| {"task_id": "scibench_atkins_065", "source": "atkins", "original_problem_id": "e1.22(a) ", "problem_text": "A certain gas obeys the van der Waals equation with $a=0.50 \\mathrm{~m}^6 \\mathrm{~Pa}$ $\\mathrm{mol}^{-2}$. Its volume is found to be $5.00 \\times 10^{-4} \\mathrm{~m}^3 \\mathrm{~mol}^{-1}$ at $273 \\mathrm{~K}$ and $3.0 \\mathrm{MPa}$. From this information calculate the van der Waals constant $b$. What is the compression factor for this gas at the prevailing temperature and pressure?", "unit": " ", "answer_number": "0.66"} |
| {"task_id": "scibench_atkins_042", "source": "atkins", "original_problem_id": " e1.7(a)", "problem_text": "In an attempt to determine an accurate value of the gas constant, $R$, a student heated a container of volume $20.000 \\mathrm{dm}^3$ filled with $0.25132 \\mathrm{g}$ of helium gas to $500^{\\circ} \\mathrm{C}$ and measured the pressure as $206.402 \\mathrm{cm}$ of water in a manometer at $25^{\\circ} \\mathrm{C}$. Calculate the value of $R$ from these data. (The density of water at $25^{\\circ} \\mathrm{C}$ is $0.99707 \\mathrm{g} \\mathrm{cm}^{-3}$; a manometer consists of a U-shaped tube containing a liquid. One side is connected to the apparatus and the other is open to the atmosphere. The pressure inside the apparatus is then determined from the difference in heights of the liquid.)", "unit": "$\\mathrm{JK}^{-1} \\mathrm{~mol}^{-1}$", "answer_number": "8.3147"} |
| {"task_id": "scibench_atkins_015", "source": "atkins", "original_problem_id": " p2.3(c)", "problem_text": "Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. A sample consisting of $2.0 \\mathrm{~mol} \\mathrm{CO}_2$ occupies a fixed volume of $15.0 \\mathrm{dm}^3$ at $300 \\mathrm{~K}$. When it is supplied with $2.35 \\mathrm{~kJ}$ of energy as heat its temperature increases to $341 \\mathrm{~K}$. Assume that $\\mathrm{CO}_2$ is described by the van der Waals equation of state, and calculate $\\Delta H$.", "unit": "$\\text{kJ}$", "answer_number": "+3.03"} |
| {"task_id": "scibench_calculus_023", "source": "calculus", "original_problem_id": " D.89", "problem_text": "Find the area of triangle $A B C$, correct to five decimal places, if\r\n$$\r\n|A B|=10 \\mathrm{~cm} \\quad|B C|=3 \\mathrm{~cm} \\quad \\angle A B C=107^{\\circ}\r\n$$", "unit": " $\\mathrm{cm^2}$", "answer_number": "14.34457"} |
| {"task_id": "scibench_calculus_036", "source": "calculus", "original_problem_id": " 9.R.19", "problem_text": "One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 5000 inhabitants, 160 people have a disease at the beginning of the week and 1200 have it at the end of the week. How long does it take for $80 \\%$ of the population to become infected?", "unit": " $\\mathrm{days}$", "answer_number": "15"} |
| {"task_id": "scibench_calculus_037", "source": "calculus", "original_problem_id": " 6.2.53", "problem_text": "Find the volume of the described solid S. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths $3 \\mathrm{~cm}$, $4 \\mathrm{~cm}$, and $5 \\mathrm{~cm}$", "unit": " $\\mathrm{cm}^3$", "answer_number": "10"} |
| {"task_id": "scibench_chemmc_018", "source": "chemmc", "original_problem_id": " D-8", "problem_text": "Evaluate the series\r\n$$\r\nS=\\sum_{n=1}^{\\infty} \\frac{(-1)^{n+1}}{2^n}\r\n$$", "unit": " ", "answer_number": "0.3333333"} |
| {"task_id": "scibench_chemmc_006", "source": "chemmc", "original_problem_id": "1-40 ", "problem_text": "Through what potential must a proton initially at rest fall so that its de Broglie wavelength is $1.0 \\times 10^{-10} \\mathrm{~m}$ ?", "unit": "V ", "answer_number": "0.082"} |
| {"task_id": "scibench_chemmc_002", "source": "chemmc", "original_problem_id": "D-7 ", "problem_text": "Evaluate the series\r\n$$\r\nS=\\sum_{n=0}^{\\infty} \\frac{1}{3^n}\r\n$$", "unit": " ", "answer_number": "1.5"} |
| {"task_id": "scibench_class_049", "source": "class", "original_problem_id": " 10.20 B.", "problem_text": "Calculate the effective gravitational field vector $\\textbf{g}$ at Earth's surface at the equator. Take account of the difference in the equatorial (6378 km) and polar (6357 km) radius as well as the centrifugal force. ", "unit": "$m/s^2$ ", "answer_number": "9.780"} |
| {"task_id": "scibench_class_051", "source": "class", "original_problem_id": "9.66 A. ", "problem_text": "In a typical model rocket (Estes Alpha III) the Estes C6 solid rocket engine provides a total impulse of $8.5$ N-s. Assume the total rocket mass at launch is $54$ g and that it has a rocket engine of mass $20$ g that burns evenly for $1.5$ s. The rocket diameter is $24$ mm. Assume a constant burn rate of the propellent mass ($11$ g), a rocket exhaust speed $800$ m/s, vertical ascent, and drag coefficient $c_w = 0.75$. Take into account the change of rocket mass with time and omit the effect of gravity. Find the rocket's speed at burn out.", "unit": "m/s ", "answer_number": "131"} |
| {"task_id": "scibench_class_010", "source": "class", "original_problem_id": " Problem 9.42", "problem_text": "A steel ball of velocity $5 \\mathrm{~m} / \\mathrm{s}$ strikes a smooth, heavy steel plate at an angle of $30^{\\circ}$ from the normal. If the coefficient of restitution is 0.8 , at what velocity does the steel ball bounce off the plate?", "unit": "$\\mathrm{~m} / \\mathrm{s}$ ", "answer_number": "4.3"} |
| {"task_id": "scibench_diff_021", "source": "diff", "original_problem_id": " Page 40 29", "problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime}+\\frac{1}{4} y=3+2 \\cos 2 t, \\quad y(0)=0\r\n$$\r\nDetermine the value of $t$ for which the solution first intersects the line $y=12$.", "unit": " ", "answer_number": "10.065778"} |
| {"task_id": "scibench_diff_010", "source": "diff", "original_problem_id": " page344-15", "problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime \\prime}+\\gamma y^{\\prime}+y=k \\delta(t-1), \\quad y(0)=0, \\quad y^{\\prime}(0)=0\r\n$$\r\nwhere $k$ is the magnitude of an impulse at $t=1$ and $\\gamma$ is the damping coefficient (or resistance).\r\nLet $\\gamma=\\frac{1}{2}$. Find the value of $k$ for which the response has a peak value of 2 ; call this value $k_1$.", "unit": " ", "answer_number": "2.8108"} |
| {"task_id": "scibench_diff_032", "source": "diff", "original_problem_id": " page344-14", "problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime \\prime}+\\gamma y^{\\prime}+y=\\delta(t-1), \\quad y(0)=0, \\quad y^{\\prime}(0)=0,\r\n$$\r\nwhere $\\gamma$ is the damping coefficient (or resistance).\r\nFind the time $t_1$ at which the solution attains its maximum value.", "unit": " ", "answer_number": "2.3613"} |
| {"task_id": "scibench_fund_017", "source": "fund", "original_problem_id": " Question 22.59", "problem_text": "How much work is required to turn an electric dipole $180^{\\circ}$ in a uniform electric field of magnitude $E=46.0 \\mathrm{~N} / \\mathrm{C}$ if the dipole moment has a magnitude of $p=3.02 \\times$ $10^{-25} \\mathrm{C} \\cdot \\mathrm{m}$ and the initial angle is $64^{\\circ} ?$\r\n", "unit": "$10^{-23} \\mathrm{~J}$ ", "answer_number": "1.22"} |
| {"task_id": "scibench_fund_004", "source": "fund", "original_problem_id": "1.02 ", "problem_text": "A heavy object can sink into the ground during an earthquake if the shaking causes the ground to undergo liquefaction, in which the soil grains experience little friction as they slide over one another. The ground is then effectively quicksand. The possibility of liquefaction in sandy ground can be predicted in terms of the void ratio $e$ for a sample of the ground:\r\n$$\r\ne=\\frac{V_{\\text {voids }}}{V_{\\text {grains }}} .\r\n$$\r\nHere, $V_{\\text {grains }}$ is the total volume of the sand grains in the sample and $V_{\\text {voids }}$ is the total volume between the grains (in the voids). If $e$ exceeds a critical value of 0.80 , liquefaction can occur during an earthquake. What is the corresponding sand density $\\rho_{\\text {sand }}$ ? Solid silicon dioxide (the primary component of sand) has a density of $\\rho_{\\mathrm{SiO}_2}=2.600 \\times 10^3 \\mathrm{~kg} / \\mathrm{m}^3$.", "unit": " $10^3 \\mathrm{~kg} / \\mathrm{m}^3$", "answer_number": "1.4"} |
| {"task_id": "scibench_fund_055", "source": "fund", "original_problem_id": "Question 21.25 ", "problem_text": "How many electrons would have to be removed from a coin to leave it with a charge of $+1.0 \\times 10^{-7} \\mathrm{C}$ ?", "unit": "$10^{11}$", "answer_number": "6.3"} |
| {"task_id": "scibench_matter_015", "source": "matter", "original_problem_id": " 35.1(a)", "problem_text": "Calculate the molar energy required to reverse the direction of an $\\mathrm{H}_2 \\mathrm{O}$ molecule located $100 \\mathrm{pm}$ from a $\\mathrm{Li}^{+}$ ion. Take the magnitude of the dipole moment of water as $1.85 \\mathrm{D}$.", "unit": " $10^3 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "answer_number": "1.07"} |
| {"task_id": "scibench_matter_023", "source": "matter", "original_problem_id": " 4.8(a)", "problem_text": "Electron diffraction makes use of electrons with wavelengths comparable to bond lengths. To what speed must an electron be accelerated for it to have a wavelength of $100 \\mathrm{pm}$ ? ", "unit": " $10^6 \\mathrm{~m} \\mathrm{~s}^{-1}$", "answer_number": "7.27"} |
| {"task_id": "scibench_matter_021", "source": "matter", "original_problem_id": " 73.4(a)", "problem_text": "The equilibrium pressure of $\\mathrm{O}_2$ over solid silver and silver oxide, $\\mathrm{Ag}_2 \\mathrm{O}$, at $298 \\mathrm{~K}$ is $11.85 \\mathrm{~Pa}$. Calculate the standard Gibbs energy of formation of $\\mathrm{Ag}_2 \\mathrm{O}(\\mathrm{s})$ at $298 \\mathrm{~K}$.", "unit": " $\\mathrm{~kJ} \\mathrm{~mol}^{-1}$", "answer_number": "-11.2"} |
| {"task_id": "scibench_quan_013", "source": "quan", "original_problem_id": " 14.35", "problem_text": "Given that $D_e=4.75 \\mathrm{eV}$ and $R_e=0.741 Å$ for the ground electronic state of $\\mathrm{H}_2$, find $U\\left(R_e\\right)$ for this state.", "unit": " $\\mathrm{eV}$", "answer_number": " -31.95"} |
| {"task_id": "scibench_quan_032", "source": "quan", "original_problem_id": " 7.56", "problem_text": "Calculate the uncertainty $\\Delta L_z$ for the hydrogen-atom stationary state: $2 p_z$.", "unit": " ", "answer_number": "0"} |
| {"task_id": "scibench_quan_003", "source": "quan", "original_problem_id": " 13.5", "problem_text": "The ${ }^7 \\mathrm{Li}^1 \\mathrm{H}$ ground electronic state has $D_0=2.4287 \\mathrm{eV}, \\nu_e / c=1405.65 \\mathrm{~cm}^{-1}$, and $\\nu_e x_e / c=23.20 \\mathrm{~cm}^{-1}$, where $c$ is the speed of light. (These last two quantities are usually designated $\\omega_e$ and $\\omega_e x_e$ in the literature.) Calculate $D_e$ for ${ }^7 \\mathrm{Li}^1 \\mathrm{H}$.", "unit": " $\\mathrm{eV}$", "answer_number": "2.5151"} |
| {"task_id": "scibench_stat_003", "source": "stat", "original_problem_id": " 5.6-3", "problem_text": "Let $\\bar{X}$ be the mean of a random sample of size 36 from an exponential distribution with mean 3 . Approximate $P(2.5 \\leq \\bar{X} \\leq 4)$", "unit": " ", "answer_number": "0.8185"} |
| {"task_id": "scibench_stat_058", "source": "stat", "original_problem_id": " Problem 1.1.7", "problem_text": "Given that $P(A \\cup B)=0.76$ and $P\\left(A \\cup B^{\\prime}\\right)=0.87$, find $P(A)$.", "unit": " ", "answer_number": "0.63"} |
| {"task_id": "scibench_stat_016", "source": "stat", "original_problem_id": " 5.8-5", "problem_text": "If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=1000$.", "unit": " ", "answer_number": "0.925"} |
| {"task_id": "scibench_thermo_014", "source": "thermo", "original_problem_id": " 2.10", "problem_text": "A muscle fiber contracts by $3.5 \\mathrm{~cm}$ and in doing so lifts a weight. Calculate the work performed by the fiber. Assume the muscle fiber obeys Hooke's law $F=-k x$ with a force constant $k$ of $750 . \\mathrm{N} \\mathrm{m}^{-1}$.", "unit": "$\\mathrm{~J}$", "answer_number": "0.46"} |
| {"task_id": "scibench_thermo_055", "source": "thermo", "original_problem_id": " 15.2", "problem_text": "For a two-level system where $v=1.50 \\times 10^{13} \\mathrm{~s}^{-1}$, determine the temperature at which the internal energy is equal to $0.25 \\mathrm{Nhv}$, or $1 / 2$ the limiting value of $0.50 \\mathrm{Nhv}$.", "unit": " $\\mathrm{~K}$", "answer_number": "655"} |
| {"task_id": "scibench_thermo_054", "source": "thermo", "original_problem_id": " 14.16", "problem_text": "Calculate the rotational partition function for $\\mathrm{SO}_2$ at $298 \\mathrm{~K}$ where $B_A=2.03 \\mathrm{~cm}^{-1}, B_B=0.344 \\mathrm{~cm}^{-1}$, and $B_C=0.293 \\mathrm{~cm}^{-1}$", "unit": " ", "answer_number": "5840"} |
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