 College Physics for AP® Courses

# Section Summary

### 13.1Temperature

• Temperature is the quantity measured by a thermometer.
• Temperature is related to the average kinetic energy of atoms and molecules in a system.
• Absolute zero is the temperature at which there is no molecular motion.
• There are three main temperature scales: Celsius, Fahrenheit, and Kelvin.
• Temperatures on one scale can be converted to temperatures on another scale using the following equations:
$T º F = 9 5 T º C + 32 T º F = 9 5 T º C + 32 size 12{T rSub { size 8{°F} } = { {9} over {5} } T rSub { size 8{°C} } +"32"} {}$
$T º C = 5 9 T º F − 32 T º C = 5 9 T º F − 32 size 12{T rSub { size 8{°C} } = { {5} over {9} } left (T rSub { size 8{°F} } - "32" right )} {}$
$T K = T º C + 273 . 15 T K = T º C + 273 . 15 size 12{T rSub { size 8{K} } =T rSub { size 8{°C} } +"273" "." "15"} {}$
$T º C = T K − 273 . 15 T º C = T K − 273 . 15 size 12{T rSub { size 8{°C} } =T rSub { size 8{K} } - "273" "." "15"} {}$
• Systems are in thermal equilibrium when they have the same temperature.
• Thermal equilibrium occurs when two bodies are in contact with each other and can freely exchange energy.
• The zeroth law of thermodynamics states that when two systems, A and B, are in thermal equilibrium with each other, and B is in thermal equilibrium with a third system, C, then A is also in thermal equilibrium with C.

### 13.2Thermal Expansion of Solids and Liquids

• Thermal expansion is the increase, or decrease, of the size (length, area, or volume) of a body due to a change in temperature.
• Thermal expansion is large for gases, and relatively small, but not negligible, for liquids and solids.
• Linear thermal expansion is
$ΔL=αLΔT,ΔL=αLΔT, size 12{ΔL=αLΔT} {}$
where $ΔLΔL size 12{ΔL} {}$ is the change in length $LL size 12{L} {}$, $ΔTΔT size 12{ΔT} {}$ is the change in temperature, and $αα size 12{α} {}$ is the coefficient of linear expansion, which varies slightly with temperature.
• The change in area due to thermal expansion is
$ΔA=2αAΔT,ΔA=2αAΔT, size 12{ΔA=2αAΔT} {}$
where $ΔAΔA size 12{ΔA} {}$ is the change in area.
• The change in volume due to thermal expansion is
$ΔV=βVΔT,ΔV=βVΔT, size 12{ΔV=βVΔT} {}$
where $ββ size 12{β} {}$ is the coefficient of volume expansion and $β≈3αβ≈3α size 12{β approx 3α} {}$. Thermal stress is created when thermal expansion is constrained.

### 13.3The Ideal Gas Law

• The ideal gas law relates the pressure and volume of a gas to the number of gas molecules and the temperature of the gas.
• The ideal gas law can be written in terms of the number of molecules of gas:
$PV=NkT,PV=NkT, size 12{ ital "PV"= ital "NkT"} {}$
where $PP size 12{P} {}$ is pressure, $VV size 12{V} {}$ is volume, $TT size 12{T} {}$ is temperature, $NN size 12{N} {}$ is number of molecules, and $kk size 12{k} {}$ is the Boltzmann constant
$k=1.38×10–23 J/K.k=1.38×10–23 J/K. size 12{k=1 "." "38" times "10" rSup { size 8{–"38"} } " J/K"} {}$
• A mole is the number of atoms in a 12-g sample of carbon-12.
• The number of molecules in a mole is called Avogadro’s number $NANA size 12{N rSub { size 8{A} } } {}$,
$NA=6.02×1023mol−1.NA=6.02×1023mol−1. size 12{N rSub { size 8{A} } =6 "." "02" times "10" rSup { size 8{"23"} } "mol" rSup { size 8{ - 1} } } {}$
• A mole of any substance has a mass in grams equal to its molecular weight, which can be determined from the periodic table of elements.
• The ideal gas law can also be written and solved in terms of the number of moles of gas:
$PV=nRT,PV=nRT, size 12{ ital "PV"= ital "nRT"} {}$
where $nn size 12{n} {}$ is number of moles and $RR size 12{R} {}$ is the universal gas constant,
$R=8.31J/mol⋅K.R=8.31J/mol⋅K. size 12{R=8 "." "31""J/mol" cdot K} {}$
• The ideal gas law is generally valid at temperatures well above the boiling temperature.

### 13.4Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature

• Kinetic theory is the atomistic description of gases as well as liquids and solids.
• Kinetic theory models the properties of matter in terms of continuous random motion of atoms and molecules.
• The ideal gas law can also be expressed as
$PV = 1 3 Nm v 2 ¯ , PV = 1 3 Nm v 2 ¯ , size 12{ ital "PV"= { {1} over {3} } ital "Nm" {overline {v rSup { size 8{2} } }} ,} {}$
where $PP size 12{P} {}$ is the pressure (average force per unit area), $VV size 12{V} {}$ is the volume of gas in the container, $NN size 12{N} {}$ is the number of molecules in the container, $mm size 12{m} {}$ is the mass of a molecule, and $v2¯v2¯ size 12{ {overline {v rSup { size 8{2} } }} } {}$ is the average of the molecular speed squared.
• Thermal energy is defined to be the average translational kinetic energy $KE¯KE¯ size 12{ {overline {"KE"}} } {}$ of an atom or molecule.
• The temperature of gases is proportional to the average translational kinetic energy of atoms and molecules.
$KE ¯ = 1 2 m v 2 ¯ = 3 2 kT KE ¯ = 1 2 m v 2 ¯ = 3 2 kT size 12{ {overline {"KE"}} = { {1} over {2} } m {overline {v rSup { size 8{2} } }} = { {3} over {2} } ital "kT"} {}$

or

$v 2 ¯ = v rms = 3 kT m . v 2 ¯ = v rms = 3 kT m . size 12{ sqrt { {overline {v rSup { size 8{2} } }} } =v rSub { size 8{"rms"} } = sqrt { { {3 ital "kT"} over {m} } } "." } {}$
• The motion of individual molecules in a gas is random in magnitude and direction. However, a gas of many molecules has a predictable distribution of molecular speeds, known as the Maxwell-Boltzmann distribution.

### 13.5Phase Changes

• Most substances have three distinct phases: gas, liquid, and solid.
• Phase changes among the various phases of matter depend on temperature and pressure.
• The existence of the three phases with respect to pressure and temperature can be described in a phase diagram.
• Two phases coexist (i.e., they are in thermal equilibrium) at a set of pressures and temperatures. These are described as a line on a phase diagram.
• The three phases coexist at a single pressure and temperature. This is known as the triple point and is described by a single point on a phase diagram.
• A gas at a temperature below its boiling point is called a vapor.
• Vapor pressure is the pressure at which a gas coexists with its solid or liquid phase.
• Partial pressure is the pressure a gas would create if it existed alone.
• Dalton’s law states that the total pressure is the sum of the partial pressures of all of the gases present.

### 13.6Humidity, Evaporation, and Boiling

• Relative humidity is the fraction of water vapor in a gas compared to the saturation value.
• The saturation vapor density can be determined from the vapor pressure for a given temperature.
• Percent relative humidity is defined to be
$percent relative humidity = vapor density saturation vapor density × 100 . percent relative humidity = vapor density saturation vapor density × 100 . size 12{ size 11{"percent relative humidity"= { { size 11{"vapor density"}} over { size 11{"saturation vapor density"}} } times "100" "." }} {}$
• The dew point is the temperature at which air reaches 100% relative humidity.
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