The diffusion coefficient of silver (Ag) in copper (Cu) has the following values:
T (K) D (m2s-1)
923 5.42 x 10-16
1173 1.34 x 10-13
(a) Calculate the activation energy for diffusion of silver in copper. Express your answer in units of J/mol.
193000
Thank you
CHEAT!
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To calculate the activation energy for diffusion of silver in copper, we will use the Arrhenius equation which relates the diffusion coefficient (D) to the temperature (T) and activation energy (Ea):
D = D0 * exp(-Ea / (RT))
Where:
D0 is the pre-exponential factor,
R is the gas constant (8.314 J/mol·K), and
T is the temperature in Kelvin.
We have two sets of data points, so we can use the equation for each set and then solve for Ea.
For the first set of data:
T1 = 923 K
D1 = 5.42 x 10^-16 m^2/s
For the second set of data:
T2 = 1173 K
D2 = 1.34 x 10^-13 m^2/s
Let's first calculate the pre-exponential factor (D0) for each set using the Arrhenius equation:
D1 = D0 * exp(-Ea / (RT1))
D0 = D1 / exp(-Ea / (RT1))
D2 = D0 * exp(-Ea / (RT2))
D0 = D2 / exp(-Ea / (RT2))
Now we can equate the values of D0 from both equations:
D1 / exp(-Ea / (RT1)) = D2 / exp(-Ea / (RT2))
To simplify the equation, let's take the natural logarithm (ln) of both sides:
ln(D1 / exp(-Ea / (RT1))) = ln(D2 / exp(-Ea / (RT2)))
Using the properties of the logarithm:
ln(D1) - Ea / (RT1) = ln(D2) - Ea / (RT2)
Now, we can isolate the activation energy (Ea) term:
Ea / (RT2) - Ea / (RT1) = ln(D2) - ln(D1)
Ea * (1 / (RT2) - 1 / (RT1)) = ln(D2) - ln(D1)
Ea = (ln(D2) - ln(D1)) / (1 / (RT2) - 1 / (RT1))
Let's substitute the values and calculate:
Ea = (ln(1.34 x 10^-13) - ln(5.42 x 10^-16)) / (1 / (8.314 J/mol·K * 1173 K) - 1 / (8.314 J/mol·K * 923 K))
Calculating this equation, we can find the value of Ea.