English
Menu

Calculation drill

Standard calculation problems from materials science courses and exams, with new numbers every time. Solve each one yourself and enter the answer with its unit: you will see whether it is right, the worked steps showing which values go into which formula, and the common mistake your number points to. You can also open the related calculator with the same numbers to check.

The numbers are practice values drawn from ranges each time. Property values come from the data of this site’s calculators (sources below) or are stated in the problem as assumptions. For real design work, use values from standards and material data sheets.

Problem types (38)

Click a type to get one problem of that type. Click a field name to start with that field selected.

Crystals

Phase diagrams

Diffusion and defects

Mechanical properties

Fracture

Thermal

Composites

Electrical, corrosion, creep

How to use

Choose a field or a problem type and start a set of 5 or 10 problems. With “More of my weak types” on, types you have got wrong more often come up more often. Clicking a type name in the list below gives one problem of that type.

Type the answer as a number and choose its unit from the menu. 1.23e-5, 1.23×10^-5 and 1.23 × 10⁻⁵ are all read. When there is a unit menu, you can answer in mm or µm or any unit listed, as long as the number matches the unit you choose (it is converted by the unit’s factor).

“Check” shows whether you are right and opens the worked steps. If you are wrong, it tells you which common mistake your number points to (radius vs diameter, °C vs K, wt% vs at%, 2θ vs θ and so on). “Show answer” opens the steps without answering.

“Copy link to this problem” gives a URL that opens the same problem with the same numbers. The URL only holds the problem type and a seed number, never the answer. Your record is kept only in this browser (localStorage) and is never sent anywhere. If your browser blocks storage, the problems still work; only the record is not kept.

Tips for solving calculation problems

Start by writing down the given values and the unknown, with symbols and units. Words such as “diameter”, “total length 2a” and “mass fraction” often differ from what the formula needs (radius, a, volume fraction), so fix them here.

Converting everything to SI (m, kg, s, K, J, Pa) before substituting is the safest habit. MPa with mm (1 MPa = 1 N/mm²) is also handy in mechanics, but with fracture toughness in MPa·√m the length must be in meters. A temperature difference ΔT is the same number in °C and K, but the T in exp(−Q/RT) or T(C + log t) must be absolute.

Check the size of your answer. Metal interplanar spacings are 0.1–0.3 nm, densities 2–22 g/cm³, the stress in a constrained steel bar heated by 100 K is a few hundred MPa, and carburized depths are 0.1 to a few mm. If you are off by a factor of 10 or more, suspect a unit conversion or a power of ten.

Keep one or two more digits in intermediate values than in the answer (storing them in the calculator memory is safest). Rounding to three digits midway and reusing the rounded value can magnify errors in subtractions, exponentials and powers enough to fall outside the tolerance.

Units

The unit menu offers other units of the same quantity (nm and Å, MPa and GPa, m²/s and cm²/s, h and s, and so on). A correct value in the chosen unit is accepted. If there is only one unit shown, answer in that unit.

Fractions can be answered as a decimal from 0 to 1 or in % (0.25 and 25 % are the same). For mass or volume fractions, either is fine.

Temperatures are answered in °C or K. Converting between them adds or subtracts 273.15 rather than multiplying, so a °C value typed with K selected is wrong. If you give a K value as °C, you will be told that you gave the absolute temperature (K) as if it were °C.

Useful conversions: 1 nm = 10 Å = 10⁻⁹ m, 1 µm = 10⁻⁶ m, 1 MPa = 1 N/mm² = 10⁶ Pa, 1 GPa = 1000 MPa, 1 cm²/s = 10⁻⁴ m²/s, 1 g/cm³ = 1000 kg/m³, 1 eV = 1.602 × 10⁻¹⁹ J, 1 kJ/mol = 1000 J/mol.

Significant figures and marking

The numbers in each problem are already rounded to the digits shown, and the reference answer is calculated from those rounded numbers. So if you use the numbers as written, rounding of the data can never make you wrong.

Answers are marked by relative error. Most types accept ±1 %. Types that go through an exponential (diffusion coefficients, carburizing, vacancy concentration, Larson–Miller time) accept ±2 %, because intermediate rounding matters more there, and types that need many digits, such as angles and lattice parameters, accept ±0.5 %. Types answered as a temperature (the highest temperature of a constrained bar, the Larson–Miller temperature) are marked by the relative error of the absolute temperature in K (±0.3 % and ±0.5 %, roughly ±1 to 6 K between 300 and 1200 K). The tolerance is shown with each problem.

Giving the answer to the number of significant figures stated (usually 3) is enough; more digits are fine. For example, if the exact value is 0.2867 nm, both 0.287 and 0.2867 are accepted. Two digits such as 0.29 may be more than 1 % off and marked wrong.

If you are just outside the tolerance (within 5 %), you will get a “Close” hint about rounding or a different constant. The method is usually right: check that you did not round intermediate values too much and that you used the constants (N_A, R, k and so on) given in the problem.

Data sources

Lattice parameters, densities, crystal structures and atomic weights
Data of the Periodic Table for Materials (lattice parameters and densities from the CRC Handbook of Chemistry and Physics; IUPAC/CIAAW standard atomic weights).
Ranges of Young’s modulus, expansion coefficient, thermal conductivity and fracture toughness
Data of the material selection chart (ranges from M. F. Ashby, “Material property data for engineering materials”, 5th ed., 2021). A value inside the range is chosen, and the problem states it as “take … as”.
Phase diagram values (Cu–Ni, Pb–Sn, Ag–Cu, Al–Si, Al–Cu, Pt–Ag) and the Fe–C eutectoid
Data of the Binary Phase Diagram Lab and the Iron–Carbon Phase Diagram Viewer (values from Callister’s Materials Science and Engineering and others; each tool page lists the source for every system). Fe–C uses eutectoid 0.76 wt% C, ferrite solubility 0.022 wt% C and Fe₃C 6.70 wt% C.
X-ray wavelengths
The same Kα1 values as the XRD Analysis Workbench (λ = hc/E from the photon energies in the X-ray Data Booklet).
Diffusion D₀ and Q, error function
The same values as the Diffusion and Surface-Treatment Simulator’s data set (Table 5.2 of W. D. Callister, Jr. and D. G. Rethwisch, Materials Science and Engineering: An Introduction, 9th ed.). The error function and its inverse are also computed as in that tool. Temperatures are chosen where the tool gives no warning (inside the phase field; above 0.75 of the melting point for substitutional diffusion), and the carburizing surface concentration is kept below the Acm line at that temperature.
Conductor resistivities
Annealed copper standard (IEC 60028: 1.7241 × 10⁻⁸ Ω·m at 20 °C, temperature coefficient 0.00393 K⁻¹) and electrical-grade aluminum (61 % IACS: 2.8264 × 10⁻⁸ Ω·m, temperature coefficient 0.00403 K⁻¹).
Physical constants
CODATA values rounded to 3–4 digits (N_A = 6.022 × 10²³ mol⁻¹, R = 8.314 J/(mol·K), k = 8.617 × 10⁻⁵ eV/K, e = 1.602 × 10⁻¹⁹ C, F = 96485 C/mol).
Assumed values
Vacancy formation energies, Poisson’s ratios, Hall–Petch constants, critical resolved shear stresses, fiber moduli, ionic radii and the Larson–Miller constant C = 20 are practice values chosen from ranges plausible for the material. The electron mobility of n-type silicon is within ±5 % of the Caughey–Thomas estimate from the donor concentration N, µ = 52.2 + 1365/[1 + (N/9.68 × 10¹⁶ cm⁻³)^0.68] cm²/(V·s) (300 K, coefficients of Jacoboni et al. 1977).

FAQ

My answer should be right, but it is marked wrong.

First check the unit menu. If the unit you chose does not match the number you typed, a correct value is marked wrong (usually with a “unit mix-up” hint). If the units match and you are slightly off, you probably rounded intermediate values too much or used a different constant. Use the numbers and constants in the problem and keep at least four digits along the way.

How are the numbers chosen?

Each type has ranges that are realistic for real materials, and values are drawn from them every time. Property values such as lattice parameters, densities, Young’s moduli and expansion coefficients come from the same data as this site’s Periodic Table for Materials, Material Selection Chart and Binary Phase Diagram Lab. For properties listed as a range (such as Young’s modulus), a value inside the range is chosen and the problem says “take … as”.

Can I solve the same problem again, or give a friend the same problem?

Yes. Open the URL from “Copy link to this problem” (?t=type&s=seed) and you get the same problem with the same numbers. The URL does not contain the answer, so you can use it to set problems for each other.

Where is my record stored? Can I delete it?

Only in this browser’s localStorage; nothing is sent to a server, and it is not shared with other devices or browsers. “Clear record” at the bottom of the page deletes it. In private browsing or when storage is blocked, no record is kept, but the problems work normally.

How does “More of my weak types” choose?

For each type, the more often you have got it wrong compared with right, the more likely it is to be chosen. Types you have not tried yet still come up at a fair rate. Within a set, the same type does not come up twice in a row.

What do I get by opening the calculator?

The related calculator opens with the same numbers filled in, so you can check the answer another way or see what happens when you change a condition. For example, the XRD Analysis Workbench shows the whole diffraction pattern for a Bragg-angle problem, and the Diffusion and Surface-Treatment Simulator shows the concentration profile for a carburizing problem. For types no tool covers, links to related terms and articles are shown instead.

The textbook answer is slightly different.

Textbooks often round intermediate values or use shortened constants (N_A = 6.02 × 10²³, R = 8.31), so the third digit can differ. The answers here are calculated without intermediate rounding from the numbers and constants given in the problem. Anything within the tolerance is accepted either way.

Do I need to remember which elements are BCC and which are FCC?

In this drill the crystal structure is given in the problem. Exams may not tell you, so it helps to remember that Fe (α), Cr, Mo, W, V, Nb and Ta are BCC, and Al, Cu, Ni, Ag, Au, Pt, Pb and Pd are FCC.

Practice questions in related topics