At 10°C below the melting point of aluminum, 0.08% of the atom sites are vacant. At 484°C only 0.01% are vacant. Given this information, determine the energy of vacancy formation (ΔHv) for aluminum.
Energy of vacancy formation (in joules / vacancy):
1.21*10^-19
Some properties of aluminum are summarized in the following list.
To determine the energy of vacancy formation (ΔHv) for aluminum, we can use the equation:
ΔHv = R * T * ln (X₂ / X₁)
where:
ΔHv is the energy of vacancy formation in Joules per vacancy,
R is the gas constant (8.314 J/mol·K),
T is the temperature in Kelvin,
X₂ is the final vacancy concentration (0.01% or 0.0001),
X₁ is the initial vacancy concentration (0.08% or 0.0008), and
ln is the natural logarithm.
First, we need to convert the temperatures from Celsius to Kelvin. The melting point of aluminum is approximately 660°C, so 10°C below that would be (660 - 10 = 650°C). Adding 273.15 to both temperatures will give us the values in Kelvin.
Initial temperature (T₁) = 650°C + 273.15 = 923.15 K
Final temperature (T₂) = 484°C + 273.15 = 757.15 K
Now we can calculate the energy of vacancy formation (ΔHv):
ΔHv = R * T * ln (X₂ / X₁)
= 8.314 J/mol·K * (757.15 K - 923.15 K) * ln (0.0001 / 0.0008)
Calculating this expression will give us the energy of vacancy formation in Joules per vacancy for aluminum.