Answer :
Answer:
first step here is to substitute the 3 of your two equations into the second;
3 Ne^(-Q_v/k(1293)) = Ne^(-Q_v/k(1566))
Since 'N' is a constant, we can remove it from both sides.
We also want to combine our two Q_v values, so we can solve for Q_v, so we should put them both on the same side:
3 = e^(-Q_v/k(1293)) / e^(-Q_v/k(1566))
3 = e^(-Q_v/k(1293) + Q_v/k(1566) ) (index laws)
ln (3) = -Q_v/k(1293) + Q_v/k(1566) (log laws)
ln (3) = -0.13Q_v / k(1566) (addition of fractions)
Q_v = ln (3)* k * 1566 / -0.13 (rearranging the equation)
Now, as long as you know Boltzmann's constant it's just a matter of substituting it for k and plugging everything into a calculator.
Energy for vacancy formation literally means the change in energy when breaking atom bonds. The energy for vacancy formation in J/mol is 68371.38 J/mol
Given that
[tex]T_1 = 1020^oC[/tex]
[tex]T_2 = 1290^oC[/tex]
Since the hypothetical metal increases by a factor of 3, then:
[tex]NV_1 = 1[/tex] --- the initial number of vacancies
[tex]NV_1 = 3[/tex] --- the final number of vacancies
The energy for vacancy formation is calculated using:
[tex]Q_v = \frac{R \times \ln(NV_1/NV_2)}{1/T_2 - 1/T_1}[/tex]
Where:
[tex]R = 8.3145[/tex] ----- molar gas constant
Convert temperatures to kelvin
[tex]T_1 = 1020+273 = 1293[/tex]
[tex]T_2 = 1290+273=1563[/tex]
So, we have:
[tex]Q_v = \frac{R \times \ln(NV_1/NV_2)}{1/T_2 - 1/T_1}[/tex]
[tex]Q_v = \frac{8.3145 \times \ln(1/3)}{1/1563 - 1/1293}[/tex]
[tex]Q_v = \frac{-9.13441187413}{-0.00013359993}[/tex]
[tex]Q_v = 68371.3821866[/tex]
Approximate
[tex]Q_v = 68371.38[/tex]
Hence, the energy for vacancy formation is 68371.38 J/mol
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