Category: Part 2: Thermodynamics

Problem Statement Find the thermal diffusivity $\chi = \kappa/(\rho c_p)$ of nitrogen at $T=300\ \text{K}$, $p=1\ \text{atm}$. ($\kappa\approx2.5\times10^{-2}\ \text{W/m·K}$, $c_p = 1040\ \text{J/kg·K}$, $\rho \approx 1.14\ \text{kg/m}^3$) Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the…

Problem Statement Solve the kinematics problem: Derive the correction to the speed of sound in a van der Waals gas compared to an ideal gas. All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) This problem draws on fundamental physical principles. The key is to identify…

Problem Statement Solve the quantum/modern physics problem: Solve the quantum/modern physics problem: Explain why the equipartition theorem fails for molecular vibrations at room temperature. The equipartition theorem requires $k_BT \gg \hbar\omega$ for each mode (the quantum of energy must be much smaller than thermal energy). For a diatomic molecule like N Given Information See problem…

Problem Statement Solve the fluid mechanics problem: Show that along the critical isotherm ($T=T_c$), the van der Waals pressure varies as $p – p_c \propto (V-V_c)^3$ near $V_c$. Expand the van der Waals pressure in reduced variables near the critical point. Setting $\phi = V/V_c = 1+x$ (small $x$) and $\tau=1$: $$\pi = \frac{8\tau/3}{\phi-1/3} –…

Problem Statement The compressibility factor is $Z = pV/(nRT)$. For an ideal gas $Z=1$. For a van der Waals gas, find $Z$ to first order in $p$. 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…

Problem Statement Solve the thermodynamics problem: One mole of ideal gas at $T$, $V_1$ undergoes Joule (free) expansion to $V_2 = 2V_1$. Find $\Delta S$ and explain why it is positive despite no heat flow. In free expansion: $Q=0$, $W=0$, $\Delta T=0$ (ideal gas). Yet entropy increases because the process is irreversible. Calculate $\Delta S$…

Problem Statement Solve the thermodynamics problem: A heat reservoir at $T_1 = 500\ \text{K}$ contains thermal energy $Q = 10\ \text{kJ}$ relative to surroundings at $T_0 = 300\ \text{K}$. Find the maximum useful work extractable. The maximum work is obtained by a Carnot engine operating between $T_1$ and $T_0$: $$W_{max} = Q\left(1-\frac{T_0}{T_1}\r Given Information See…

Problem Statement Solve the thermodynamics problem: An ideal gas Carnot engine uses 1 mol gas, $T_1=400\ \text{K}$, $T_2=200\ \text{K}$, absorbs $Q_1=5\ \text{kJ}$ per cycle. Find max work and heat rejected. $$\eta_{Carnot} = 1-\frac{T_2}{T_1} = 1-\frac{200}{400} = 0.5 = 50\%$$ $$W = \eta Q_1 = 0.5\times5000 = 2500\ \text{J} = 2.5\ \text{kJ}$$ $$Q_2 Given Information See…

Problem Statement A Stirling engine operates between $T_1=800\ \text{K}$ and $T_2=300\ \text{K}$ with regenerator efficiency $\epsilon=0.9$. Find the actual efficiency. 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 Solve the thermodynamics problem: Two large bodies at $T_1 = 500\ \text{K}$ and $T_2 = 300\ \text{K}$ are connected by a rod. Heat $Q = 1000\ \text{J}$ flows from hot to cold. Find the entropy produced. $$\Delta S = \frac{Q}{T_2} – \frac{Q}{T_1} = Q\left(\frac{1}{T_2}-\frac{1}{T_1}\right) = 1000\left(\frac{1}{300}-\frac{1}{500}\righ Given Information See problem statement for all…