Category: Part 2: Thermodynamics

Problem Statement Solve the thermodynamics problem: Derive the speed of sound $v_s$ in an ideal gas and calculate it for air at $T = 273\ \text{K}$. Sound propagation is an adiabatic process. The adiabatic bulk modulus is $B_s = -V(\partial p/\partial V)_s = \gamma p$. The wave speed: $$v_s = \sqrt{\frac{B_s}{\rho}} = \sqrt{\frac{\gamma p}{\rho}} =…

Problem Statement Show that the heat absorbed in a reversible isobaric process equals the enthalpy change: $Q_p = \Delta H$. 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: Find the work done when $\nu=1\ \text{mol}$ of a van der Waals gas expands isothermally from $V_1$ to $V_2$ at temperature $T$. From the van der Waals equation: $p = \frac{\nu RT}{V-\nu b} – \frac{a\nu^2}{V^2}$. Work (with $\nu=1$): $$W = \int_{V_1}^{V_2} p\,dV = RT\ln\frac{V_2-b}{V_1-b} + a\left(\f Given Information See…

Problem Statement Solve the thermodynamics problem: Derive the relation $TV^{\gamma-1}=\text{const}$ for a reversible adiabatic process in an ideal gas. For an adiabatic process $dQ=0$, so the first law gives $dU = -p\,dV$: $$\nu C_v\,dT = -p\,dV$$ Using $p = \nu RT/V$: $$\nu C_v\,dT = -\frac{\nu RT}{V}dV \implies \frac{C_v\,dT}{T} = -\frac{R\,dV}{V} Given Information See problem statement…

Problem Statement Solve the thermodynamics problem: Show that the Joule-Thomson coefficient for a van der Waals gas is $\mu_{JT} = \frac{1}{C_p}\left(\frac{2a}{RT}-b\right)$. Find the inversion temperature $T_i$. The Joule-Thomson coefficient $\mu_{JT} = (\partial T/\partial p)_H$. For a van der Waals gas to first order in $a$ and $b$: $$\mu_{JT} = \ Given Information See problem statement…

Problem Statement An ideal gas undergoes a cycle: isothermal expansion ($1\to2$), isochoric cooling ($2\to3$), adiabatic compression ($3\to1$). Find the net work per cycle in terms of $T_1, T_2, \nu$. 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…

Problem Statement Solve the work-energy problem: The internal energy of a van der Waals gas is $U = \nu C_v T – a\nu^2/V$ (plus a constant). Find $(\partial U/\partial V)_T$. $$\left(\frac{\partial U}{\partial V}\right)_T = \frac{a\nu^2}{V^2}$$ This is the internal pressure — it arises because molecules must do work against intermolecular attracti Given Information See problem…

Problem Statement Solve the thermodynamics problem: Find $C_v$ and $C_p$ of a mixture of $\nu_1=1\ \text{mol}$ He and $\nu_2=2\ \text{mol}$ N₂. Total internal energy: $U = \nu_1\frac{3}{2}RT + \nu_2\frac{5}{2}RT = (1.5+5.0)RT = 6.5RT$. Total molar heat capacity at constant volume (per total mole $\nu=3$): $$C_v^{mix} = \frac{1}{\nu}\frac{dU}{dT} = \f Given Information $U = \nu_1\frac{3}{2}RT +…

Problem Statement Solve the fluid mechanics problem: One mole of water is vaporised at $100°\text{C}$ and $1\ \text{atm}$. Find the work done by the vapour against atmospheric pressure. The vapour expands from essentially zero liquid volume to the gas volume $V_{gas}$. Work at constant pressure: $$W = p\Delta V \approx p V_{gas} = \nu RT…

Problem Statement Solve the thermodynamics problem: Show that for an ideal gas with $f$ degrees of freedom: $C_v = fR/2$, $C_p = (f+2)R/2$, and $\gamma = (f+2)/f$. By the equipartition theorem, each quadratic degree of freedom has average energy $\frac{1}{2}k_BT$ per molecule. The molar internal energy: $$U = \nu\frac{f}{2}RT \implies C_v = \frac{1}{ Given Information…