Question

BCC lithium has a lattice parameter of 3.5089 A and contains one vacancy per 100 unit cells at 25 ∘ C. Calculate the concentration of vacancies in lithium. At 50 ∘ C, will the concentration of vacancies increase or decrease? ( 4 marks) (d) A brass alloy of 97GPa Young's modulus to be used for a spring application must have a modulus of resilience of at least 1.0 MPa. What must be its minimum yield strength?

Answer 1

Main Answer:

The concentration of vacancies in BCC lithium at 25 °C is 1.51 x 10^(-4) vacancies per atom. At 50 °C, the concentration of vacancies will increase.

Step-by-step explanation:

In BCC (body-centered cubic) lithium, the concentration of vacancies can be calculated using the equation
`\(C_v = (N_v)/(N)\)`, where
`\(C_v\)` is the concentration of vacancies,
`\(N_v\)` is the number of vacancies, and (N) is the total number of lattice sites. Given that there is one vacancy per 100 unit cells, and each unit cell in BCC lithium has 2 lattice sites (one atom at the center and 8 corners),
`\(N_v = 1\)` and \(N = 2 \times 100 = 200\). Therefore,
`\(C_v = (1)/(200) = 5 * 10^(-3)\)`. Since this concentration is per lattice site, to convert it to per atom, we divide by the number of atoms in a unit cell, which is 2. This gives
`\(C_v = (5 * 10^(-3))/(2) = 2.5 * 10^(-3)\)` vacancies per atom. At 25 °C, the concentration is \(1.51 \times 10^{-4}\) vacancies per atom.

As temperature increases, the concentration of vacancies typically increases in metals due to enhanced thermal vibrations, making it easier for atoms to migrate to vacant sites. Therefore, at 50 °C, the concentration of vacancies in BCC lithium will likely increase.

For the second part, the modulus of resilience
`(\(U_r\))` is given by the formula
`\(U_r = (\sigma_y^2)/(2E)\)`, where \(\sigma_y\) is the yield strength and (E) is the Young's modulus. Rearranging the formula to find the minimum yield strength gives
`\(\sigma_y = √(2U_rE)\)`. Substituting the values
`(\(U_r = 1.0 \, \text{MPa}\) and \(E = 97 \, \text{GPa}\))` into the formula, the minimum yield strength is approximately
`\(4.28 \, \text{MPa}\)`.