Study with Chalk Inverse
Learn the concept.
Work through the problem.
A small, carefully worked collection for college math and science. Read the reasoning, then change the inputs in a real tool.
Concepts
Factoring and zeros
Rewrite a polynomial as a product, then use its factors to understand where its graph meets the axis.
Understand the concept Math · Calculus ILimits and holes in a graph
Separate a function’s nearby behavior from its value at the point you are approaching.
Understand the concept Chemistry · General chemistryDimensional analysis
Use units to choose conversion factors, organize a calculation and check whether the answer makes sense.
Understand the concept Chemistry · General chemistryStoichiometry and theoretical yield
Use a balanced reaction and the available reactants to predict how much product can form.
Understand the concept Biology · Introductory biologyInheritance probabilities
Connect parental alleles, possible gametes and offspring probabilities without confusing genotype with phenotype.
Understand the concept Biology · Introductory biologyRestriction digests and gels
Connect DNA cut positions to fragment lengths, then interpret the pattern those fragments make on a teaching gel.
Understand the concept Math · CalculusFrom secant to derivative
A derivative is the limiting rate seen as two nearby inputs come together.
Understand the concept Math · CalculusWhy the product rule has two terms
A changing rectangle gains two edge strips and one smaller corner.
Understand the concept Math · CalculusRates through a composition
A change in x passes through an inner function before reaching the outer one.
Understand the concept Math · CalculusFrom rectangles to an integral
Signed sums approximate accumulation by adding contributions over small intervals.
Understand the concept Math · CalculusAccumulation and the Fundamental Theorem
A running integral connects area accumulated so far with the rate being accumulated.
Understand the concept Math · CalculusBuild a Taylor approximation
Match a function’s value and successive derivatives with a polynomial centered at a.
Understand the concept Math · College algebraWhy binomial coefficients appear
An expansion counts choices from repeated factors.
Understand the concept Chemistry · General chemistry IITesting a reaction mechanism with its rate law
Decide whether a proposed mechanism fits an observed rate law with a short if–then–else test: add the steps, write the slow step’s rate law, replace any intermediate, then compare.
Understand the concept Chemistry · General chemistry IIFinding reaction orders and the rate constant from experimental data
Find each reactant’s order and the rate constant k from initial-rate data, then use integrated rate law plots and half-life to find the order and k from concentration–time data.
Understand the concept Chemistry · General chemistry IIActivation energy and frequency factor from experimental data
Use rate constants measured at several temperatures to find the activation energy Ea and the frequency factor A from an Arrhenius plot, or Ea from two temperatures with the two-point form.
Understand the concept Chemistry · General chemistry IIColligative properties: freezing point, boiling point, vapor pressure and osmotic pressure
Calculate freezing-point depression, boiling-point elevation, vapor pressure with nonvolatile or volatile solutes, and osmotic pressure, and use each to find a molar mass or a van ’t Hoff factor.
Understand the concept Chemistry · General chemistry IIPotential energy diagrams: exothermic, endothermic and catalyzed reactions
Read a potential energy diagram: where ΔH, the forward and reverse activation energies and the transition state are, how exothermic and endothermic reactions differ, and what a catalyst changes and what it cannot.
Understand the concept Chemistry · General chemistry IIEnthalpy of solution: lattice energy, hydration and mixing
Build an energy cycle for dissolving: separate the solute, separate the solvent and mix, or for an ionic solid, break the lattice and hydrate the ions. Then use the cycle to find ΔH(soln) and predict whether a solution warms or cools.
Understand the conceptWorked Examples
Find both zeros of a quadratic
Solve x² − 5x + 6 = 0, then connect the answer to a graph.
Work through the example Math · Calculus IFind a limit at an excluded input
Find the limit of (x² − 4)/(x − 2) as x approaches 2.
Work through the example Math · Calculus IRewriting a trigonometric limit
Use the special sine limit to evaluate this expression. Angles are in radians; the denominator is the sine of twenty-one times θ.
Work through the example Chemistry · General chemistryConvert a mass using density
A sample occupies 25.0 mL and has density 1.20 g/mL. What is its mass?
Work through the example Chemistry · General chemistryPredict CO₂ from calcium carbonate
Model 5.00 g of pure CaCO₃ with 0.200 mol HCl. What is the theoretical mass of CO₂?
Work through the example Biology · Introductory biologyInterpret an Aa × Aa cross
Under complete dominance, what genotype and phenotype probabilities follow from two heterozygous parents?
Work through the example Biology · Introductory biologyPredict a digest and its gel pattern
A linear 6000 bp molecule has cuts at 1000 and 3500 bp. Find its fragments and predict the visible band positions.
Work through the example Math · CalculusDerive the slope of x² at 2
Find the tangent slope for f(x)=x² at x=2.
Work through the example Math · CalculusDifferentiate x²(x+1)
Find the derivative in product form and check by expansion.
Work through the example Math · CalculusDifferentiate (3x+1)²
Follow the inner and outer rates.
Work through the example Math · CalculusCompare sums for x² on [0,1]
Use two rectangles and compare with the integral.
Work through the example Math · CalculusAccumulate a changing signed rate
Find F(x)=∫₀ˣ(t−1)dt and explain when it decreases.
Work through the example Math · CalculusApproximate e^0.2 with a quadratic
Build the degree-two approximation at zero and bound its error.
Work through the example Math · College algebraExpand (x−2)³
Use coefficients, powers and signs together.
Work through the example Chemistry · General chemistry IIRule a mechanism in or out for 2NO + Br₂
Nitric oxide and bromine form nitrosyl bromide, 2NO(g) + Br₂(g) → 2NOBr(g). Experiments give rate = k[NO]²[Br₂]. Which of these two-step mechanisms is consistent with that rate law? Mechanism A: NO + Br₂ → NOBr₂ (slow), then NOBr₂ + NO → 2NOBr (fast). Mechanism B: NO + Br₂ ⇌ NOBr₂ (fast), then NOBr₂ + NO → 2NOBr (slow).
Work through the example Chemistry · General chemistry IIFind the orders and k for 2NO + O₂ from initial rates
For 2NO(g) + O₂(g) → 2NO₂(g) at a fixed temperature, three trials give these initial rates. Find the order in NO, the order in O₂, the rate law and k with units.
Work through the example Chemistry · General chemistry IIFind the order and k from concentration–time data
A reactant A decomposes at constant temperature. Its concentration is measured every 20 minutes. Use integrated rate law plots to find the order, k and the half-life, and predict [A] at 120 minutes.
Work through the example Chemistry · General chemistry IIFind Ea and A from an Arrhenius plot
A first-order reaction has these rate constants at four temperatures. Find the activation energy and the frequency factor.
Work through the example Chemistry · General chemistry IIFind Ea from two rate constants and predict a third
A reaction has k = 2.5 × 10⁻³ s⁻¹ at 298 K and k = 1.1 × 10⁻² s⁻¹ at 318 K. Find Ea, then predict k at 338 K.
Work through the example Chemistry · General chemistry IIFreezing and boiling points of an ethylene glycol solution
25.0 g of ethylene glycol, C₂H₆O₂, is dissolved in 100.0 g of water. Find the freezing and boiling points of the solution. For water, Kf = 1.86 °C·kg/mol and Kb = 0.512 °C·kg/mol.
Work through the example Chemistry · General chemistry IIVapor pressure of a glucose solution
50.0 g of glucose, C₆H₁₂O₆, is dissolved in 250.0 g of water at 25 °C, where pure water’s vapor pressure is 23.8 torr. Find the solution’s vapor pressure and the lowering.
Work through the example Chemistry · General chemistry IIVapor pressure and vapor composition of a benzene–toluene mixture
A liquid mixture contains 1.00 mol benzene and 2.00 mol toluene at 25 °C, where the pure vapor pressures are 95.1 torr (benzene) and 28.4 torr (toluene). Treating the mixture as ideal, find the total vapor pressure and the mole fraction of benzene in the vapor.
Work through the example Chemistry · General chemistry IIMolar mass of a protein from osmotic pressure
A solution of 0.500 g of a protein in enough water to make 100.0 mL has an osmotic pressure of 1.86 torr at 25.0 °C. Find the protein’s molar mass.
Work through the example Chemistry · General chemistry IIRead an energy diagram with and without a catalyst
A reaction has ΔH = −40 kJ/mol and Ea = 50 kJ/mol. A catalyst lowers the forward barrier to 35 kJ/mol. Find the reverse barriers with and without the catalyst, say what happens to ΔH and K, and estimate how much faster the catalyzed reaction is at 298 K.
Work through the example Chemistry · General chemistry IIEnthalpy of solution of sodium chloride from an energy cycle
Using a data table with lattice energy −771 kJ/mol for NaCl and hydration enthalpies −406 kJ/mol for Na⁺ and −364 kJ/mol for Cl⁻, build the energy cycle and find ΔH(soln).
Work through the example