First law of Thermodynamics: Thermal Energy and Work, Thermodynamics | Physics

Learning objectives

Section Key Terms

Boltzmann constant

first law of thermodynamics

ideal gas law

internal energy

pressure

Pressure, Volume, Temperature, and the Ideal Gas Law

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

P = \frac{F}{A},

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_02_Shot.

View original figure (external; image not reproduced in this text edition)

Original figure description: In part (a), a finger pokes a man’s arm. In part (b), a needle pokes a man’s arm with the same force but over a smaller area, resulting in greater pressure.

(a) Although the person being poked with the finger might be irritated, the force has little lasting effect. (b) In contrast, the same force applied to an area the size of the sharp end of a needle is great enough to break the skin.

The SI unit for pressure is the pascal, where $1 Pa = {1 N/m}^{2} .$

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

P V = N k T .

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 $k = 1.38 \times 10^{−23} J/K,$ 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.

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_02_TirePressure).

View original figure (external; image not reproduced in this text edition)

Original figure description: In part (a), air is added to a tire, measured using a pressure gauge, and the tire increases in volume. In part (b), more air is added but the volume remains the same, resulting in greater pressure as indicated by the gauge. In part (c), the volume remains nearly constant, so that an increase in temperature increases the pressure.

(a) When air is pumped into a deflated tire, its volume first increases without much increase in pressure. (b) When the tire is filled to a certain point, the tire walls resist further expansion, and the pressure increases as more air is added. (c) Once the tire is inflated fully, its pressure increases with temperature.

Pressure–Volume Work

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_03_Expansion), an increase in volume decreases the internal pressure of a system as the system does work.

View original figure (external; image not reproduced in this text edition)

Original figure description: A piston in an engine cylinder with cross-sectional area A moves up a distance d as energy is added, causing the volume to expand by Ad.

An expansion of a gas requires energy transfer to keep the pressure constant. Because pressure is constant, the work done is

P Δ V

Recall that the formula for work is $W = F d .$ We can rearrange the definition of pressure, $P = \frac{F}{A},$ to get an expression for force in terms of pressure.

F = P A

Substituting this expression for force into the definition of work, we get

W = P A d .

Because area multiplied by displacement is the change in volume, $W = P Δ V$ , the mathematical expression for pressure–volume work is

W = P Δ V .

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.

The First Law of Thermodynamics

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.

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

Δ U = Q - W .

Here, $Δ U$ is the change in internal energy, U, of the system. As shown in Figure_12_02_FirstThermo, 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.

View original figure (external; image not reproduced in this text edition)

Original figure description: Q is shown adding energy to the system, while W causes energy to leave the system. For the system, the following equations are shown:

The first law of thermodynamics is the conservation of energy principle stated for a system, where heat and work are the methods of transferring energy to and from a system. Q represents the net heat transfer—it is the sum of all transfers of energy by heat into and out of the system. Q is positive for net heat transfer into the system.

W_{out}

is the work done by the system, and

W_{in}

is the work done on the system. W is the total work done on or by the system. W is positive when more work is done by the system than on it. The change in the internal energy of the system,

Δ U

, is related to heat and work by the first law of thermodynamics:

Δ U = Q - W .

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.

Biology: Biological Thermodynamics

We often think about thermodynamics as being useful for inventing or testing machinery, such as engines or steam turbines. However, thermodynamics also applies to living systems, such as our own bodies. This forms the basis of the biological thermodynamics (Figure_12_02_FoodEnergy).

View original figure (external; image not reproduced in this text edition)

Original figure description: A man takes in energy from food, gives off heat, and does work in part (a). A flower absorbs heat from the Sun and gives off heat in part (b).

(a) The first law of thermodynamics applies to metabolism. Heat transferred out of the body (Q) and work done by the body (W) remove internal energy, whereas food intake replaces it. (Food intake may be considered work done on the body.) (b) Plants convert part of the radiant energy in sunlight into stored chemical energy, a process called photosynthesis.

Life itself depends on the biological transfer of energy. Through photosynthesis, plants absorb solar energy from the sun and use this energy to convert carbon dioxide and water into glucose and oxygen. Photosynthesis takes in one form of energy—light—and converts it into another form—chemical potential energy (glucose and other carbohydrates).

Human metabolism is the conversion of food into energy given off by heat, work done by the body’s cells, and stored fat. Metabolism is an interesting example of the first law of thermodynamics in action. Eating increases the internal energy of the body by adding chemical potential energy; this is an unromantic view of a good burrito.

The body metabolizes all the food we consume. Basically, metabolism is an oxidation process in which the chemical potential energy of food is released. This implies that food input is in the form of work. Exercise helps you lose weight, because it provides energy transfer from your body by both heat and work and raises your metabolic rate even when you are at rest.

Biological thermodynamics also involves the study of transductions between cells and living organisms. Transduction is a process where genetic material—DNA—is transferred from one cell to another. This often occurs during a viral infection (e.g., influenza) and is how the virus spreads, namely, by transferring its genetic material to an increasing number of previously healthy cells. Once enough cells become infected, you begin to feel the effects of the virus (flu symptoms—muscle weakness, coughing, and congestion).

Energy is transferred along with the genetic material and so obeys the first law of thermodynamics. Energy is transferred—not created or destroyed—in the process. When work is done on a cell or heat transfers energy to a cell, the cell’s internal energy increases. When a cell does work or loses heat, its internal energy decreases. If the amount of work done by a cell is the same as the amount of energy transferred in by heat, or the amount of work performed on a cell matches the amount of energy transferred out by heat, there will be no net change in internal energy.

Based on what you know about heat transfer and the first law of thermodynamics, do you need to eat more or less to maintain a constant weight in colder weather? Explain why.

more; as more energy is lost by the body in colder weather, the need to eat increases so as to maintain a constant weight
more; eating more food means accumulating more fat, which will insulate the body from colder weather and will reduce the energy loss
less; as less energy is lost by the body in colder weather, the need to eat decreases so as to maintain a constant weight
less; eating less food means accumulating less fat, so less energy will be required to burn the fat, and, as a result, weight will remain constant

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?

Solution

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

Q = 40 .00 J - 25 .00 J = 15 .00 J.

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

W = 10 .00 J - 4 .00 J = 6 .00 J.

The change in internal energy is given by the first law of thermodynamics.

Δ U = Q - W = 15 .00 J - 6 .00 J = 9 .00 J

Discussion

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

Δ U_{1} = Q_{1} - W_{1} = 40 .00 J - 10 .00 J = 30 .00 J.

For 25.00 J of heat out and 4.00 J of work in, the change in internal energy is

Δ U_{2} = Q_{2} - W_{2} = - 25. 00 J - (- 4.00 J) = - 21. 00 J.

The total change is

Δ U = Δ U_{1} + Δ U_{2} = 30 .00 J + (- 21. 00 J) = 9. 00 J.

No matter whether you look at the overall process or break it into steps, the change in internal energy is the same.

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?

Solution

Here, the net heat and total work are given directly as $Q = - 150 .00 J and W = - 159.00 J,$ so that

Δ U = Q - W = - 150. 00 J - (- 159. 00 J) = 9. 00 J.

Discussion

View original figure (external; image not reproduced in this text edition)

Original figure description: Two different processes produce the same change in a system. (a) Heat transfers a total of 15.00 J into the system, while work takes out a total of 6.00 J. The change in internal energy is ΔU = Q - W = 9.00 J. (b) Heat removes 150.00 J from the system while work puts 159.00 J into it, producing an increase in internal energy of 9.00 J.

Two different processes produce the same change in a system. (a) A total of 15.00 J of heat transfer occurs into the system, while work takes out a total of 6.00 J. The change in internal energy is ΔU = Q – W = 9.00 J. (b) Heat transfer removes 150.00 J from the system while work puts 159.00 J into it, producing an increase of 9.00 J in internal energy. If the system starts out in the same state in (a) and (b), it will end up in the same final state in either case—its final state is related to internal energy, not how that energy was acquired.

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 $Δ U$ 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 eip-idm379187600 above, $W_{out}$ is work done by the system, and $W_{in}$ is work done on the system.

Practice Problems

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Check Your Understanding

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What is the SI unit for pressure?

pascal, or N/m³
coulomb
newton
pascal, or N/m²

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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.

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Section Summary

Pressure is the force per unit area over which the force is applied perpendicular to the area.
Thermal expansion is the increase, or decrease, of the size (length, area, or volume) of a body due to a change in temperature.
The ideal gas law relates the pressure and volume of a gas to the number of gas particles (atoms or molecules) and the absolute temperature of the gas.
Heat and work are the two distinct methods of energy transfer.
Heat is energy transferred solely due to a temperature difference.
The first law of thermodynamics is given as $Δ U = Q - W$ , where $Δ U$ is the change in internal energy of a system, Q is the net energy transfer into the system by heat (the sum of all transfers by heat into and out of the system), and W is the net work done by the system (the sum of all energy transfers by work out of or into the system).
Both Q and W represent energy in transit; only $Δ U$ represents an independent quantity of energy capable of being stored.
The internal energy U of a system depends only on the state of the system, and not how it reached that state.

Key Equations

ideal gas law	$P V = N k T$
first law of thermodynamics	$Δ U = Q - W$
pressure	$P = \frac{F}{A}$
pressure–volume work	$W = P Δ V$

Concept Items

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How does pressure-volume work relate to heat and internal energy of a system?

The energy added to a system by heat minus the change in the internal energy of that system is equal to the pressure-volume work done by the system.
The sum of the energy released by a system by heat and the change in the internal energy of that system is equal to the pressure-volume work done by the system.
The product of the energy added to a system by heat and the change in the internal energy of that system is equal to the pressure-volume work done by the system.
If the energy added to a system by heat is divided by the change in the internal energy of that system, the quotient is equal to the pressure-volume work done by the system.

On what does internal energy depend?

The path of energy changes in the system
The state of the system
The size of the system
The shape of the system

The first law of thermodynamics helps us understand the relationships among which three quantities?

Heat, work, and internal energy
Heat, work, and external energy
Heat, work, and enthalpy
Heat, work, and entropy

Critical Thinking

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Air is pumped into a car tire, causing its temperature to increase. In another tire, the temperature increase is due to exposure to the sun. Is it possible to tell what caused the temperature increase in each tire? Explain your answer.

No, because it is a chemical change, and the cause of that change does not matter; the final state of both systems are the same.
Although the final state of each system is identical, the source is different in each case.
No, because the changes in energy for both systems are the same, and the cause of that change does not matter; the state of each system is identical.
Yes, the changes in the energy for both systems are the same, but the causes of that change are different, so the states of each system are not identical.

How does the transfer of energy from the sun help plants?

Plants absorb solar energy from the sun and utilize it during the fertilization process.
Plants absorb solar energy from the sun and utilize it during the process of photosynthesis to turn it into plant matter.
Plants absorb solar energy from the sun and utilize it to increase the temperature inside them.
Plants absorb solar energy from the sun and utilize it during the shedding of their leaves and fruits.

Problems

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Test Prep Multiple Choice

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Test Prep Short Answer

If a fixed quantity of an ideal gas is held at a constant volume, which variable relates to pressure, and what is that relation?

Temperature; inverse proportionality $(P \propto \frac{1}{T})$
Temperature, direct proportionality to square root $(P \propto \sqrt{T})$
Temperature; direct proportionality $(P \propto T)$
Temperature; direct proportionality to square $(P \propto T^{2})$

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By convention, if Q is positive, what is the direction in which heat transfers energy with regard to the system?

The direction of the heat transfer of energy depends on the changes in W, regardless of the sign of Q.
The direction of Q cannot be determined from just the sign of Q.
The direction of net heat transfer of energy will be out of the system.
The direction of net heat transfer of energy will be into the system.

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Test Prep Extended Response

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Under which conditions will the work done by the gas in a system increase?

It will increase when a large amount of energy is added to the system, and that energy causes an increase in the gass volume, its pressure, or both.
It will increase when a large amount of energy is extracted from the system, and that energy causes an increase in the gass volume, its pressure, or both.
It will increase when a large amount of energy is added to the system, and that energy causes a decrease in the gass volume, its pressure, or both.
It will increase when a large amount of energy is extracted from the system, and that energy causes a decrease in the gass volume, its pressure, or both.

How does energy transfer by heat aid in body metabolism?

The energy is given to the body through the work done by the body (W) and through the intake of food, which may also be considered as the work done on the body. The transfer of energy out of the body is by heat (−Q) .
The energy given to the body is by the intake of food, which may also be considered as the work done on the body. The transfer of energy out of the body is by heat (−Q) and the work done by the body (W) .
The energy given to the body is by the transfer of energy by heat (Q) into the body, which may also be considered as the work done on the body. The transfer of energy out of the body is the work done by the body (W) .
The energy given to the body is by the transfer of energy by heat (Q) inside the body. The transfer of energy out of the body is by the intake of food and the work done by the body (W) .

Two distinct systems have the same amount of stored internal energy. Five hundred joules are added by heat to the first system, and 300 J are added by heat to the second system. What will be the change in internal energy of the first system if it does 200 J of work? How much work will the second system have to do in order to have the same internal energy?

700 J; 0 J
300 J; 300 J
700 J; 300 J
300 J; 0 J

Glossary

Boltzmann constant: constant with the value k = 1.38×10⁻²³ J/K, which is used in the ideal gas law
first law of thermodynamics: states that the change in internal energy of a system equals the net energy transfer by heat into the system minus the net work done by the system
ideal gas law: physical law that relates the pressure and volume of a gas to the number of gas molecules or atoms, or number of moles of gas, and the absolute temperature of the gas
internal energy: sum of the kinetic and potential energies of a system’s constituent particles (atoms or molecules)
pressure: force per unit area perpendicular to the force, over which the force acts

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