6.3: The First Law of Thermodynamics

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The First Law of Thermodynamics

One of the great achievements of the 19^th century was the recognition that heat and work are two forms of the same thing (energy), and heat and work are the only ways in which we can transfer energy from one object to another. We can summarize these statements as follows:

In any process, energy can never be created or destroyed; it can only be transferred

from one object to another in the form of heat and/or work.

This statement is called the First Law of Thermodynamics, and it can also be written as a mathematical equation:

\[\Delta E = q + w\]

Where

\(\Delta E\) is the change in the internal energy of a system (a collection of matter)

\(q\) is the amount of heat transferred into or out of the system

\(w\) is the amount of work that is done on or by the system

First Law of Thermodynamics and the Law of Conservation of Energy

The first law of thermodynamics is actually the law of conservation of energy stated in a form most useful in thermodynamics. The first law gives the relationship between heat transfer, work done, and the change in internal energy of a system.

Here are three examples. Note that in each case, we must be careful to specify our system before we assign signs to \(q\) and \(w\).

You heat a beaker of water: If we choose the water to be our system, \(q\) is a positive number, because the heat is moving into the water. The energy of the water increases when we heat it, so \(\Delta E\) is positive, and \(q\) must agree with this.
You lift a suitcase: If we consider you to be the system, \(w\) is a negative number, because you are doing the work. Your body’s energy decreases as you lift the suitcase, so \(\Delta E\) is negative, and \(w\) must agree with this. On the other hand, if we consider the suitcase to be the system, \(w\) is a positive number, because the surroundings are doing the work on the suitcase. (If you aren’t the system, you are part of the surroundings.)
You drop an ice cube into a beaker of hot water: If we consider the ice cube to be the system, \(q\) is a positive number, because the ice cube absorbs heat (and gains energy). If we consider the hot water to be the system. \(q\) is a negative number, because the water loses heat (and loses energy).

Heat and Work in the First Law

Heat \(q\) and work \(w\) are the two everyday means of bringing energy into or taking energy out of a system. The processes are quite different. Heat transfer, a disorganized process, is driven by temperature differences. Work, a quite organized process, involves a macroscopic force exerted through a distance. Nevertheless, heat and work can produce identical results. For example, both can cause a temperature increase. Heat transfer into a system, such as when the Sun warms the air in a bicycle tire, can increase its temperature, and so can work done on the system, as when the bicyclist pumps air into the tire. Once the temperature increase has occurred, it is impossible to tell whether it was caused by heat transfer or by doing work. This uncertainty is an important point. Heat and work are both energy in transit—neither is stored as such in a system. However, both can change the internal energy \(E\) of a system.

Internal Energy in the First Law

The internal energy \(E\) of a system is the sum of the kinetic and potential energies of its atoms and molecules. Because it is impossible to keep track of all individual atoms and molecules, we must deal with averages and distributions.

The internal energy \(E\) of a system depends only on the state of the system and not how it reached that state. More specifically, \(E\) is found to be a function of a few macroscopic quantities (pressure, volume, and temperature, for example), independent of past history such as whether there has been heat transfer or work done. This independence means that if we know the state of a system, we can calculate changes in its internal energy \(U\) from a few macroscopic variables.

To get a better idea of how to think about the internal energy of a system, let us examine a system going from State 1 to State 2. The system has internal energy \(E_1\) in State 1, and it has internal energy \(E_2\) in State 2, no matter how it got to either state. So the change in internal energy

\[\Delta E = E_2 - E_1\]

is independent of what caused the change. In other words, \(\delta E\) is independent of path. By path, we mean the method of getting from the starting point to the ending point. Why is this independence important? Both \(q\) and \(w\) depend on path, but \(\Delta E\) does not (Equation 6.3.1). This path independence means that internal energy \(E\) is easier to consider than either heat transfer or work done.

Example \(\PageIndex{1}\): Calculating Change in Internal Energy - The Same Change in \(E\) is Produced by Two Different Processes

Question:

Suppose there is heat transfer of 40.00 J to a system, while the system does 10.00 J of work. Later, there is heat transfer of 25.00 J out of the system while 4.00 J of work is done on the system. What is the net change in internal energy of the system?
What is the change in internal energy of a system when a total of 150.00 J of heat transfer occurs out of (from) the system and 159.00 J of work is done on the system

Strategy:

In part (a), we must first find the net heat transfer and net work done from the given information. Then the first law of thermodynamics (Equation 6.3.1).

can be used to find the change in internal energy. In part (b), the net heat transfer and work done are given, so the equation can be used directly.

Solution for (a)

The net heat transfer is the heat transfer into the system minus the heat transfer out of the system, or

\[ q = q_{in} + q_{out} = 40.00 J + (- 25.00 J) = 15.00 J \nonumber\]

Similarly, the total work is the work done by the system minus the work done on the system, or

\[w = w_{in}+q_{out} = 4.00 J + (-10.00 J) = -6.00 J \nonumber\]

Adding the net heat transfer and total work gives

\[\Delta E = q + w = 15 J + (-6.00 J) = 9.00 J. \nonumber\]

Solution for (b)

Here the net heat transfer and total work are given directly to be \(q = -150.00 J\) and \(w = +159.00 J\), so that

\[\Delta E= q + w = -150.00 + 159.00 = 9.00 J \nonumber\]

Discussion

A very different process in part (b) produces the same 9.00-J change in internal energy as in part (a). Note that the change in the system in both parts is related to \(\Delta E\) and not to the individual \(q\)s or \(w\)s involved. The system ends up in the same state in both (a) and (b). Parts (a) and (b) present two different paths for the system to follow between the same starting and ending points, and the change in internal energy for each is the same—it is independent of path.

Summary

The table presents a summary of terms relevant to the first law of thermodynamics.

Term	Definition
\(E\)	Internal energy—the sum of the kinetic and potential energies of a system’s atoms and molecules. Can be divided into many subcategories, such as thermal and chemical energy. Depends only on the state of a system (such as its \(P\), \(V\) and \(T\), not on how the energy entered the system. Change in internal energy is path independent.
\(q\)	Heat—energy transferred because of a temperature difference. Characterized by random molecular motion. Highly dependent on path. \(q\) entering a system is positive.
\(w\)	Work—energy transferred by a force moving through a distance. An organized, orderly process. Path dependent. \(w\) done by a system (either against an external force or to increase the volume of the system) is positive.

The first law of thermodynamics is given as \(\Delta E = q + w\), where \(\Delta E\) is the change in internal energy of a system, \(q\) is the net heat transfer (the sum of all heat transfer into and out of the system), and \(w\) is the net work done (the sum of all work done on or by the system).
Both \(q\) and \(w\) are energy in transit; only \(\Delta E\) represents an independent quantity capable of being stored.
The internal energy \(E\) of a system depends only on the state of the system and not how it reached that state.

Glossary

first law of thermodynamics: states that the change in internal energy of a system equals the net heat transfer into the system minus the net work done by the system

internal energy: the sum of the kinetic and potential energies of a system’s atoms and molecules