Infinitesimal Changes of State

In the preceding examples the initial and final states differ by a finite amount. Later we will consider infinitesimal changes of state in which a small amount of heat is added to the system, the system does a small amount of work , and its internal energy changes by an amount . For such a process we state the first law in differential form as

first law in thermodynamics, infinitesimal process (11)

(12)

Understanding the First Law of Thermodynamics

At the beginning of this discussion we tentatively defined internal energy in terms of microscopic kinetic and potential energies. Actually calculating internal energy in this way for any real system would be hopelessly complicated.

So let's look at internal energy in another way. Starting over, we define the change in internal energy during any change of a system as the quantity given by Eq. (8), . This is an operational definition because we can measure and . It does not define itself, only. This is analogous to our treatment of potential energy, in which we arbitrarily defined the potential energy of a mechanical system to be zero at a certain position.

This new definition trades one difficulty for another. If we define by Eq. (8), then when the system goes from state 1 to state 2 by two different paths, how do we know that is the same for the two paths? We have already seen that and are, in general, not the same for different paths. If , which equals , is also path dependent, then is ambiguous.

The only way to answer this question is through experiment. For various materials we measure and for various changes of state and various paths to learn whether is or is not path dependent. The results of many such investigations are clear and unambiguous: While and depend on the path, is independent of path. The change in internal energy of a system during any thermodynamic process depends only on the initial and final states, not on the path leading from one to the other.

Experiment, then, is the ultimate justification for believing that a thermodynamic system in a specific state has a unique internal energy that depends only on that state.

Cyclic Processes and Isolated Systems

Two special cases of the first law of thermodynamics are worth mentioning. A process that eventually returns a system to its initial state is called a cyclic process. For such a process, the final state is the same as the initial state, and so the total internal energy change must be zero. Then

and

If a net quantity of work is done by the system during this process, an equal amount of energy must have flowed into the system as heat . But there is no reason either or individually has to be zero.

Another special case occurs in an isolated system, one that does no work on its surroundings and has no heat flow to or from its surroundings. For any process taking place in an isolated system,

and therefore

In other words, the internal energy of an isolated system is constant.

First law of thermodynamics for different kinds of thermodynamic processes

In this section we describe four specific kinds of thermodynamic processes that occur often in practical situations. These can be summarized briefly as "no heat transfer" or adiabatic, "constant volume" or isochoric, "constant pressure" or isobaric, and "constant temperature" or isothermal. For some of these processes we can use a simplified form of the first law of thermodynamics .

(a) Adiabatic Process

An adiabatic process is defined as one with no heat transfer into or out of a system; . We can prevent heat flow either by surrounding the system with thermally insulating material or by carrying out the process so quickly that there is not enough time for appreciable heat flow. From the first law we find that for every adiabatic process

.

When a system expands adiabatically, is positive (the system does work on its surroundings), so is negative and the internal energy decreases. When a system is compressed adiabatically, is negative (work is done on the system by its surroundings) and increases. In many (but not all) systems an increase of internal energy is accompanied by a rise in temperature, and a decrease in internal energy with a drop in temperature.

The compression stroke in an internal-combustion engine is an approximately adiabatic process. The temperature rises as the air-fuel mixture in the cylinder is compressed. The expansion of the burned fuel during the power stroke is also an approximately adiabatic expansion with a drop in temperature

(b) Isochoric Process

An isochoric process is a constant-volume process. When the volume of a thermodynamic system is constant, it does no work on its surroundings. Then and

(isochoric process)

In an isochoric process, all the energy added as heat remains in the system as an increase in internal energy. Heating a gas in a closed constant-volume container is an example of an isochoric process.

(c) Isobaric Process

An isobaric process is a constant-pressure process. In general, none of the three quantities , , and is zero in an isobaric process, but calculating is easy nonetheless:

(d) Isothermal Process

An isothermal process is a constant-temperature process. For a process to be isothermal, any heat flow into or out of the system must occur slowly enough that thermal equilibrium is maintained. In general , and first law of thermodynamics has form:

.

For ideal gas, if the temperature is constant, the internal energy is also constant; and . That is, any energy entering the system as heat must leave it again as work done by the system

Figure 10

Figure 10 shows a -diagram for these four processes for a constant amount of an ideal gas. The path followed in an adiabatic process (a to 1) is called an adiabat. A vertical line (constant volume) is an isochor, a horizontal line (constant pressure) is an isobar, and a curve of constant temperature is an isotherm.

Specific Heat Capacity

The quantity of heat required to increase the temperature of a mass of a certain material from to is found to be approximately proportional to the temperature change . It is also proportional to the mass of material. When you're heating water to make tea, you need twice as much heat for two cups as for one if the temperature change is the same. The quantity of heat needed also depends on the nature of the material; raising the temperature of 1 kilogram of water by 1 C° requires 4190 J of heat, but only 910 J is needed to raise the temperature of 1 kilogram of aluminum by 1 C°. Putting all these relationships together, we have

(13)

Here - heat required for temperature change of mass , is a quantity, different for different materials, called the specific heat of the material. For an infinitesimal temperature change and corresponding quantity of heat :

, (14)

(specific heat) (15)

In Eqs. (13), (14), and (15), (or ) and (or ) can be either positive or negative. When they are positive, heat enters the body and its temperature increases; when they are negative, heat leaves the body and its temperature decreases.

Molar Heat Capacity

Sometimes it's more convenient to describe a quantity of substance in terms of the number of moles rather than the mass of material. Recall from your study of chemistry that a mole of any pure substance always contains the same number of molecules. The molar mass of any substance, denoted by , is the mass per mole. (The quantity is sometimes called molecular weight, but molar mass is preferable; the quantity depends on the mass of a molecule, not its weight.) For example, the molar mass of water is 18.0 g/mol = 18.0x10^-3 kg/mol; 1 mole of water has a mass of 18.0 g = 0.0180 kg. The total mass of material is equal to the mass per mole times the number of moles :

(16)

Replacing the mass in Eq. (17.13) by the product , we find

(17)

The product is called the molar heat capacity (or molar specific heat) and is denoted by (capitalized). With this notation we rewrite Eq. (17) as

(heat required for temperature change of moles) (18)

Comparing to Eq. (15), we can express the molar heat capacity (heat per mole per temperature change) in terms of the specific heat (heat per mass per temperature change) and the molar mass (mass per mole):

(molar heat capacity) ( 19)

For. example, the molar heat capacity of water is

= (0.0180 kg/mol) (4190 j/kg • K) = 75.4 j/mol • K

CAUTION The meaning of "heat capacity". The term "heat capacity" is unfortunate because it gives the erroneous impression that a body contains a certain amount of heat. Remember, heat is energy in transit to or from a body, not the energy residing in the body.