Question 3 of 8: BCC Vanadium — Lattice Constant and Density; Diffusion Factors

Question 3: BCC Vanadium — Lattice Constant and Density; Diffusion Factors (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. BCC vanadium; $d_{110}=0.215$ nm; $M_V=50.95$ g/mol (page 1 table); $N_A=6.02\times10^{23}$ mol$^{-1}$.

Find. (a) Lattice constant $a$ and density $\rho$. (b) Factors governing solid-state diffusion rate.

Approach

For a cubic lattice the general interplanar-spacing formula is $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$; since (110) is an allowed BCC reflection ($h+k+l=2$, even), this formula applies directly without any correction, so the lattice constant follows immediately from the given $d_{110}$. Density then follows from the standard mass-per-unit-cell over volume-per-unit-cell relation, using the BCC atom count of 2 per cell.

1. Lattice constant from $d_{110}$. $$d_{110}=\frac{a}{\sqrt{1^2+1^2+0^2}}=\frac{a}{\sqrt2}\ \Rightarrow\ a=d_{110}\sqrt2=0.215\times\sqrt2=\boxed{0.3041\ \text{nm}}.$$
2. Density. A BCC unit cell contains $n=2$ atoms (8 corners$\times\tfrac18$ + 1 body-centre); with $a=3.041\times10^{-8}$ cm and $M_V=50.95$ g/mol, $$\rho=\frac{nM_V}{N_Aa^3}=\frac{2\times50.95}{6.02\times10^{23}\times(3.041\times10^{-8})^3} =\boxed{6.02\ \text{g/cm}^3}.$$ (The accepted density of vanadium is $6.11\ \text{g/cm}^3$ — the close agreement confirms the BCC/$d_{110}$ method.)
3. (b) Factors affecting diffusion rate. Solid-state diffusion rate (the diffusivity $D$ in Fick's law) is governed by:
   • Temperature — the dominant factor, through the Arrhenius relation $D=D_0\exp(-Q_d/RT)$; diffusivity can change by orders of magnitude over a few hundred degrees.
   • Diffusion mechanism — interstitial diffusion (small solute atoms, e.g. C or N in Fe, hopping between interstitial sites) is intrinsically faster than vacancy (substitutional) diffusion, which requires an adjacent vacancy to be available.
   • Crystal structure / packing — more open structures (BCC, packing factor 0.68) diffuse faster than close-packed structures (FCC, packing factor 0.74) at a comparable temperature, because there is more interstitial free volume to move through.
   • Concentration gradient — the driving force itself; steeper gradients give a larger diffusive flux for the same $D$ (Fick's first law, $J=-D\,\Delta c/\Delta x$).
   • Diffusing species / host combination — atomic size mismatch, bond strength, and the activation energy $Q_d$ specific to that solute–solvent pair.
   • Short-circuit paths — grain boundaries, dislocations, and free surfaces offer lower-activation-energy diffusion paths than the bulk lattice, so grain size and dislocation density influence the effective (bulk-averaged) diffusion rate, especially at lower temperatures where bulk diffusion is slow.