Category: Part 2: Thermodynamics

Problem Statement Solve the thermodynamics problem: One mole of diatomic ideal gas ($\gamma=1.4$) expands adiabatically from $T_1=400\ \text{K}$ to $T_2=250\ \text{K}$. Find the work done. In an adiabatic process $Q=0$, so by the first law $W = -\Delta U$: $$W = -\nu C_v(T_2-T_1) = -1.0\times\frac{5}{2}\times8.314\times(250-400)$$ $$= -1.0\times20.78 Given Information See problem statement for all given…

Problem Statement Solve the thermodynamics problem: An ideal gas undergoes a polytropic process $pV^n = \text{const}$ with $n=1.3$. For $\nu=1\ \text{mol}$, $T_1=300\ \text{K}$, $T_2=400\ \text{K}$, find $W$ and $Q$. Work done in a polytropic process: $$W = \frac{\nu R(T_1-T_2)}{n-1} = \frac{1.0\times8.314\times(300-400)}{1.3-1} = \frac{-831.4}{0.3} Given Information See problem statement for all given quantities. Physical Concepts &…

Problem Statement Solve the thermodynamics problem: One mole of monatomic ideal gas is heated at constant pressure from $T_1=300\ \text{K}$ to $T_2=600\ \text{K}$. Find $W$, $\Delta U$, and $Q$. Work at constant pressure: $W = \nu R\Delta T = 1.0\times8.314\times300 = 2494\ \text{J} \approx 2.5\ \text{kJ}$. Internal energy change (monatomic: $C_v = \ Given Information…

Problem Statement Solve the work-energy problem: Two moles of diatomic gas ($\gamma=1.4$) are heated at constant volume from 200 K to 500 K. Find $Q$ and $\Delta U$. For a diatomic gas: $C_v = \frac{5}{2}R$. At constant volume $W=0$, so $Q=\Delta U$. $$Q = \Delta U = \nu C_v\Delta T = 2.0\times\frac{5}{2}\times8.314\times300 = 12{,}471\ \text{J}…

Problem Statement Solve the thermodynamics problem: One mole of ideal gas expands isothermally at $T = 300\ \text{K}$ from $V_1 = 1.0\ \text{L}$ to $V_2 = 5.0\ \text{L}$. Find $Q$, $W$, and $\Delta U$. For an ideal gas, $\Delta U = 0$ in an isothermal process (internal energy depends only on $T$). Work done by…

Problem Statement Find the Boyle temperature $T_B$ for a van der Waals gas. Calculate for N₂ ($a = 0.136\ \text{J·m}^3/\text{mol}^2$, $b = 38.5\ \text{cm}^3/\text{mol}$). 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…

Problem Statement Find $T_c$, $p_c$, $V_c$ for a van der Waals gas in terms of $a$, $b$, $R$. Calculate for CO₂ ($a = 0.364$, $b = 42.9\ \text{cm}^3/\text{mol}$). 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…

Problem Statement What fraction of ideal gas molecules have speeds below the most probable speed $v_p$? 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 relevant conservation laws and equations of motion, then solving systematically with…

Problem Statement Find the number of molecules hitting unit area of a wall per second for N₂ at $T = 300\ \text{K}$, $p = 1.0\ \text{atm}$. 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…

Problem Statement Find the ratio $n_0/n(h)$ for air at $h = 6.0\ \text{km}$, $T = 273\ \text{K}$. 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…