Category: Part 2: Thermodynamics

Problem Statement Describe the behaviour of van der Waals isotherms below the critical temperature and explain the Maxwell construction. 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 $a$ accounts…

Problem Statement Solve the fluid mechanics problem: The latent heat of vaporization of water is $L=2.26\ \text{MJ/kg}$ at $100°\text{C}$. Find how much the boiling point changes if pressure decreases by $\Delta p = 1000\ \text{Pa}$. From the Clausius-Clapeyron equation: $dp/dT = L/(T\Delta V_m)$. Molar latent heat: $L_m = 2.26\times10^6\times0.018 = Given Information See problem statement…

Problem Statement Describe the pressure-temperature phase diagram of a pure substance, identifying the triple point, critical point, and coexistence curves. 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,…

Problem Statement Derive the Clausius-Clapeyron equation for the slope of a coexistence curve in a $p$-$T$ phase diagram. 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: Derive the Kelvin equation relating the vapour pressure over a curved liquid surface to its radius of curvature. A small droplet of radius $R$ has excess internal pressure $\Delta p = 2\sigma/R$ (Young-Laplace). This increases the chemical potential of the liquid. Equating chemical potentials of liq Given Information…

Problem Statement Solve the fluid mechanics problem: Find the depression of mercury in a glass capillary of radius $r=1.0\ \text{mm}$. ($\sigma_{Hg}=0.50\ \text{N/m}$, $\theta=140°$, $\rho_{Hg}=13600\ \text{kg/m}^3$) Mercury does not wet glass ($\theta>90°$, $\cos\theta $$h = \frac{2\sigma\cos\theta}{\rho g r}$$ $\cos140° = -0.766$. $$h = \frac{2\time Given Information See problem statement for all given quantities. Physical Concepts &…

Problem Statement Solve the fluid mechanics problem: Show that the capillary rise height is inversely proportional to the tube radius (Jurin’s law). Calculate $h$ for water in capillaries of $r=0.1\ \text{mm}$ and $r=1.0\ \text{mm}$. $$h = \frac{2\sigma\cos\theta}{\rho g r} \propto \frac{1}{r}$$ This is Jurin’s law (1718): narrower tubes pull liquid h Given Information See problem…

Problem Statement Solve the fluid mechanics problem: Water ($\sigma=0.073\ \text{N/m}$, $\theta=0°$) is in a capillary of radius $r=0.5\ \text{mm}$. Find the pressure difference across the meniscus. The meniscus is approximately spherical with radius of curvature $R_c = r/\cos\theta = r$ (for $\theta=0°$). Young-Laplace: $$\Delta p = \frac{2\sigma}{R_ Given Information See problem statement for all given…

Problem Statement Solve the work-energy problem: A liquid film is stretched on a wire frame of width $l=0.10\ \text{m}$ by a distance $x=0.05\ \text{m}$. If the force needed is $F=0.012\ \text{N}$, find $\sigma$. The film has two surfaces. The force to stretch the film: $$F = 2\sigma l \implies \sigma = \frac{F}{2l} = \frac{0.012}{2\times0.10} =…

Problem Statement Solve the fluid mechanics problem: Find the work done to blow a soap bubble from radius $R_1=0$ to $R_2 = 3.0\ \text{cm}$. ($\sigma=0.040\ \text{N/m}$) A soap bubble has two surfaces. The surface energy is $E = \sigma\times2\times4\pi R^2 = 8\pi\sigma R^2$. Work done to create the bubble (against surface tension, ignoring gas compres…