Category: Part 2: Thermodynamics

Problem Statement State Boltzmann’s $H$-theorem and its significance for the arrow of time. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to…

Problem Statement Solve the fluid mechanics problem: How does the mean free path change if pressure is doubled at constant temperature? $$\langle l\rangle = \frac{1}{\sqrt{2}\pi d^2 n}, \quad n = \frac{p}{k_BT}$$ $$\langle l\rangle = \frac{k_BT}{\sqrt{2}\pi d^2 p} \propto \frac{1}{p}$$ If pressure is doubled ($p\to2p$), the number density doubles ($n\ Given Information See problem statement for…

Problem Statement Solve the momentum/collision problem: Find the total number of molecular collisions per unit volume per unit time in nitrogen at $T=273\ \text{K}$, $p=1\ \text{atm}$. The number of collisions per molecule per second: $z = \sqrt{2}\pi d^2 n\bar{v}$. Total collision rate per unit volume (dividing by 2 to avoid double-counting): $$Z = \fra Given…

Problem Statement At what pressure does the mean free path of nitrogen equal the diameter of a vessel $D = 0.10\ \text{m}$ at $T = 300\ \text{K}$? ($d = 0.37\ \text{nm}$) Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The…

Problem Statement Solve the nuclear physics problem: Two chambers of volume $V = 1.0\ \text{L}$ separated by a membrane with a hole of area $A = 1\ \text{mm}^2$ and thickness $l = 1\ \text{mm}$. One chamber has $n_1 = 10^{25}\ \text{m}^{-3}$ N₂, the other is empty. Find the initial diffusion rate at $T = 300\…

Problem Statement Solve the thermodynamics problem: Show that the viscosity of an ideal gas scales as $\eta \propto \sqrt{T}$ and is independent of pressure. Compare to liquids. $\eta = \frac{1}{3}\rho\bar{v}\langle l\rangle$. Substituting $\rho = nm$ and $\langle l\rangle = 1/(\sqrt{2}\pi d^2 n)$: $$\eta = \frac{m\bar{v}}{3\sqrt{2}\pi d^2}$$ Since $ Given Information $\langle l\rangle = 1/$ Physical…

Problem Statement Solve the thermodynamics problem: Calculate the Prandtl number $Pr = \eta c_p/\kappa$ for a monatomic ideal gas using kinetic theory. From kinetic theory: $\eta = \frac{1}{3}\rho\bar{v}\langle l\rangle$ and $\kappa = \frac{1}{3}\rho\bar{v}\langle l\rangle c_v$, so: $$\kappa = \eta c_v$$ Therefore: $$Pr = \frac{\eta c_p}{\kappa} = \f Given Information See problem statement for all given…

Problem Statement Solve the nuclear physics problem: Derive the diffusion coefficient $D$ for a gas from kinetic theory. Fick’s first law: $J = -D\,dn/dx$. From kinetic theory, molecules carry concentration information a distance $\langle l\rangle$ between collisions: $$D = \frac{1}{3}\bar{v}\langle l\rangle$$ Since $\langle l\rangle = 1/(\sqrt{2}\pi Given Information $\langle l\rangle = 1/$ Physical Concepts &…

Problem Statement Solve the nuclear physics problem: Show that the self-diffusion coefficient $D$ and dynamic viscosity $\eta$ of a gas are related by $D = \eta/\rho$. From kinetic theory: $$D = \frac{1}{3}\bar{v}\langle l\rangle, \qquad \eta = \frac{1}{3}\rho\bar{v}\langle l\rangle$$ Dividing: $$\frac{D}{\eta/\rho} = \frac{\frac{1}{3}\bar{v}\langle l Given Information See problem statement for all given quantities. Physical Concepts &…

Problem Statement Derive the thermal conductivity $\kappa$ of an ideal gas from kinetic theory and show it is independent of pressure. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of…