The peritectic decomposition of the compound ZrZn,- at 545°C may be represented as ZrZn14 =ZrZn6+8^^ allQy The alloy at the peritectic temperature contains 99.2 atom % zinc. Calculate the activity coefficient of zinc in the alloy. Given: At 545°C (818 K), (AG°ZrZn ~ AG°ZrZn ) = "6 7 8 c a l (~2837 J>·
Question
The peritectic decomposition of the compound ZrZn,- at 545°C may be represented as ZrZn14 =ZrZn6+8^^ allQy The alloy at the peritectic temperature contains 99.2 atom % zinc. Calculate the activity coefficient of zinc in the alloy. Given: At 545°C (818 K), (AG°ZrZn ~ AG°ZrZn ) = "6 7 8 c a l (~2837 J>·
Solution
The activity coefficient (γ) of a component in a solution can be calculated using the Gibbs free energy change (ΔG) for the reaction at equilibrium. The equation is:
ΔG = -RTlnK
where R is the gas constant (8.314 J/mol·K), T is the temperature in Kelvin, and K is the equilibrium constant.
The equilibrium constant K can be expressed in terms of the activities (a) of the components:
K = (a_ZrZn6 * a_Zn^8) / a_ZrZn14
Given that the alloy contains 99.2 atom % zinc, the activity of zinc (a_Zn) can be approximated as 0.992. The activities of ZrZn6 and ZrZn14 can be approximated as 1, because they are pure substances.
Substituting these values into the equation for K gives:
K = (1 * 0.992^8) / 1 = 0.992^8
Substituting this value for K into the equation for ΔG gives:
ΔG = -RTlnK = -8.314 * 818 * ln(0.992^8) = -2837 J
The activity coefficient γ can be calculated from the activity a using the equation:
a = γx
where x is the mole fraction. In this case, x = 0.992.
Solving the equation for γ gives:
γ = a / x = 0.992 / 0.992 = 1
Therefore, the activity coefficient of zinc in the alloy is 1.
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