Question

A mafic magma with 'Sr' concentration of 400 ppm has undergone 60% fractional crystallization. If 'Sr' is incompatible in the mineral assemblage that crystallizes from the magma with a bulk distribution coefficient of 0.2, the concentration of 'Sr' in the residual liquid is ppm (rounded off to nearest integer).

Hint:

Use the Rayleigh fractionation equation C_L = C_0 times F to the power (D-1), where F is the fraction of melt still remaining, not the fraction crystallized.

Answer 1

Correct Answer: 833

Step 1: Recall the Rayleigh fractionation equation.
When a mineral assemblage keeps crystallizing out of a magma and is removed from further contact with the melt, an incompatible trace element becomes concentrated in the remaining liquid. This behavior is described by the Rayleigh fractionation equation
\[ C_L = C_0\,F^{(D-1)} \]
where \(C_L\) is the concentration in the residual liquid, \(C_0\) is the starting concentration, \(F\) is the fraction of melt still remaining, and \(D\) is the bulk distribution coefficient of the element between the crystallizing minerals and the melt.

Step 2: Identify the given values.
The starting concentration of Sr is
\[ C_0 = 400\ \text{ppm} \]
Since 60% of the magma has crystallized, the fraction of melt remaining is
\[ F = 1 - 0.60 = 0.40 \]
The bulk distribution coefficient is
\[ D = 0.2 \]

Step 3: Substitute into the Rayleigh equation.
\[ C_L = 400 \times (0.4)^{(0.2 - 1)} = 400 \times (0.4)^{-0.8} \]

Step 4: Evaluate the exponential term.
\[ (0.4)^{-0.8} = \frac{1}{(0.4)^{0.8}} \]
Taking logarithms, \(\ln(0.4) \approx -0.9163\), so
\[ 0.8 \times (-0.9163) = -0.7330 \]
\[ (0.4)^{0.8} = e^{-0.7330} \approx 0.4805 \]
\[ (0.4)^{-0.8} \approx \frac{1}{0.4805} \approx 2.081 \]

Step 5: Compute the residual liquid concentration.
\[ C_L = 400 \times 2.081 \approx 832.5\ \text{ppm} \]

Step 6: Round off and conclude.
Rounded to the nearest integer, the Sr concentration in the residual liquid is about 833 ppm, since Sr behaves incompatibly (D less than 1) and gets enriched in the melt as crystallization proceeds.
\[ \boxed{833} \]