By the end of this section, you will be able to do the following:
- Describe how pressure, volume, and temperature relate to one another and to work, based on the ideal gas law
- Describe pressure–volume work
- Describe the first law of thermodynamics verbally and mathematically
- Solve problems involving the first law of thermodynamics
The learning objectives in this section will help your students master the following standards:
- (6) Science concepts. The student knows that changes occur within a physical system and applies the laws of conservation of energy and momentum. The student is expected to:
- (G) analyze and explain everyday examples that illustrate the laws of thermodynamics, including the law of conservation of energy and the law of entropy.
Section Key Terms
|Boltzmann constant||first law of thermodynamics||ideal gas law||internal energy||pressure|
Pressure, Volume, Temperature, and the Ideal Gas Law
[BL][OL][AL] Review the concept of force.
[OL] Ask students how much force it would take to hammer a nail into a wall. Would they achieve the same result if the nail were blunt instead of pointed? Why or why not?
Before covering the first law of thermodynamics, it is first important to understand the relationship between pressure, volume, and temperature. Pressure, P, is defined as
where F is a force applied to an area, A, that is perpendicular to the force.
Depending on the area over which it is exerted, a given force can have a significantly different effect, as shown in Figure 12.3.
The SI unit for pressure is the pascal, where
Pressure is defined for all states of matter but is particularly important when discussing fluids (such as air). You have probably heard the word pressure being used in relation to blood (high or low blood pressure) and in relation to the weather (high- and low-pressure weather systems). These are only two of many examples of pressures in fluids.
The relationship between the pressure, volume, and temperature for an ideal gas is given by the ideal gas law. A gas is considered ideal at low pressure and fairly high temperature, and forces between its component particles can be ignored. The ideal gas law states that
where P is the pressure of a gas, V is the volume it occupies, N is the number of particles (atoms or molecules) in the gas, and T is its absolute temperature. The constant k is called the Boltzmann constant and has the value For the purposes of this chapter, we will not go into calculations using the ideal gas law. Instead, it is important for us to notice from the equation that the following are true for a given mass of gas:
- When volume is constant, pressure is directly proportional to temperature.
- When temperature is constant, pressure is inversely proportional to volume.
- When pressure is constant, volume is directly proportional to temperature.
This last point describes thermal expansion—the change in size or volume of a given mass with temperature. What is the underlying cause of thermal expansion? An increase in temperature means that there’s an increase in the kinetic energy of the individual atoms. Gases are especially affected by thermal expansion, although liquids expand to a lesser extent with similar increases in temperature, and even solids have minor expansions at higher temperatures. This is why railroad tracks and bridges have expansion joints that allow them to freely expand and contract with temperature changes.
[BL][OL][AL]Inform students that the ideal gas law is only strictly true for gases that are ideal. Real gases differ from these in certain ways. For instance, it is assumed that in an ideal gas, there are no intermolecular forces, all collisions between molecules are perfectly elastic, and so on. However, the ideal gas law is useful in calculations, because under standard conditions of temperature and pressure, many real gases, such as oxygen, nitrogen, hydrogen, and noble gases exhibit qualitative behavior that is very close to the behavior of an ideal gas.
To get some idea of how pressure, temperature, and volume of a gas are related to one another, consider what happens when you pump air into a deflated tire. The tire’s volume first increases in direct proportion to the amount of air injected, without much increase in the tire pressure. Once the tire has expanded to nearly its full size, the walls limit volume expansion. If you continue to pump air into tire (which now has a nearly constant volume), the pressure increases with increasing temperature (see Figure 12.4).
Pressure–volume work is the work that is done by the compression or expansion of a fluid. Whenever there is a change in volume and external pressure remains constant, pressure–volume work is taking place. During a compression, a decrease in volume increases the internal pressure of a system as work is done on the system. During an expansion (Figure 12.5), an increase in volume decreases the internal pressure of a system as the system does work.
Recall that the formula for work is We can rearrange the definition of pressure, to get an expression for force in terms of pressure.
Substituting this expression for force into the definition of work, we get
Because area multiplied by displacement is the change in volume, , the mathematical expression for pressure–volume work is
Just as we say that work is force acting over a distance, for fluids, we can say that work is the pressure acting through the change in volume. For pressure–volume work, pressure is analogous to force, and volume is analogous to distance in the traditional definition of work.
Work from Expansion
This video describes work from expansion (or pressure–volume work). Sal combines the equations and to get .
If the volume of a system increases while pressure remains constant, is the value of work done by the system W positive or negative? Will this increase or decrease the internal energy of the system?
- Positive; internal energy will decrease
- Positive; internal energy will increase
- Negative; internal energy will decrease
- Negative; internal energy will increase
The First Law of Thermodynamics
[BL] Review heat transfer. When and how does heat transfer energy between two bodies? What happens when energy is transferred into or out of a system by heat?
Heat (Q) and work (W) are the two ways to add or remove energy from a system. The processes are very different. Heat is driven by temperature differences, while work involves a force exerted through a distance. Nevertheless, heat and work can produce identical results. For example, both can cause a temperature increase. Heat transfers energy into a system, such as when the sun warms the air in a bicycle tire and increases the air’s temperature. Similarly, work can be 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 or work. Heat and work are both energy in transit—neither is stored as such in a system. However, both can change the internal energy, U, of a system.
Internal energy is the sum of the kinetic and potential energies of a system’s atoms and molecules. It can be divided into many subcategories, such as thermal and chemical energy, and depends only on the state of a system (that is, P, V, and T), not on how the energy enters or leaves the system.
In order to understand the relationship between heat, work, and internal energy, we use the first law of thermodynamics. The first law of thermodynamics applies the conservation of energy principle to systems where heat and work are the methods of transferring energy into and out of the systems. It can also be used to describe how energy transferred by heat is converted and transferred again by work.
[OL][AL] Ask students to give examples of processes where energy is converted from one form into another. Analyze whether energy transfer by heat occurs in each case.
Recall that the principle of conservation of energy states that energy cannot be created or destroyed, but it can be altered from one form to another.
The first law of thermodynamics states that the change in internal energy of a closed system equals the net heat transfer into the system minus the net work done by the system. In equation form, the first law of thermodynamics is
Here, is the change in internal energy, U, of the system. As shown in Figure 12.6, Q is the net heat transferred into the system—that is, Q is the sum of all heat transfers into and out of the system. W is the net work done by the system—that is, W is the sum of all work done on or by the system. By convention, if Q is positive, then there is a net heat transfer into the system; if W is positive, then there is net work done by the system. So positive Q adds energy to the system by heat, and positive W takes energy from the system by work. Note that if heat transfers more energy into the system than that which is done by work, the difference is stored as internal energy.
It follows also that negative Q indicates that energy is transferred away from the system by heat and so decreases the system’s internal energy, whereas negative W is work done on the system, which increases the internal energy.
First Law of Thermodynamics/Internal Energy
This video explains the first law of thermodynamics, conservation of energy, and internal energy. It goes over an example of energy transforming between kinetic energy, potential energy, and heat transfer due to air resistance.
[BL][OL] Demonstrate that the equation may be rearranged as This shows that any energy added by heat to a system is either converted into work or stored as internal energy.
More on Internal Energy
This video goes into further detail, explaining internal energy and how to use the equation Note that Sal uses the equation , where W is the work done on the system, whereas we use W to represent work done by the system.
Solving Problems Involving the First Law of Thermodynamics
Calculating Change in Internal Energy
Suppose 40.00 J of energy is transferred by heat to a system, while the system does 10.00 J of work. Later, heat transfers 25.00 J out of the system, while 4.00 J is done by work on the system. What is the net change in the system’s internal energy?
You must first calculate the net heat and net work. Then, using the first law of thermodynamics, find the change in internal energy.
The net heat is the transfer into the system by heat minus the transfer out of the system by heat, or
The total work is the work done by the system minus the work done on the system, or
The change in internal energy is given by the first law of thermodynamics.
A different way to solve this problem is to find the change in internal energy for each of the two steps separately and then add the two changes to get the total change in internal energy. This approach would look as follows:
For 40.00 J of heat in and 10.00 J of work out, the change in internal energy is
For 25.00 J of heat out and 4.00 J of work in, the change in internal energy is
The total change is
No matter whether you look at the overall process or break it into steps, the change in internal energy is the same.
[BL] Make sure students are clear on the use of negative signs for heat transfer and work. Work done by an isolated system means an increase in volume, so W is positive and decreases or is negative.
Calculating Change in Internal Energy: The Same Change in U is Produced by Two Different Processes
What is the change in the internal energy of a system when a total of 150.00 J is transferred by heat from the system and 159.00 J is done by work on the system?
The net heat and work are already given, so simply use these values in the equation
Here, the net heat and total work are given directly as so that
A very different process in this second worked example produces the same 9.00 J change in internal energy as in the first worked example. Note that the change in the system in both parts is related to and not to the individual Q’s or W’s involved. The system ends up in the same state in both problems. Note that, as usual, in Figure 12.8 above, is work done by the system, and is work done on the system.
Check Your Understanding
Use these questions to assess student achievement of the section’s learning objectives. If students are struggling with a specific objective, these questions will help identify which and direct students to the relevant content.
What is the SI unit for pressure?
- pascal, or N/m3
- pascal, or N/m2
When is pressure-volume work said to be done ON a system?
- When there is an increase in both volume and internal pressure.
- When there is a decrease in both volume and internal pressure.
- When there is a decrease in volume and an increase in internal pressure.
- When there is an increase in volume and a decrease in internal pressure.