Category: Part 2: Thermodynamics

Problem Statement Solve the fluid mechanics problem: Derive the expression for the dynamic viscosity of an ideal gas from kinetic theory. Consider a gas with a velocity gradient $dv_x/dz$. Molecules carry momentum across planes. The momentum flux (shear stress) is: $$\tau = \eta\frac{dv_x}{dz}$$ From kinetic theory, the viscosity is: $$\eta = \frac{1} Given Information See…

Problem Statement Solve the fluid mechanics problem: A piston of area $A=0.1\ \text{m}^2$ moves with velocity $u=10\ \text{m/s}$ into a gas at $p=1\ \text{atm}$, $T=300\ \text{K}$. Find the extra force on the piston due to molecular impacts compared to rest. When the piston moves toward the gas at speed $u \ll \bar{v}$, molecules hitting the…

Problem Statement Solve the nuclear physics problem: Show that for effusion through a small hole, the flux ratio of two gases equals the inverse square root of their molar mass ratio (Graham’s law). The effusion flux (particles per unit area per unit time): $$J = \frac{1}{4}n\bar{v} = \frac{1}{4}\frac{p}{k_BT}\sqrt{\frac{8k_BT}{\pi m}} = \frac{p}{\sqr Given Information See problem…

Problem Statement Solve the work-energy problem: Using the Maxwell-Boltzmann energy distribution, find the fraction of molecules with energy exceeding $\varepsilon_0 = k_BT$. The fraction with $\varepsilon > \varepsilon_0$: $$F(\varepsilon>\varepsilon_0) = \int_{\varepsilon_0}^{\infty} g(\varepsilon)\,d\varepsilon$$ In terms of $x = \varepsilon/(k Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental…

Problem Statement Solve the kinematics problem: Find the mean relative speed $\langle v_{rel} \rangle$ of gas molecules of mass $m$ at temperature $T$. For identical molecules, the relative velocity $\mathbf{v}_{rel} = \mathbf{v}_1 – \mathbf{v}_2$ has magnitude distributed as Maxwell with an effective mass $\mu = m/2$ (reduced mass): $$\langle v_ Given Information See problem statement…

Problem Statement Find the efficiency of an ideal Stirling cycle operating between $T_1=600\ \text{K}$ and $T_2=300\ \text{K}$ with ideal regeneration. 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 work-energy problem: Write the Maxwell distribution in terms of molecular energy $\varepsilon = \frac{1}{2}mv^2$ and find the most probable energy. The speed distribution $f(v)dv$ in terms of energy $\varepsilon$: since $\varepsilon = \frac{1}{2}mv^2$, $d\varepsilon = mv\,dv$, $v = \sqrt{2\varepsilon/m}$: $$g(\varepsilo Given Information See problem statement for all given quantities. Physical Concepts…

Problem Statement Solve the thermodynamics problem: State Boltzmann’s entropy formula and explain its physical meaning. How does it relate to the thermodynamic definition of entropy? Boltzmann’s formula: $$S = k_B\ln\Omega$$ where $\Omega$ is the number of microstates (statistical weight) corresponding to the macroscopic state. $k_B = 1.38\times10^{- Given Information See problem statement for all given…

Problem Statement Solve the thermodynamics problem: A block at $T_1=400\ \text{K}$ transfers $Q=1000\ \text{J}$ of heat to a block at $T_2=300\ \text{K}$ directly (irreversibly). Find $\Delta S_{universe}$. $$\Delta S_1 = -\frac{Q}{T_1} = -\frac{1000}{400} = -2.5\ \text{J/K}$$ $$\Delta S_2 = +\frac{Q}{T_2} = +\frac{1000}{300} = +3.33\ \text{J/K}$$ $$ Given Information See problem statement for all given quantities.…

Problem Statement Solve the thermodynamics problem: Describe the Carnot cycle on a $T$-$S$ diagram and derive the efficiency from the diagram. On a $T$-$S$ diagram the four Carnot processes appear as: 1→2 Isothermal expansion at $T_1$: horizontal line at $T_1$, $S$ increases from $S_1$ to $S_2$. Heat absorbed $Q_1 = T_1(S_2-S_1)$. 2→3 Adiabatic expan Given…