The melting point of cobalt is 1480°C (1753 K). Calculate the free energy change for the transfer of one g-atom of cobalt from pure liquid to a 1 wt % solution in liquid iron at 1500°C (1773 K). Assume that cobalt behaves ideally in iron at that temperature. Atomic weights of Co and Fe are 58.9 and 55.85 respectively.
Question
The melting point of cobalt is 1480°C (1753 K). Calculate the free energy change for the transfer of one g-atom of cobalt from pure liquid to a 1 wt % solution in liquid iron at 1500°C (1773 K). Assume that cobalt behaves ideally in iron at that temperature. Atomic weights of Co and Fe are 58.9 and 55.85 respectively.
Solution
To solve this problem, we need to use the formula for the free energy change (ΔG) of a solution, which is given by:
ΔG = RT ln X
where R is the gas constant, T is the temperature in Kelvin, and X is the mole fraction of the solute.
Step 1: Calculate the mole fraction of cobalt in the solution.
First, we need to convert the weight percent to a mole fraction. The weight percent is given as 1 wt %, which means that there is 1 gram of cobalt in 100 grams of solution.
The number of moles of cobalt (nCo) is given by its mass divided by its atomic weight:
nCo = 1 g / 58.9 g/mol = 0.017 mol
The mass of iron in the solution is 99 g (since the total mass of the solution is 100 g and the mass of cobalt is 1 g). The number of moles of iron (nFe) is given by its mass divided by its atomic weight:
nFe = 99 g / 55.85 g/mol = 1.77 mol
The mole fraction of cobalt (XCo) is then given by the number of moles of cobalt divided by the total number of moles:
XCo = nCo / (nCo + nFe) = 0.017 mol / (0.017 mol + 1.77 mol) = 0.0095
Step 2: Calculate the free energy change.
Now we can substitute the values into the formula for ΔG:
ΔG = RT ln X = 8.314 J/(mol·K) * 1773 K * ln(0.0095) = -34,600 J/mol
Therefore, the free energy change for the transfer of one g-atom of cobalt from pure liquid to a 1 wt % solution in liquid iron at 1500°C is -34,600 J/mol.
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