Magma Crystallization: Nickel Concentration Analysis

This solution calculates the difference in nickel (Ni) concentration in a basaltic magma after 25% crystallization, comparing equilibrium and fractional crystallization processes.

Case I: Equilibrium Crystallization

In equilibrium crystallization, the remaining liquid is assumed to be in equilibrium with the crystallized solid. The concentration of Ni in the remaining liquid ($C_{L,eq}$) is calculated using the formula:

$C_{L,eq} = \frac{C_0}{1 + (K_{D,Ni} - 1) \times x}$

Where:

• $C_0$ = Initial Ni concentration = 300 ppm
• $K_{D,Ni}$ = Partition coefficient (olivine/melt) = 10
• $x$ = Fraction of solid crystallized = 0.25

Calculation:

$C_{L,eq} = \frac{300}{1 + (10 - 1) \times 0.25}$

$C_{L,eq} = \frac{300}{1 + (9 \times 0.25)}$

$C_{L,eq} = \frac{300}{1 + 2.25}$

$C_{L,eq} = \frac{300}{3.25} \approx 92.31 \text{ ppm}$

Case II: Fractional Crystallization

In fractional crystallization, the solid is continuously removed, and the concentration in the remaining liquid ($C_{L,frac}$) follows the Rayleigh equation:

$C_{L,frac} = C_0 \times F^{(K_{D,Ni} - 1)}$

Where:

• $C_0$ = Initial Ni concentration = 300 ppm
• $F$ = Fraction of liquid remaining = $1 - x = 1 - 0.25 = 0.75$
• $K_{D,Ni}$ = Partition coefficient (olivine/melt) = 10

Calculation:

$C_{L,frac} = 300 \times (0.75)^{(10 - 1)}$

$C_{L,frac} = 300 \times (0.75)^9$

$C_{L,frac} \approx 300 \times 0.07508$

$C_{L,frac} \approx 22.52 \text{ ppm}$

Difference in Nickel Concentrations

The absolute difference between the Ni concentrations in the remaining liquids is:

$\Delta C_{Ni} = |C_{L,eq} - C_{L,frac}|$

$\Delta C_{Ni} = |92.31 \text{ ppm} - 22.52 \text{ ppm}|$

$\Delta C_{Ni} \approx 69.79 \text{ ppm}$

This value lies between 69 and 71.