The Hook
A student derives that the period of a pendulum is \(T = 2\pi\sqrt{g/L}\).
Something is wrong. How do you know without doing any physics?
Check the units: \([g/L] = (m/s^2)/m = 1/s^2\), so \([\sqrt{g/L}] = 1/s\).
But period should have units of seconds, not \(1/seconds\)!
The correct formula must be \(T = 2\pi\sqrt{L/g}\).
Physical equations must be dimensionally consistent. You cannot add meters to seconds. Every term in an equation must have the same dimensions. This is not just a check — it's a constraint that limits what equations are possible.
1. Dimension Checker
Enter the dimensions of each side to verify dimensional consistency.
Common Quantity Dimensions
| Quantity | Dimensions | SI Unit |
|---|---|---|
| Energy | M L² T⁻² | J = kg·m²/s² |
| Force | M L T⁻² | N = kg·m/s² |
| Pressure | M L⁻¹ T⁻² | Pa = N/m² |
| Velocity | L T⁻¹ | m/s |
| Acceleration | L T⁻² | m/s² |
| Frequency | T⁻¹ | Hz = 1/s |
| Diffusion coef. | L² T⁻¹ | m²/s |
| Viscosity | M L⁻¹ T⁻¹ | Pa·s |
2. Scaling Laws
Dimensional analysis reveals how quantities scale with size.
Key insight: Small objects have higher surface-to-volume ratio.
• Nanoparticles: more reactive (more surface per volume)
• Catalysts: finely divided for maximum surface
• Small animals: lose heat faster
3. Diffusion Time
Time to diffuse distance \(L\): \(t \sim L^2/D\)
4. Unit Converter
Convert between common chemistry energy units.
Common Chemistry Conversions
| Quantity | Conversion |
|---|---|
| Energy | 1 eV = 96.485 kJ/mol = 8065.5 cm⁻¹ |
| Energy | 1 Hartree = 27.21 eV = 627.5 kcal/mol |
| Length | 1 Å = 10⁻¹⁰ m = 100 pm |
| Pressure | 1 atm = 101.325 kPa = 760 Torr |
| Amount | 1 mol = 6.022 × 10²³ particles |
5. Linearization
Transform nonlinear equations for easier analysis.
6. Fermi Estimation
Combine dimensional analysis with rough estimates.
Summary
Dimensional Analysis Uses
| Check equations | All terms same dimensions |
| Derive relationships | Match exponents |
| Identify scaling | Form dimensionless groups |
| Catch errors | Quick sanity check |
| Estimate | Order of magnitude |
Formula Manipulation
| Logarithm | Linearize exponentials |
| Reciprocal | Linearize Langmuir, M-M |
| Rearrange | Isolate desired variable |
| Substitute | Connect different forms |
Key Insight: Units are not just labels — they carry information. Respecting dimensional consistency is the first line of defense against errors and a powerful tool for discovery.
Exercises
- Verify that \(PV = nRT\) is dimensionally consistent.
- Derive how the frequency of a vibrating string depends on length, tension, and linear density.
- How long to diffuse 1 cm in water (\(D \approx 10^{-9}\) m²/s)?
- Convert 2.5 eV to kJ/mol.