Category: Part 2: Thermodynamics

Problem Statement Solve the fluid mechanics problem: Find the height to which water rises in a glass capillary of radius $r = 0.20\ \text{mm}$. ($\sigma = 0.073\ \text{N/m}$, contact angle $\theta = 0°$, $\rho = 1000\ \text{kg/m}^3$) At equilibrium, the surface tension force balances the weight of the liquid column: $$2\pi r\sigma\cos\theta = \pi r^2\…

Problem Statement Derive Young’s equation for the contact angle $\theta$ of a liquid drop on a solid surface. 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…

Problem Statement Solve the fluid mechanics problem: Find the excess pressure inside a soap bubble of radius $R = 2.0\ \text{cm}$ if $\sigma = 0.040\ \text{N/m}$. A soap bubble has two surfaces (inner and outer). The pressure-radius relation from energy minimization (or Young-Laplace equation with two interfaces): $$\Delta p = \frac{4\sigma}{R}$$ (Fac Given Information See…

Problem Statement Solve the fluid mechanics problem: Find the excess pressure inside a water drop of radius $R = 1.0\ \text{mm}$. ($\sigma_{water} = 0.073\ \text{N/m}$) A liquid drop has a single curved surface. The Young-Laplace equation for a spherical surface: $$\Delta p = \frac{2\sigma}{R}$$ $$\Delta p = \frac{2\times0.073}{1.0\times10^{-3}} = \fr Given Information See problem statement…

Problem Statement Solve the thermodynamics problem: In the Einstein model, each atom oscillates at a fixed frequency $\omega$. Write the energy and $C_v$ expressions. Each atom is a quantum harmonic oscillator. The mean energy per oscillator (3D): $$\bar{U} = 3\hbar\omega\left(\frac{1}{e^{\hbar\omega/k_BT}-1}+\frac{1}{2}\right)$$ Molar heat capacity Given Information See problem statement for all given quantities. Physical Concepts &…

Problem Statement Solve the fluid mechanics problem: Define surface tension $\sigma$ and give its units. Explain the molecular origin of surface tension. Definition: Surface tension is the force per unit length acting along a liquid surface (or equivalently, the work per unit area to increase the surface area): $$\sigma = \frac{dW}{dA} = \frac{F}{l}$$ Given Information…

Problem Statement Solve the thermodynamics problem: State the third law of thermodynamics (Nernst heat theorem) and its consequences. Statement (Nernst 1906): As $T\to0$, the entropy of any perfect crystalline substance approaches zero: $S\to0$ as $T\to0$. Equivalently: it is impossible to reach absolute zero temperature by any finite sequence of ope Given Information See problem statement…

Problem Statement Solve the thermodynamics problem: Derive the Dulong-Petit law for the molar heat capacity of a solid and state its limitation. In a solid, each atom oscillates in three dimensions. Each oscillation has 2 degrees of freedom (kinetic + potential), giving $f=6$ per atom. By the equipartition theorem: $$U = \nu\times6\times\frac{1}{2}RT Given Information See…

Problem Statement Explain why entropy increases using the statistical interpretation of thermodynamics. 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 units…

Problem Statement Describe Maxwell’s demon and explain why it does not violate the second law. 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…