Thermodynamics

9.1

Kinetic Theory of Temperature and Pressure

Syllabus

Learning Objective	Essential Knowledge
9.1.A Describe the pressure a gas exerts on its container in terms of atomic motion within that gas.	9.1.A.1 Atoms in a gas collide with and exert forces on other atoms in the gas and with the container in which the gas is contained. 9.1.A.1.i Collisions involving pairs of atoms or an atom and a fixed object can be described and analyzed using conservation of momentum principles. 9.1.A.1.ii The pressure exerted by a gas on a surface is the ratio of the sum of the magnitudes of the perpendicular components of the forces exerted by the gas's atoms on the surface to the area of the surface. Equation: $P = \dfrac{F_{\perp}}{A}$ 9.1.A.1.iii Pressure exists throughout the gas itself, not just at the boundary between the gas and the container.
9.1.B Describe the temperature of a system in terms of the atomic motion within that system.	9.1.B.1 The temperature of a system is characterized by the average kinetic energy of the atoms within that system. 9.1.B.1.i The Maxwell–Boltzmann distribution provides a graphical representation of the energies and speeds of atoms at a given temperature. 9.1.B.1.ii The root-mean-square speed corresponding to the average kinetic energy for an ideal gas is related to the temperature of the gas by Equation: $K_{\text{avg}} = \dfrac{3}{2} k_B T = \dfrac{1}{2} m v_{\text{rms}}^2$ Boundary statement: AP Physics 2 only expects students to perform qualitative and quantitative analysis of collisions in one and two dimensions. Students are not expected to know the functional form of the Maxwell-Boltzmann distribution but are expected to be familiar with how features of the distribution are related to the temperature of the gas.

Learning Objective

Essential Knowledge

9.1.A
Describe the pressure a gas exerts on its container in terms of atomic motion within that gas.

9.1.A.1 Atoms in a gas collide with and exert forces on other atoms in the gas and with the container in which the gas is contained.
- 9.1.A.1.i Collisions involving pairs of atoms or an atom and a fixed object can be described and analyzed using conservation of momentum principles.
- 9.1.A.1.ii The pressure exerted by a gas on a surface is the ratio of the sum of the magnitudes of the perpendicular components of the forces exerted by the gas's atoms on the surface to the area of the surface.
  - Equation: $P = \dfrac{F_{\perp}}{A}$
- 9.1.A.1.iii Pressure exists throughout the gas itself, not just at the boundary between the gas and the container.

9.1.B
Describe the temperature of a system in terms of the atomic motion within that system.

9.1.B.1 The temperature of a system is characterized by the average kinetic energy of the atoms within that system.
- 9.1.B.1.i The Maxwell–Boltzmann distribution provides a graphical representation of the energies and speeds of atoms at a given temperature.
- 9.1.B.1.ii The root-mean-square speed corresponding to the average kinetic energy for an ideal gas is related to the temperature of the gas by
  - Equation: $K_{\text{avg}} = \dfrac{3}{2} k_B T = \dfrac{1}{2} m v_{\text{rms}}^2$

Boundary statement: AP Physics 2 only expects students to perform qualitative and quantitative analysis of collisions in one and two dimensions. Students are not expected to know the functional form of the Maxwell-Boltzmann distribution but are expected to be familiar with how features of the distribution are related to the temperature of the gas.

Source: College Board AP Course and Exam Description

Kinetic theory: gas pressure

Thermodynamics 热力学 studies heat and energy in large collections of particles. The kinetic theory 分子运动论 explains gas behavior from the random motion of its molecules:

The key assumptions of the kinetic theory of an ideal gas

Temperature 温度 is a measure of the average kinetic energy of the molecules: $K_{\text{avg}}=\tfrac{3}{2}k_B T$ (with $T$ in kelvin). Hotter means faster-moving molecules.
Pressure 压强 comes from molecules colliding with the container walls – more frequent or harder collisions give more pressure.

The Maxwell-Boltzmann speed distribution shifts right and flattens at higher temperature

Worked example. Find the average kinetic energy of a gas molecule at $300\ \text{K}$, using $k_B=1.38\times10^{-23}\ \text{J/K}$:

$$K_{\text{avg}}=\tfrac32 k_B T=1.5\times1.38\times10^{-23}\times300=6.2\times10^{-21}\ \text{J}.$$

It depends only on temperature, not on the type of gas – at the same $T$, light and heavy molecules share the same average kinetic energy (so the light ones move faster).

Explore

Explore the spread of molecular speeds

Raise the temperature and watch the whole speed distribution shift right and flatten — the particles move faster on average, which is exactly what $\bar K = \tfrac{3}{2} k_B T$ means.

Vocabulary Train

English	Chinese	Pinyin
Thermodynamics	热力学	rè lì xué
kinetic theory	分子运动论	fēn zǐ yùn dòng lùn
Temperature	温度	wēn dù
Pressure	压强	yā qiáng

Exercise sheet

9.2

The Ideal Gas Law

Syllabus

Learning Objective	Essential Knowledge
9.2.A Describe the properties of an ideal gas.	9.2.A.1 The classical model of an ideal gas assumes that the instantaneous velocities of atoms are random, the volumes of the atoms are negligible compared to the total volume occupied by the gas, the atoms collide elastically, and the only appreciable forces on the atoms are those that occur during collisions. 9.2.A.2 An ideal gas is one in which the relationships between pressure, volume, the number of moles or number of atoms, and temperature of a gas can be modeled using the equation Equation: $PV = nRT = N k_B T$ 9.2.A.3 Graphs modeling the pressure, temperature, and volume of gases can be used to describe or determine properties of that gas. 9.2.A.4 A temperature at which an ideal gas has zero pressure can be extrapolated from a graph of pressure as a function of temperature.

Learning Objective

Essential Knowledge

9.2.A
Describe the properties of an ideal gas.

9.2.A.1 The classical model of an ideal gas assumes that the instantaneous velocities of atoms are random, the volumes of the atoms are negligible compared to the total volume occupied by the gas, the atoms collide elastically, and the only appreciable forces on the atoms are those that occur during collisions.
9.2.A.2 An ideal gas is one in which the relationships between pressure, volume, the number of moles or number of atoms, and temperature of a gas can be modeled using the equation
- Equation: $PV = nRT = N k_B T$
9.2.A.3 Graphs modeling the pressure, temperature, and volume of gases can be used to describe or determine properties of that gas.
9.2.A.4 A temperature at which an ideal gas has zero pressure can be extrapolated from a graph of pressure as a function of temperature.

Source: College Board AP Course and Exam Description

An ideal gas 理想气体 obeys

$$PV=nRT \qquad(\text{or } PV=Nk_B T),$$

linking pressure $P$, volume $V$, amount ($n$ moles or $N$ molecules), and absolute temperature $T$. Use it to predict how a gas responds when you change one quantity and hold others fixed (e.g. heating at constant volume raises pressure).

Boyle's law: at constant temperature, pressure times volume is constant

Worked example. A sealed rigid container of gas is at $1.0\times10^{5}\ \text{Pa}$ and $300\ \text{K}$. It is heated to $450\ \text{K}$. Because the volume and amount are fixed, $P/T$ is constant:

$$P_2=P_1\frac{T_2}{T_1}=1.0\times10^{5}\times\frac{450}{300}=1.5\times10^{5}\ \text{Pa}.$$

Always convert temperatures to kelvin before using a gas law – using Celsius here would give nonsense.

A hot air balloon rising into the sky Heating the gas inside a balloon lowers its density, so the balloon rises — the ideal gas law in action

Explore

Explore squeezing a gas

Slide the piston in to shrink the volume. The same particles are crammed into less space, so they hit the walls more often and the pressure climbs — while $PV$ stays constant at fixed temperature.

Vocabulary Train

English	Chinese	Pinyin
ideal gas	理想气体	lǐ xiǎng qì tǐ

Exercise sheet

9.3

Thermal Energy Transfer and Equilibrium

Syllabus

Learning Objective	Essential Knowledge
9.3.A Describe the transfer of energy between two systems in thermal contact due to temperature differences of those two systems.	9.3.A.1 Two systems are in thermal contact if the systems may transfer energy by thermal processes. 9.3.A.1.i Heating is the transfer of energy into a system by thermal processes. 9.3.A.1.ii Cooling is the transfer of energy out of a system by thermal processes. 9.3.A.2 The thermal processes by which energy may be transferred between systems at different temperatures are conduction, convection, and radiation. 9.3.A.3 Energy is transferred through thermal processes spontaneously from a higher-temperature system to a lower-temperature system. 9.3.A.3.i In collisions between atoms from different systems, energy is most likely to be transferred from higher-energy atoms to lower-energy atoms. 9.3.A.3.ii After many collisions of atoms from different systems, the most probable state is one in which both systems have the same temperature. 9.3.A.4 Thermal equilibrium results when no net energy is transferred by thermal processes between two systems in thermal contact with each other.

Learning Objective

Essential Knowledge

9.3.A
Describe the transfer of energy between two systems in thermal contact due to temperature differences of those two systems.

9.3.A.1 Two systems are in thermal contact if the systems may transfer energy by thermal processes.
- 9.3.A.1.i Heating is the transfer of energy into a system by thermal processes.
- 9.3.A.1.ii Cooling is the transfer of energy out of a system by thermal processes.
9.3.A.2 The thermal processes by which energy may be transferred between systems at different temperatures are conduction, convection, and radiation.
9.3.A.3 Energy is transferred through thermal processes spontaneously from a higher-temperature system to a lower-temperature system.
- 9.3.A.3.i In collisions between atoms from different systems, energy is most likely to be transferred from higher-energy atoms to lower-energy atoms.
- 9.3.A.3.ii After many collisions of atoms from different systems, the most probable state is one in which both systems have the same temperature.
9.3.A.4 Thermal equilibrium results when no net energy is transferred by thermal processes between two systems in thermal contact with each other.

Source: College Board AP Course and Exam Description

Heat 热量 $Q$ is energy transferred because of a temperature difference; it flows from hot to cold. Two objects in contact reach thermal equilibrium 热平衡 when they share the same temperature, and net heat flow stops. The three transfer methods are conduction 传导, convection 对流, and radiation 辐射.

Explore

Add heat and watch the temperature

Adding heat usually raises temperature, but during a phase change the temperature holds flat while the energy breaks bonds. Two bodies in contact settle at one temperature — thermal equilibrium.

Vocabulary Train

English	Chinese	Pinyin
Heat	热量	rè liàng
thermal equilibrium	热平衡	rè píng héng
conduction	传导	chuán dǎo
convection	对流	duì liú
radiation	辐射	fú shè

Exercise sheet

9.4

The First Law of Thermodynamics

Syllabus

Learning Objective	Essential Knowledge
9.4.A Describe the internal energy of a system.	9.4.A.1 The internal energy of a system is the sum of the kinetic energy of the objects that make up the system and the potential energy of the configuration of those objects. 9.4.A.1.i The atoms in an ideal gas do not interact with each other via conservative forces, and the internal structure is not considered. Therefore, an ideal gas does not have internal potential energy. 9.4.A.1.ii The internal energy of an ideal monatomic gas is the sum of the kinetic energies of the constituent atoms in the gas. Equation: $U = \dfrac{3}{2} nRT = \dfrac{3}{2} N k_B T$ 9.4.A.2 Changes to a system's internal energy can result in changes to the internal structure and internal behavior of that system without changing the motion of the system's center of mass.
9.4.B Describe the behavior of a system using thermodynamic processes.	9.4.B.1 The first law of thermodynamics is a restatement of conservation of energy that accounts for energy transferred into or out of a system by work, heating, or cooling. 9.4.B.1.i For an isolated system, the total energy is constant. 9.4.B.1.ii For a closed system, the change in internal energy is the sum of energy transferred to or from the system by heating, or work done on the system. Equation: $\Delta U = Q + W$ 9.4.B.1.iii The work done on a system by a constant or average external pressure that changes the volume of that system (for example, a piston compressing a gas in a container) is defined as Equation: $W = -P \Delta V$ 9.4.B.2 Pressure-volume graphs (also known as PV diagrams) are representations used to represent thermodynamic processes. 9.4.B.2.i Lines of constant temperature on a PV diagram are called isotherms. 9.4.B.2.ii The absolute value of the work done on a gas when the gas expands or compresses is equal to the area underneath the curve of a plot of pressure vs. volume for the gas. 9.4.B.3 Special cases of thermal processes depend on the relationship between the configuration of the system, the nature of the work done on the system, and the system's surroundings. These include constant volume (isovolumetric), constant temperature (isothermal), and constant pressure (isobaric), as well as processes where no energy is transferred to or from the system through thermal processes (adiabatic).

Learning Objective

Essential Knowledge

9.4.A
Describe the internal energy of a system.

9.4.A.1 The internal energy of a system is the sum of the kinetic energy of the objects that make up the system and the potential energy of the configuration of those objects.
- 9.4.A.1.i The atoms in an ideal gas do not interact with each other via conservative forces, and the internal structure is not considered. Therefore, an ideal gas does not have internal potential energy.
- 9.4.A.1.ii The internal energy of an ideal monatomic gas is the sum of the kinetic energies of the constituent atoms in the gas.
  - Equation: $U = \dfrac{3}{2} nRT = \dfrac{3}{2} N k_B T$
9.4.A.2 Changes to a system's internal energy can result in changes to the internal structure and internal behavior of that system without changing the motion of the system's center of mass.

9.4.B
Describe the behavior of a system using thermodynamic processes.

9.4.B.1 The first law of thermodynamics is a restatement of conservation of energy that accounts for energy transferred into or out of a system by work, heating, or cooling.
- 9.4.B.1.i For an isolated system, the total energy is constant.
- 9.4.B.1.ii For a closed system, the change in internal energy is the sum of energy transferred to or from the system by heating, or work done on the system.
  - Equation: $\Delta U = Q + W$
- 9.4.B.1.iii The work done on a system by a constant or average external pressure that changes the volume of that system (for example, a piston compressing a gas in a container) is defined as
  - Equation: $W = -P \Delta V$
9.4.B.2 Pressure-volume graphs (also known as PV diagrams) are representations used to represent thermodynamic processes.
- 9.4.B.2.i Lines of constant temperature on a PV diagram are called isotherms.
- 9.4.B.2.ii The absolute value of the work done on a gas when the gas expands or compresses is equal to the area underneath the curve of a plot of pressure vs. volume for the gas.
9.4.B.3 Special cases of thermal processes depend on the relationship between the configuration of the system, the nature of the work done on the system, and the system's surroundings. These include constant volume (isovolumetric), constant temperature (isothermal), and constant pressure (isobaric), as well as processes where no energy is transferred to or from the system through thermal processes (adiabatic).

Source: College Board AP Course and Exam Description

The first law 热力学第一定律 is conservation of energy for a gas:

$$\Delta U = Q + W,$$

where $\Delta U$ is the change in internal energy 内能 (tied to temperature), $Q$ is heat added to the gas, and $W$ is work done on the gas. On a pressure–volume diagram, the work done by the gas is the area under the process curve. Watch signs: compressing a gas does positive work on it.

A gas pushing a piston out by a small distance does work equal to p times the volume change

The four thermodynamic processes drawn from a common starting state on a PV diagram

The work done during a volume change equals the area under the P-V curve

Worked example. A gas absorbs $500\ \text{J}$ of heat and, as it expands, does $200\ \text{J}$ of work on its surroundings. Find the change in its internal energy. Work done on the gas is $W=-200\ \text{J}$ (it does work, so it loses that energy):

$$\Delta U=Q+W=500+(-200)=300\ \text{J}.$$

The internal energy rises by $300\ \text{J}$, so the gas ends up hotter. Getting the sign of $W$ right is the whole game in first-law problems.

Vocabulary Train

English	Chinese	Pinyin
first law	热力学第一定律	rè lì xué dì yí dìng lǜ
internal energy	内能	nèi néng

Exercise sheet

9.5

Specific Heat and Thermal Conductivity

Syllabus

Learning Objective	Essential Knowledge
9.5.A Describe the energy required to change the temperature of an object by a certain amount.	9.5.A.1 The amount of energy required to change the temperature of a material is related to the material's specific heat. Equation: $Q = mc\Delta T$ 9.5.A.2 The specific heat of a material is an intrinsic property of that material that depends on the arrangement and interactions of the atoms that make up the material.
9.5.B Describe the rate at which energy is transferred by conduction through a given material.	9.5.B.1 The rate at which energy is transferred by conduction through a given material is related to the thermal conductivity, the physical dimensions of the material, and the temperature difference across the material. Equation: $\dfrac{Q}{\Delta t} = \dfrac{kA\Delta T}{L}$ 9.5.B.2 The thermal conductivity of a material is an intrinsic property of that material that depends on the arrangement and interactions of the atoms that make up the material. Boundary statement: AP Physics 2 will model specific heat as independent of temperature.

Learning Objective

Essential Knowledge

9.5.A
Describe the energy required to change the temperature of an object by a certain amount.

9.5.A.1 The amount of energy required to change the temperature of a material is related to the material's specific heat.
- Equation: $Q = mc\Delta T$
9.5.A.2 The specific heat of a material is an intrinsic property of that material that depends on the arrangement and interactions of the atoms that make up the material.

9.5.B
Describe the rate at which energy is transferred by conduction through a given material.

9.5.B.1 The rate at which energy is transferred by conduction through a given material is related to the thermal conductivity, the physical dimensions of the material, and the temperature difference across the material.
- Equation: $\dfrac{Q}{\Delta t} = \dfrac{kA\Delta T}{L}$
9.5.B.2 The thermal conductivity of a material is an intrinsic property of that material that depends on the arrangement and interactions of the atoms that make up the material.

Boundary statement: AP Physics 2 will model specific heat as independent of temperature.

Source: College Board AP Course and Exam Description

Specific heat 比热容 $c$ is the heat needed to raise one kilogram by one degree: $Q=mc\,\Delta T$. A high specific heat (like water's) means a substance resists temperature change.
Thermal conductivity 热导率 measures how fast heat conducts through a material; the rate of conduction rises with area and temperature difference and falls with thickness.

Conduction: vibrating particles pass energy along a metal bar

Worked example. How much heat raises the temperature of $2.0\ \text{kg}$ of water from $20\,{}^{\circ}\text{C}$ to $80\,{}^{\circ}\text{C}$? Water's specific heat is $c=4200\ \text{J/(kg}\,{}^{\circ}\text{C)}$:

$$Q=mc\,\Delta T=2.0\times4200\times(80-20)=5.0\times10^{5}\ \text{J}.$$

Water's large specific heat is why it is used as a coolant and why coastal climates are mild.

Explore

Explore how much energy heats a material

Pick a material, set the mass and temperature rise, and read the energy from $Q = mc\,\Delta T$. Water needs far more energy than the metals to warm by the same amount.

Vocabulary Train

English	Chinese	Pinyin
Specific heat	比热容	bǐ rè róng
Thermal conductivity	热导率	rè dǎo lǜ

Exercise sheet

9.6

Entropy and the Second Law of Thermodynamics

Syllabus

Learning Objective	Essential Knowledge
9.6.A Describe the change in entropy for a given system over time.	9.6.A.1 The second law of thermodynamics states that the total entropy of an isolated system can never decrease and is constant only when all processes the system undergoes are reversible. 9.6.A.2 Entropy can be qualitatively described as the tendency of energy to spread or the unavailability of some of the system's energy to do work. 9.6.A.2.i Localized energy will tend to disperse and spread out. 9.6.A.2.ii Entropy is a state function and therefore only depends on the current state or configuration of a system, not how the system reached that state. 9.6.A.2.iii Maximum entropy occurs when a system is in thermodynamic equilibrium. 9.6.A.3 The change in a system's entropy is determined by the system's interactions with its surroundings. 9.6.A.3.i Isolated systems spontaneously move toward thermodynamic equilibrium. 9.6.A.3.ii The entropy of an isolated system never decreases, but the entropy of a closed system can decrease because energy can be transferred into or out of the system. Boundary statement: Only qualitative treatment of the second law of thermodynamics is within the scope of AP Physics 2.

Learning Objective

Essential Knowledge

9.6.A
Describe the change in entropy for a given system over time.

9.6.A.1 The second law of thermodynamics states that the total entropy of an isolated system can never decrease and is constant only when all processes the system undergoes are reversible.
9.6.A.2 Entropy can be qualitatively described as the tendency of energy to spread or the unavailability of some of the system's energy to do work.
- 9.6.A.2.i Localized energy will tend to disperse and spread out.
- 9.6.A.2.ii Entropy is a state function and therefore only depends on the current state or configuration of a system, not how the system reached that state.
- 9.6.A.2.iii Maximum entropy occurs when a system is in thermodynamic equilibrium.
9.6.A.3 The change in a system's entropy is determined by the system's interactions with its surroundings.
- 9.6.A.3.i Isolated systems spontaneously move toward thermodynamic equilibrium.
- 9.6.A.3.ii The entropy of an isolated system never decreases, but the entropy of a closed system can decrease because energy can be transferred into or out of the system.

Boundary statement: Only qualitative treatment of the second law of thermodynamics is within the scope of AP Physics 2.

Source: College Board AP Course and Exam Description

Entropy 熵 measures the disorder, or the number of ways to arrange, a system. The second law 热力学第二定律: the total entropy of an isolated system never decreases – energy naturally spreads out. This sets the direction of processes: heat flows hot→cold on its own, never the reverse, and no engine can convert heat entirely into work.

Vocabulary Train

English	Chinese	Pinyin
Entropy	熵	shāng
second law	热力学第二定律	rè lì xué dì èr dìng lǜ

Exercise sheet

9.6

Exam tips

Always convert temperatures to kelvin ($T_{\text{K}}=T_{\text{C}}+273$) before using any gas law.
In the first law $\Delta U = Q + W$, get the sign of $W$ right: work done on the gas is positive; when the gas expands and does work on its surroundings, $W$ is negative.
Temperature measures the average kinetic energy of particles, so at the same $T$ lighter molecules move faster.
On a PV diagram the work done by the gas is the area under the process curve.
Use $Q=mc\,\Delta T$ for temperature change; the second law fixes the direction — heat flows hot→cold and total entropy never decreases.

Kinetic Theory of Temperature and Pressure

Explore the spread of molecular speeds

The Ideal Gas Law

Explore squeezing a gas

Thermal Energy Transfer and Equilibrium

Add heat and watch the temperature

The First Law of Thermodynamics

Specific Heat and Thermal Conductivity

Explore how much energy heats a material

Entropy and the Second Law of Thermodynamics

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