Category: Part 2: Thermodynamics

Problem Statement Solve the quantum/modern physics problem: Solve the quantum/modern physics problem: At what temperature does quantum degeneracy become important for electrons in copper? Quantum effects become important when $k_BT \lesssim E_F$. Define the Fermi temperature: $$T_F = \frac{E_F}{k_B} = \frac{7.04\times1.6\times10^{-19}}{1.38\times10^{-23}} = Given Information See problem statement for all given quantities. Physical Concepts & Formulas…

Problem Statement Solve the work-energy problem: Solve the work-energy problem: Find the Fermi energy of conduction electrons in copper. (Density $\rho=8900\ \text{kg/m}^3$, $M=63.5\ \text{g/mol}$, one conduction electron per atom) Number density of electrons: $$n_e = \frac{\rho N_A}{M} = \frac{8900\times6.022\times10^{23}}{0.0635} = 8.44\times10^ Given Information See problem statement for all given quantities. Physical Concepts & Formulas This…

Problem Statement Solve the fluid mechanics problem: Solve the fluid mechanics problem: Describe the lambda transition in liquid $^4$He and its thermodynamic signature. Liquid $^4$He undergoes a phase transition at $T_\lambda = 2.17\ \text{K}$ (at SVP) from normal He-I to superfluid He-II. Name: The heat capacity diverges logarithmically, and its shap Given Information See problem…

Problem Statement Solve the thermodynamics problem: Find the Bose-Einstein condensation temperature $T_c$ for an ideal Bose gas of $^{87}$Rb atoms at density $n=10^{20}\ \text{m}^{-3}$. ($m=87\times1.66\times10^{-27}\ \text{kg}$) All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts sec Given Information See problem statement for all given quantities. Physical Concepts…

Problem Statement Sketch the Landau theory of second-order phase transitions. What determines whether a transition is first- or second-order? 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…

Problem Statement Solve the Newton’s Laws / mechanics problem: Solve the Newton’s Laws / mechanics problem: A water drop on a tilted surface will slide when the tilt angle $\alpha$ exceeds a critical value. Derive the condition in terms of advancing ($\theta_a$) and receding ($\theta_r$) contact angles. A drop slides when gravitational force exceeds the…

Problem Statement Solve the magnetic field/force problem: Solve the magnetic field/force problem: Describe the Ising model of a ferromagnet and state the main features of its phase transition. The 2D Ising model: spins $s_i = \pm1$ on a lattice with nearest-neighbour coupling $J > 0$: $$H = -J\sum_{\langle ij\rangle} s_i s_j – h\sum_i s_i$$ Phase…

Problem Statement Find the critical free energy barrier $\Delta G^*$ for homogeneous nucleation of water droplets in supersaturated vapour at supersaturation ratio $S=5$ and $T=300\ \text{K}$. ($\sigma=0.073\ \text{N/m}$, $v_l=3\times10^{-29}\ \text{m}^3$) Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution…

Problem Statement Solve the fluid mechanics problem: Solve the Newton’s Laws / mechanics problem: Water rises to $h=10.5\ \text{cm}$ in a capillary of $r=0.14\ \text{mm}$. Find $\sigma$ for water. ($\rho=1000\ \text{kg/m}^3$, $\theta=0°$) $$h = \frac{2\sigma\cos\theta}{\rho g r} \implies \sigma = \frac{h\rho g r}{2\cos\theta}$$ $$\sigma = \frac{0.105\ Given Information See problem statement for all given quantities.…

Problem Statement Write the reduced van der Waals equation and state what corresponding states implies for real-gas behaviour near critical point. Given Information See problem statement for all given quantities. Physical Concepts & Formulas The van der Waals equation of state corrects the ideal gas law for finite molecular volume and intermolecular attractions. The parameter…