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:
$T_{º F} = \frac{9}{5} T_{º C} + 32$
$T_{º C} = \frac{5}{9} (T_{º F} - 32)$
$T_{K} = T_{º C} + 273 . 15$
$T_{º C} = T_{K} - 273 . 15$
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.

13.2 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
$Δ L = αL Δ T,$
where $Δ L$ is the change in length $L$ , $Δ T$ is the change in temperature, and $α$ is the coefficient of linear expansion, which varies slightly with temperature.
The change in area due to thermal expansion is
$Δ A = 2 αA Δ T,$
where $Δ A$ is the change in area.
The change in volume due to thermal expansion is
$Δ V = βV Δ T,$
where $β$ is the coefficient of volume expansion and $β \approx 3α$ . Thermal stress is created when thermal expansion is constrained.

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:
$PV = NkT,$
where $P$ is pressure, $V$ is volume, $T$ is temperature, $N$ is number of molecules, and $k$ is the Boltzmann constant
$k = 1 . 38 \times 10^{- 23} J/K .$
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 $N_{A}$ ,
$N_{A} = 6 . 02 \times 10^{23} {mol}^{- 1} .$
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:
$PV = nRT,$
where $n$ is number of moles and $R$ is the universal gas constant,
$R = 8 . 31 J/mol \cdot K .$
The ideal gas law is generally valid at temperatures well above the boiling 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
$PV = \frac{1}{3} Nm \bar{v^{2}},$
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 $\bar{v^{2}}$ is the average of the molecular speed squared.
Thermal energy is defined to be the average translational kinetic energy $\bar{KE}$ of an atom or molecule.
The temperature of gases is proportional to the average translational kinetic energy of atoms and molecules.
$\bar{KE} = \frac{1}{2} m \bar{v^{2}} = \frac{3}{2} kT$

or

$\sqrt{\bar{v^{2}}} = v_{rms} = \sqrt{\frac{3 kT}{m}} .$
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.

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.

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
$percent relative humidity = \frac{vapor density}{saturation vapor density} \times 100 .$
The dew point is the temperature at which air reaches 100% relative humidity.