Definition: Temperature and Heat

Temperature and heat are related by the heat capacitance of an object: $$\Delta Q=C\Delta T$$ The heat capacitance of some material can sometimes be measured per unit mass or per mole.

Theorem: Thermal Expansion

When an object is heated, each of its dimensions expand according to $$\Delta L=\alpha L\Delta T,$$ where $\alpha$ is a constant with $\alpha \ll 1$. Consequently, area and volume scale approximately as $$\Delta A=2\alpha A\Delta T\qquad \Delta V=3\alpha V\Delta T$$

Definition: First Law of Thermodynamics

The First Law is conservation of energy. If heat $\Delta Q$ is inputted into a system, the heat becomes $\Delta E$ of internal energy and $\Delta W$ of work done by the system on its surroundings: $$\Delta Q=\Delta E+\Delta W$$

Definition: Ideal Gas Law

An ideal gas is an approximation for a real gas by ignoring the interactions between molecules of the gas (electrostatic, polarization, chemical reactions). An ideal gas follows the ideal gas law, which can be written in two forms: $$PV=nRT\qquad PV=Nk_BT$$ where $R=8.31\, \unit{J/mol\cdot K}$ is the ideal gas constant and $k_B=1.38\times 10^{-23}\, \unit{J/K}$ is the Boltzmann constant.

Definition: Internal Energy of Ideal Gas

We characterize the molecules of an ideal gas with its degrees of freedom $f$. A monatomic molecule has $f=3$, a diatomic molecule has $f=5$, and a polyatomic molecule has $f=6$. The constant volume molar heat capacity of an ideal gas is $$c_V=\frac{f}{2}R$$The internal energy of an ideal gas is $$E=nc_VT$$

Definition: Adiabatic Constant

The constant pressure molar heat capacity of an ideal gas is $$c_P=\frac{f+2}{2}R$$The adiabatic constant $\gamma$ of the gas is then defined as $$\gamma=\frac{c_P}{c_V}=\frac{f+2}{f}$$

Theorem: Adiabatic Processes

An adiabatic process is that in which there is no heat transfer into a gas, i.e. $\Delta Q=0$. Under these conditions, $$PV^\gamma=\text{const}$$and the work done by the gas can be shown to be $$W=\frac{1}{1-\gamma}(P_fV_f-P_0V_0)$$