S.H.
(cal × gram-1 × oC-1)
Air
1.01
0.241
Cu(s)
0.385
0.0920
Al(s)
0.900
0.215
Au(s)
0.129
0.0308
Fe(s)
0.444
0.106
H2O(s)
2.03
0.485
H2O(l)
4.18
1.00
H2O(g)
2.01
0.480






An electrostatic potential map for water shows where electron density congregates. Regions in blue are electron-poor, varying to regions in red which are electron-rich.


Heat

The terms heat and temperature are often used in the same context, but this practice is misleading (temperature was defined for us in a previous lesson). Consider inputting exactly the same amount of heat energy into two separate bodies. The temperature change will most likely differ in the two, even if the bodies are made of the same material (but are different size). The source of this temperature difference is called the heat capacity (C) of the material. Heat capacity measures the ability of a material to store or release heat without changing temperature- in other words a substance with a low heat capacity (small C value) will experience a sharper increase in temperature upon input of a fixed amount of heat relative to a substance with a high heat capacity (large C value). Trends in heat capacity have an interesting correllation to trends in conductivity- materials with high heat capacity are generally good electrical insulators and those with low heat capacity are good conductors of electricity. The expression for heat capacity is:

Two factors contribute to heat capacity. As the mass (m) of material present increases, C increases. Heat capacity is also dependent upon the internal makeup of matter- its intermolecular forces (defining its phase) and intramolecular forces (defining its bonding). These internal contributions are intensive or independent of the amount of matter present and are called the specific heat (S.H.) of the substance. Specific heat measures the amount of energy required to raise one gram of material one degree in temperature. Depending on application, S.H. is presented using one of two preferred energy units. Some examples are displayed to the left. These values can be converted to molar specific heats by multiplying by the molar mass of the material, for instance for liquid water: 4.184 (J × g-1 × oC-1) × 18.016 (g × mol-1) = 75.38 (J × mol-1 × oC-1).

The high specific heat of H2O(l) arises from the exceptionally strong intermolecular forces which bind water molecules into a liquid. This effect, called hydrogen bonding arises from the significant difference in electronegativities of O that is bonded to H (this phenomenon also occurs when other highly electronegative elements, namely F and N, are bonded to H). The more electronegative oxygen atom draws electron density, creating a net negative region in the molecule. This leaves the hydrogens electron-deficient, so they constitute a net postive region. In a bulk sample, negative ends of water molecules attract and align with the positive ends of their neighbors, creating a highly structured environment. A large amount of energy (44 kJ per mole) is required to disrupt these so-called hydrogen bonds, freely sendng water from the liquid to gaseous state. Prior to the onset of unrestricted phase change (which we call boiling), the input heat energy is distributed in the liquid by increasing the rotations, vibrations, and (mainly) the translation of individual water molecules. Because so many of these storage modes are available, water is a very effective temperature regulator (taking in huge quantities of heat with small temperature change), and is therefore essential to temperature control in processes as diverse as internal combustion engines and climate systems. The vast amount of water on Earth limits the daytime temperature changes to roughly a maximum of 20 oC, in spite of the planet's massive solar energy exposure. At night, the opposite process occurs; the heat energy stored by Earth's oceans is slowly released into the atmosphere to keep temperatures relatively stable without the presence of sunlight. Contrast this to the Moon, which is the same distance as Earth from the Sun, but is devoid of bodies of water. Temperature changes from night to day vary as much as several hundred oC. If such large quantities of water were not present, it is hard to discern what impact such drastically varying temperatures would have had on the development of life on Earth.

Heat energy is symbolized by q. There is an important sign convention assigned to q to differentiate between the input or removal of heat. When q is negative, a system is losing heat energy. This process is called exothermic. When q is positive, energy is input into the system and the process is called endothermic. Systems in thermal contact with their surroundings are able to exchange heat energy. Energetic changes in the surroundings result from thermochemical processes in the system, according to the First Law's conservation of energy. An exothermic system is depositing its released energy into the surroundings (qsys < 0, qsurr > 0) and an endothermic system is absorbing energy from the surroundings (qsys > 0, qsurr < 0). Heat exchange is measured in the system using the relation:

Because m and S.H. are inherently positive, an endothermic process (q > 0) is always accompanied by an increase in temperature (Tf > Ti) and an exothermic process (q < 0) always results in a decrease (Tf < Ti). This formula also shows that a given amount of heat energy will cause a temperature change that is inversely proportional to the mass of sample. For example, consider the input of 725 J of heat into each of three copper blocks with the following masses: Block 1- 10.0 g, Block 2- 20.0 g, and Block 3- 40.0 g. The blocks will experience temperature changes of:
(Note that inputting heat implies a positive sign to the energy.) The above example shows that doubling the mass of material cuts its temperature change by one-half.

Consider next the removal of 1.00 kJ of heat from three separate materials of the same mass: 50.0 g of H2O(l), 50.0 g of Al(s), and 50.0 g of Au(s). Because of the differences in their respective heat capacities, each will experience a different temperature change:

(Note that removal of heat implies a negative q value, and that for proper dimensional analysis, q must be expressed in J.) The final temperature is found according to the relationship: . Supposing that all three are intially at a room temperature of 25.0 oC, removing 1 kJ of heat from 50.0 g of each would result in Tf values of 20.2 oC, 2.8 oC, and -130 oC for H2O(l), Al(s), and Au(s), respectively.

Suppose now that a 125 g block of iron heated to 65.0 oC is placed in thermal contact with a 215 g block of copper at 20.0 oC in an adiabatic container. In accordance with the first law:

qsys = qFe + qCu = 0
so that heat is exchanged only between them.
qFe = - qCu
The hotter block cools and the colder block warms until an equilibrium temperature Tf is reached. If the system is perfectly insulated, the two blocks would remain at Tf for eternity. By substituting the heat capacity formula, we can solve for this equilibrium temperature:





Two isolated metal blocks under the given conditions would come to an equilbrium temperature of 38.1 oC.





Schematic Diagram of a Bomb Calorimeter



Naphthalene is a hydrocarbon often used in moth balls

Calorimetry

The science of calorimetry is used to determine the heat energy (or caloric) content of a material (in familiar applications, the energy content of foods). Using a device called a bomb calorimeter, a sample is placed in a chamber (known as a bomb) of known heat capacity which is immersed in water. The water bath is insulated to prevent heat loss to the outside. Oxygen is pumped into the chamber, and a spark is used to ignite the sample. After complete combustion, the temperature inside the calorimeter reaches a maximum and this can be used to compute the heat energy content of the sample. Traditionally, the calorie energy unit is used and is defined to be the amount of energy required to raise one gram of water one degree Celsius.

Considering the calorimeter as an isolated system, the heat content of a sample burned in the bomb can be found from:

qsample = -(qwater + qbomb)
Suppose that a 1.00 g sample of napthalene (C10H8 (s) ) is placed in an aluminum bomb weighing 175 g. The calorimeter chamber is filled with 245 g of water and the initial temperature is measured at 22.0 oC. The bomb is ignited and the final temperature inside the calorimeter rises to a maximum of 54.9 oC. The heat content of the combustion reaction is found to be:


(Note that the combustion reaction is exothermic). Using a molar mass of 128.16 g/mole, the heat of combustion of napthalene is:
Applications of calorimetry in the determination of food energy content use the Food Calorie unit:

1.00 Food Calorie = 1.00 kcal = 1,000 calories






Work and Internal Energy

Concerted motion (particles with net movements in a fixed direction) can be harvested to provide energy used as work (w). Conversely, energy can be used to induce net movement, or do work on a system. Electrons moving through a potential, coiling or releasing a spring, and squeezing fluids (hydraulic action) are examples of processes which can either produce or require work. We will examine a fourth type of work in this lesson; that which results in expansion or contraction of a gas against an external resistance called PV-work.

Some clarification on the above notation may be helpful. The external pressure is applied to a gas sample as an opposing force to its internal pressure. In many instances Pext is supplied by the atmosphere, for instance as it resists the evolution of a H2O(g) from H2O(l) in evaporation. A more instructive example is given by the example to the left, a gas confined within a cylinder by a sliding piston. When Pint > Pext the piston is pushed back, increasing the gas volume (while its Pint decreases according to Boyles' Law). The piston continues until Pint = Pext. Since Vf > Vi, w < 0 and we say that work is done by the gas. Now consider the case where Pint < Pext. The piston moves inward, increasing internal pressure until the two pressures match. Here Vf < Vi and w > 0. In this case we say that work is done on the gas. Unit analysis of the pressure-volume product shows that it has the same dimensions as energy:
Free expansion occurs when a gas expands into a vacuum (Pext = 0). The name stems from the fact that a gas does no work (w = -0 × V) while undergoing free expansion.


Volume behaves as a state function. Independent of any or all intermediate steps, the change in volume only depends on the initial and final readings.


Internal Energy (E) measures the energy state of a system as it undergoes chemical and/or physical processes. Like other thermodynamic variables, internal energy exhibits two important properties: 1.) it is a state function and 2.) it scales as an extensive. Being a state function means that E has the following property:

E = Ef - Ei
State functions are path independent, meaning the same outcome results independent of the route taken from the initial to final state. Take as an example measuring volume changes. There are literally an infinite number of ways, by adding and/or removing water in different aliquots, that the volume in a jar can be changed from its intial to a final volume. However the change in volume each time will be exactly the same no matter how many intermediate steps are taken. It is quite remarkable how much of chemistry is measured in a relative sense, that is, as a difference between two absolute values. The significance of state functions in difference measurements is profound. Take for example measuring the energetics of a chemical reaction. If energy were not a state function, measurements would need to be made on each step of the process, including breaking the bonds of the reactants and reforming the bonds of the products. Energy is also extensive (or extrinsic), meaning it scales proportionally to the amount of material present. Other extensives include mass, volume, and pressure. A property is intensive (or intrinsic) if it is independent of the amount of material. Intensives include the temperature and density of matter.


Using a state function to analyze a gas being heated while being compressed.

The relationship between the internal energy of a system and its heat and work exchange with the surroundings is:

E = q + w
(The form of work will be restricted to gaseous, PV-type for this discussion.) Interestingly, both q and w are not state functions. They are path-dependent, meaning their values vary in magnitude according to how the work and heat are exchanged. They combine however, to form E, which is invariant to how the system is prepared. Consequently there is an interdependence between heat and work that we will now attempt to explore further. If a system is heated at constant volume, there is no chance for expansion work to occur and the internal energy expression simplifies to:
E = qV
where qV is the heat at constant volume:
qV = CVT
Note that the constant-volume restriction has required that heat capacity be expressed by a situational-dependent value: the constant volume heat capacity (CV). Under conditions of constant pressure we will likewise define the constant pressure heat capacity (CP). Laboratory chemistry takes place in an environment most suitable to study under the influence of the constant pressure supplied by the atmosphere, where we are required to include the work term in the internal energy expression. When heating a gas against constant pressure there is a natural tendency for expansion. As a result, a portion of endothermic heat energy input into a system is not deposited as internal energy, but is returned to the surroundings as expansion work. Take for example a 1.00 mole sample of argon gas having a molar constant pressure heat capacity of 20.79 J/(oC mole) which fills a balloon at STP (standard temperature and pressure of 0.00 oC and 1.00 atm pressure). Suppose the balloon is heated to a temperature of 50 o The change in internal energy of the gas is:




Likewise during a constant-pressure exothermic process a system cools and contracts, so not all of the heat lost is stolen from internal energy, some is returned as the surroundings does work on the system to contract it. Because they lack the expansion/contraction mechanism, materials heated or cooled at constant volume will experience a greater temperature change than they do at constant pressure. Based on this discussion, CP > CV in all cases.

So why do we need a constant-volume heat capacity at all? To answer this, let us look at the special case of an ideal gas undergoing a change in its volume and heat content. We will break the process up into two parts (which we are allowed to do because of the properties of a state function). 1.) isothermal expansion followed by 2.) constant volume heating. First we do an isothermal (constant temperature) change in volume. The ideal gas obeys kinetic theory, so its constituent particles have no potential energy of interaction of any kind. The kinetic energy they possess is a function of the temperature (and mass) of the particles. If the gas particles are allowed to expand or contract isothermally, they will be moved closer together or spread further apart while maintaining their same average velocities. Since they have no potential energy (energy of position), there is no change of internal energy due to their relocation. There is thus no change in internal energy as the ideal gas changes volume at constant temperature. Now let's examine the constant volume leg. The energy change here can be computed using the constant-volume heat capacity. Since the only contribution to the internal energy comes along this leg, we conclude for an ideal gas:

Eideal gas = CVT
For the special case of an ideal gas we can therefore use the constant-volume heat capacity to monitor all its energetic processes. As an ideal gas expands adiabatically (allowing no heat exchange), it must cool as it expends its work energy. The temperature change in a gas having CV = 12.47 J/oC as it doubles its volume adiabatically against atmospheric pressure is:
Eideal gas = CVT = wideal gas
T = -(PextV)/CV = - ( (1.00 atm)×(2.00 L)×(101.3 J/L atm))/12.47 J/oC = - 16.2 oC
Now consider the special case of an ideal gas undergoing an isothermal process. Since T = 0, we conclude that E = 0. In other words, unless the ideal gas undergoes a change of temperature, its internal energy remains the same. Because an ideal gas undergoing isothermal expansion or contraction experiences no change in internal energy:
Eideal gas = 0 for isothermal processes.
qideal gas = -wideal gas
As the last statement above shows, heat and work act in opposite senses to maintain the internal energy of the ideal gas at the same level. If the gas is expanding, it is expending energy as it does work to push back its opposing force, so heat energy must be added to offset this loss (and maintain its temperature at the same value). For example an ideal gas isothermally doubling its volume against atmospheric pressure must gain the following amount of heat:
qideal gas = - wideal gas = -(-PextV) = + 1.00 atm × 2.00 L × 101.3 J/(L atm) = +203 J

The following table lists some possible experimental designs and their given outcomes:

Process q w E
Isothermal expansion against a vacuum 0 0 0
Isothermal expansion against a vacuum 0 0 0



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