ENSTP Access Contest - independent subject practice cheat sheet

Control quantifiers and the domain in pointwise versus uniform convergence.

Verify hypotheses before exchanging limits, integrals or derivatives.

A positive Hessian direction rules out a local maximum.

Theorem normalizations, constraints and topology cannot be omitted.

Distinguish similarity of maps from congruence of forms.

Projection coefficients follow residual orthogonality and the actual Gram matrix.

Repeated eigenvalues do not automatically imply diagonalizability.

Norm-spectral-radius equality is not an if-and-only-if test for normality.

Separate conditioning of a problem from stability of an algorithm.

Bisection needs continuity and a retained sign-changing bracket.

Newton convergence is conditional rather than globally guaranteed.

Euler accuracy and absolute stability are distinct requirements.

Define coordinates, frames, constraints and the system boundary.

Small-oscillation frequencies depend on mass as well as stiffness.

Momentum and kinetic-energy conservation have different conditions.

Fluid and rotating-frame formulas require their actual hypotheses.

Use geometry and boundary conditions, not symmetry by assertion.

An ideal magnetic force changes direction without doing work.

Reversible adiabatic and arbitrary adiabatic processes are different.

Thermal efficiency limits use absolute temperature and stated reservoir assumptions.

n = m/M, n = cV (V in dm3) and n = V/24.0 for gases at RTP are the three key conversions.

The limiting reagent is the one with the smallest moles ÷ coefficient; product amount depends on it.

Percentage yield measures practical success; atom economy measures how much reactant mass is useful.

Dilution uses c1V1 = c2V2 because the moles of solute do not change.

Always balance the equation before applying any molar ratio.

ΔH is negative for exothermic reactions; standard enthalpy of formation of an element in its standard state is zero.

q = mcΔT uses the total mass of solution, and ΔH per mole = −q/n.

Hess's law lets you combine any route; combustion data are used in the reverse order to formation data.

ΔH = Σ bonds broken − Σ bonds formed, and mean bond enthalpies are averages.

Feasibility is determined by ΔG = ΔH − TΔS, not by ΔH alone.

Orders are determined experimentally, not from stoichiometric coefficients.

Rate constant units depend on overall order: for second order overall, dm3 mol−1 s−1; for third order overall, dm6 mol−2 s−1.

Relative atomic mass is an abundance-weighted mean of isotope masses, calculated from mass-spectrum data.

Across a period effective nuclear charge rises, so radius falls and ionisation energy and electronegativity rise.

Large jumps in successive ionisation energies reveal the number of electrons in the outer shell.

Cr and Cu adopt 3d5 4s1 and 3d10 4s1 configurations; 4s electrons are removed first on ionisation.

Transition metals require a partially filled d subshell in at least one common oxidation state.

VSEPR: count all electron domains, then apply lone pair–lone pair > lone pair–bond pair > bond pair–bond pair repulsion.

Molecular polarity requires both polar bonds and an asymmetric arrangement of those bonds.

Higher bond order gives a shorter, stronger bond; resonance spreads the order over equivalent positions.

Only period 3 elements and below can expand their octet; N, O, F and C cannot.

Ionic lattice strength increases with ion charge and decreases with ion radius.

Convert to K and m3 before using PV = nRT; R = 8.314 J mol−1 K−1.

Real gases deviate from ideality at high pressure and low temperature.