AE/ME 6765 - Important Definitions and Laws in Thermodynamics and Statistical Mechanics

Comprehensive practice flashcards covering fundamental thermodynamics, quantum foundations, statistical mechanics, and gas kinetic theory based on the lecture transcript.

How is a thermodynamic 'System' defined in these lecture notes?

A finite portion of matter or a restricted portion of space upon which attention is focused (e.g., gas in a bottle or the inside of a pipe).

According to the State Postulate, how many independent, intensive thermodynamic properties are required to uniquely define the state of a specified substance?

The number of relevant reversible work modes ($n$) plus one.

What temperature value is 'declared' for the triple-point of water to serve as an easily reproducible standard in the SI scale?

$$273.16\,K$$

State the First Law of Thermodynamics in differential form for a closed system (control mass) with negligible kinetic and potential energy changes.

$$dU = \delta Q + \delta W$$

Provide the thermodynamic definition of temperature ($T$) as derived from the Second Law for a simple compressible substance in equilibrium.

$$\frac{1}{T} \equiv \left(\frac{\partial S}{\partial U}\right)_V$$

What is the thermodynamic definition of pressure ($p$) expressed in terms of entropy ($S$), internal energy ($U$), and volume ($V$)?

$$p \equiv T \left(\frac{\partial S}{\partial V}\right)_U$$

State the fundamental Gibbs Equation for a simple compressible, nonreacting substance.

$$dU = TdS - pdV$$

Define the chemical potential ($\mu_i$) for the i$th substance in a mixture in terms of the Gibbs Free Energy (G$$).

$$\mu_i \equiv \left(\frac{\partial G}{\partial n_i}\right)_{T,p,n_{j \neq i}}$$

What is the relationship between constant pressure specific heat ($C_p$) and constant volume specific heat ($C_v$) for a perfect gas?

$C_p - C_v = nR$ (or $c_p - c_v = R$ on a per mole basis)

What is the Van't Hoff equation used to describe the temperature dependence of the equilibrium constant $K_p$?

$$\frac{d(\ln(K_p))}{dT} = \frac{\Delta \hat{H}_R}{RT^2}$$

In the Bohr model of the atom, what is the frequency ($\nu$) of electromagnetic radiation emitted or absorbed during an orbital change?

$\nu = \frac{\Delta E}{h}$, where $h$ is Planck's constant.

What are the energy eigenvalues ($E$) for a particle in a 1-D box of length $L$?

$E_n = \frac{n^2 h^2}{8mL^2}$ where $n = 1, 2, 3, \dots$

State Boltzmann's Relation, which connects entropy ($S$) to the number of microstates ($\Omega$).

$$S = k \ln(\Omega)$$

Define the molecular partition function ($Q$) in the Boltzmann limit.

$Q = \sum_i g_i e^{-\frac{\epsilon_i}{kT}}$, where $g_i$ is the degeneracy and $\epsilon_i$ is the energy of the $$i$th level.

According to Kinetic Theory, what is the expression for gas pressure ($p$) in terms of number density ($n$), molecular mass ($m$), and mean square speed ($\overline{C^2}$)?

$$p = \frac{1}{3} n m \overline{C^2}$$

What is the Arrhenius expression for the reaction rate constant ($k_f$) in terms of activation energy ($\epsilon_a$)?

$$k_f = A e^{-\frac{\epsilon_a}{kT}}$$

Define the 'Mean Free Path' ($\lambda$) for a single-species gas of hard spheres.

$\lambda = \frac{1}{\sqrt{2} n \sigma}$, where $n$ is the number density and $\sigma$ is the collision cross-section ($\pi d^2$).

What is the value of the 'Fine Structure Constant' ($\alpha$) mentioned in the Bohr-Sommerfeld model?

$\alpha \approx 0.0073$ (or $\frac{2\pi e^2}{hc}$)

Provide the translational energy ($E_{tr}$) for an ideal gas per molecule as derived from Kinetic Theory.

$$E_{tr} = \frac{3}{2} kT$$

Which distribution describes the probability of finding a molecule with a specific speed ($C$) in an equilibrium gas?

The Maxwell-Boltzmann Speed Distribution: $\chi(C) = 4\pi \left(\frac{m}{2\pi kT}\right)^{\frac{3}{2}} C^2 e^{-\frac{mC^2}{2kT}}$