### 13.1 Temperature

- 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:
$${T}_{\text{\xba}\text{F}}=\frac{9}{5}{T}_{\text{\xba}\text{C}}+\text{32}$$$${T}_{\text{\xba}\text{C}}=\frac{5}{9}\left({T}_{\text{\xba}\text{F}}-\text{32}\right)$$$${T}_{\text{K}}={T}_{\text{\xba}\text{C}}+\text{273}\text{.}\text{15}$$$${T}_{\text{\xba}\text{C}}={T}_{\text{K}}-\text{273}\text{.}\text{15}$$
- 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.

### 13.2 Thermal 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
$$\text{\Delta}L=\mathrm{\alpha L}\text{\Delta}T,$$where $\text{\Delta}L$ is the change in length $L$, $\text{\Delta}T$ is the change in temperature, and $\alpha $ is the coefficient of linear expansion, which varies slightly with temperature.
- The change in area due to thermal expansion is
$$\text{\Delta}A=2\mathrm{\alpha A}\text{\Delta}T,$$where $\text{\Delta}A$ is the change in area.
- The change in volume due to thermal expansion is
$$\text{\Delta}V=\mathrm{\beta V}\text{\Delta}T,$$where $\beta $ is the coefficient of volume expansion and $\beta \approx \mathrm{3\alpha}$. Thermal stress is created when thermal expansion is constrained.

### 13.3 The 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:
$$\text{PV}=\text{NkT},$$where $P$ is pressure, $V$ is volume, $T$ is temperature, $N$ is number of molecules, and $k$ is the Boltzmann constant$$k=1\text{.}\text{38}\times {\text{10}}^{\u2013\text{23}}\phantom{\rule{0.25em}{0ex}}\text{J/K}.$$
- 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 ${N}_{\text{A}}$,
$${N}_{\text{A}}=6\text{.}\text{02}\times {\text{10}}^{\text{23}}\phantom{\rule{0.25em}{0ex}}{\text{mol}}^{-1}.$$
- 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:
$$\text{PV}=\text{nRT},$$where $n$ is number of moles and $R$ is the universal gas constant,$$R=8\text{.}\text{31}\phantom{\rule{0.25em}{0ex}}\text{J/mol}\cdot \text{K}.$$
- The ideal gas law is generally valid at temperatures well above the boiling temperature.

### 13.4 Kinetic 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
$$\text{PV}=\frac{1}{3}\text{Nm}\overline{{v}^{2}},$$where $P$ is the pressure (average force per unit area), $V$ is the volume of gas in the container, $N$ is the number of molecules in the container, $m$ is the mass of a molecule, and $\overline{{v}^{2}}$ is the average of the molecular speed squared.
- Thermal energy is defined to be the average translational kinetic energy $\overline{\text{KE}}$ of an atom or molecule.
- The temperature of gases is proportional to the average translational kinetic energy of atoms and molecules.
$$\overline{\text{KE}}=\frac{1}{2}m\overline{{v}^{2}}=\frac{3}{2}\text{kT}$$
or

$$\sqrt{\overline{{v}^{2}}}={v}_{\text{rms}}=\sqrt{\frac{3\text{kT}}{m}}\text{.}$$ - 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*.

### 13.5 Phase 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.

### 13.6 Humidity, 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
$$\begin{array}{c}\text{percent relative humidity}=\\ \frac{\text{vapor density}}{\text{saturation vapor density}}\times \text{100}\text{.}\end{array}$$
- The dew point is the temperature at which air reaches 100% relative humidity.