Thermodynamics

Ideal Gas Law

The ideal gas law states the relationship of ideal gas qualities:

$$\begin{aligned} PV = nRT = n N_A k_B T = N k_B T \end{aligned}$$

where: $P$: Pressure $V$: Volume $T$: Temperature $n$: Number of moles $N$: Number of particles $R$: Gas constant, where $R = N_A \cdot k_B$ $N_A$: Avogadro constant $k_B$: Boltzmann constant

Average Translational Kinetic Energy

Kinetic theory states the average pressure ($P$) of an ideal gas is:

$$\begin{aligned} P = \frac{N}{3V} m\bar{v}^2\\ P\frac{3V}{2N} = \frac{1}{2}m\bar{v}^2\\ \left(\frac{N k_B T}{V}\right)\frac{3V}{2N} = \frac{1}{2}m\bar{v}^2\\ \bar{E_k} = \frac{3}{2}k_B T \end{aligned}$$

First Law of Thermodynamics

The first law of thermodynamics states the law of conservation of energy where the change of internal energy ($\Delta U$) is the heat transfer ($Q$) to the system subtract the work done ($W$) from the process:

$$\begin{aligned} \Delta U = Q - W \end{aligned}$$

Second Law of Thermodynamics

The second law of thermodynamics states in any energy transform, the total entropy of an isolated system can never decrease over time.

Internal energy

The internal energy ($U$) is the accumulation of potential energy and kinetic energy. Therefore the change of internal energy is directly effected by the change of average translational kinetic energy ($\bar{E_k}$) by the change of temperature ($\Delta T$):

$$\begin{aligned} \Delta U = N \Delta \bar{E_k} = \frac{3}{2}Nk_B\Delta T = \frac{3}{2} nR\Delta T \end{aligned}$$

Thermodynamic Processes

Adiabatic Process

Adiabatic process is a thermodynamic process where there is no heat transfer ($Q=0$) between the system and surroundings:

$$\begin{aligned} \Delta U = -W \end{aligned}$$

Where for monatomic ideal gas, adiabatic process satisfy the condition:

$$\begin{aligned} \Delta PV^{\frac{5}{3}} = 0 \end{aligned}$$

Isothermal Process

Isothermal process is a thermodynamic process that occurs at constant temperature ($\Delta T = 0$), normally occurs when heat transfer cause a change in volume that the average kinetic energy of particles does not increase:

$$\begin{aligned} \Delta \left(\frac{PV}{nR}\right) = 0 \end{aligned}$$

Isovolumetric Process

Isovolumetric process is a thermodynamic process that occurs at constant volume ($\Delta V = 0$), since work done rely on the change of volume ($W = P \Delta V$ = 0), heat transfer ($Q$) direct result in the change of internal energy:

$$\begin{aligned} \Delta U = Q - W = Q \\ \Delta \left(\frac{P}{nRT}\right) = 0 \end{aligned}$$

Isobaric Process

Isobaric Process is a thermodynamic process that occurs at constant pressure ($\Delta P = 0$):

$$\begin{aligned} \Delta \left(\frac{V}{nRT}\right) = 0 \end{aligned}$$