Estimate the fraction of of unoccupied atomic sites in lead (Pb) at a temperature 10K below its melting point.
DATA:
Vacancy Fraction, nv/N T (degrees C)
1.8*10^-7 100
7.2*10^-6 200
Express your answer in terms of vacancy fraction, nv/N:
1*10^-4
1*10^-4
To estimate the fraction of unoccupied atomic sites in lead (Pb) at a temperature 10K below its melting point, we can use the given data points and interpolate between them.
First, let's convert the temperature given (10K below the melting point) into degrees Celsius. If the melting point of lead is approximately 327°C, then 10K below that would be 327°C - 10°C = 317°C.
Now, we need to find the vacancy fraction corresponding to a temperature of 317°C. To do this, we can use linear interpolation.
Linear interpolation formula:
nv/N = m(T - T1) + nv1/N1
Where:
nv/N = vacancy fraction at 317°C
m = slope of the line between the two data points
T = temperature in degrees Celsius (317°C)
T1 = temperature in degrees Celsius for the first data point (100°C)
nv1/N1 = vacancy fraction for the first data point
Calculating the slope:
m = (nv2/N2 - nv1/N1) / (T2 - T1)
m = (7.2*10^-6 - 1.8*10^-7) / (200 - 100)
m = (7.0*10^-6) / 100
m = 7.0*10^-8
Plugging the values into the linear interpolation formula:
nv/N = (7.0*10^-8)(317 - 100) + 1.8*10^-7
nv/N = 7.0*10^-8 * 217 + 1.8*10^-7
nv/N = 1.519*10^-5 + 1.8*10^-7
nv/N ≈ 1.537*10^-5
Therefore, the estimated fraction of unoccupied atomic sites in lead (Pb) at a temperature 10K below its melting point is approximately 1.537*10^-5, expressed in terms of vacancy fraction, nv/N.