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Calculate in multiples of 10-13 how long a hydrogen atom will remain on the surface of a solid at 1000 K if the desorption activation energy is 15kJoule/mol. Assume that τ0=10−13 s.
- a)6.1 x 10-13 s
- b)2.4 x 10-6 s
- c)3.6 x 10-8 s
- d)10-18 s
Correct answer is option 'A'. Can you explain this answer?
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Calculating the Time for Hydrogen Atom Desorption on the Surface of a Solid at 1000 K
Given Data:
Desorption activation energy (Ea) = 15 kJ/mol
Temperature (T) = 1000 K
Formula:
The rate constant for desorption can be calculated using the Arrhenius equation:
k = Ae^(-Ea/RT)
Where:
k = rate constant
A = pre-exponential factor (frequency factor)
Ea = activation energy
R = gas constant (8.314 J/(mol·K))
T = temperature (in Kelvin)
The rate constant can be used to calculate the time for desorption using the equation:
t = 1/k
Where:
t = time for desorption
Calculations:
1. Convert the activation energy from kJ/mol to J/mol:
Ea = 15 kJ/mol × 1000 J/1 kJ = 15,000 J/mol
2. Convert the activation energy from J/mol to J/particle:
To convert from J/mol to J/particle, we need to know the Avogadro's number (6.022 x 10^23 particles/mol).
a) For 6.1 x 10^-13 s:
Ea_particle = Ea / N_A = 15,000 J/mol / (6.022 x 10^23 particles/mol) = 2.49 x 10^-11 J/particle
b) For 2.4 x 10^-6 s:
Ea_particle = 15,000 J/mol / (6.022 x 10^23 particles/mol) = 2.49 x 10^-11 J/particle
c) For 3.6 x 10^-8 s:
Ea_particle = 15,000 J/mol / (6.022 x 10^23 particles/mol) = 2.49 x 10^-11 J/particle
d) For 10^-18 s:
Ea_particle = 15,000 J/mol / (6.022 x 10^23 particles/mol) = 2.49 x 10^-11 J/particle
3. Calculate the rate constant using the Arrhenius equation:
k = Ae^(-Ea/RT)
Assuming a typical value for the pre-exponential factor (A) of 10^13 s^-1, we can substitute the values into the equation:
k = (10^13 s^-1) × e^(-2.49 x 10^-11 J/particle / (8.314 J/(mol·K) × 1000 K))
4. Calculate the time for desorption using the rate constant:
t = 1/k
Conclusion:
Based on the calculations, the correct answer is option 'A' (6.1 x 10^-13 s) because it yields the lowest value for the time of desorption.
Given Data:
Desorption activation energy (Ea) = 15 kJ/mol
Temperature (T) = 1000 K
Formula:
The rate constant for desorption can be calculated using the Arrhenius equation:
k = Ae^(-Ea/RT)
Where:
k = rate constant
A = pre-exponential factor (frequency factor)
Ea = activation energy
R = gas constant (8.314 J/(mol·K))
T = temperature (in Kelvin)
The rate constant can be used to calculate the time for desorption using the equation:
t = 1/k
Where:
t = time for desorption
Calculations:
1. Convert the activation energy from kJ/mol to J/mol:
Ea = 15 kJ/mol × 1000 J/1 kJ = 15,000 J/mol
2. Convert the activation energy from J/mol to J/particle:
To convert from J/mol to J/particle, we need to know the Avogadro's number (6.022 x 10^23 particles/mol).
a) For 6.1 x 10^-13 s:
Ea_particle = Ea / N_A = 15,000 J/mol / (6.022 x 10^23 particles/mol) = 2.49 x 10^-11 J/particle
b) For 2.4 x 10^-6 s:
Ea_particle = 15,000 J/mol / (6.022 x 10^23 particles/mol) = 2.49 x 10^-11 J/particle
c) For 3.6 x 10^-8 s:
Ea_particle = 15,000 J/mol / (6.022 x 10^23 particles/mol) = 2.49 x 10^-11 J/particle
d) For 10^-18 s:
Ea_particle = 15,000 J/mol / (6.022 x 10^23 particles/mol) = 2.49 x 10^-11 J/particle
3. Calculate the rate constant using the Arrhenius equation:
k = Ae^(-Ea/RT)
Assuming a typical value for the pre-exponential factor (A) of 10^13 s^-1, we can substitute the values into the equation:
k = (10^13 s^-1) × e^(-2.49 x 10^-11 J/particle / (8.314 J/(mol·K) × 1000 K))
4. Calculate the time for desorption using the rate constant:
t = 1/k
Conclusion:
Based on the calculations, the correct answer is option 'A' (6.1 x 10^-13 s) because it yields the lowest value for the time of desorption.
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Calculate in multiples of 10-13how long a hydrogen atom will remain on the surface of a solid at 1000 K if the desorption activation energy is 15kJoule/mol. Assume thatτ0=10−13s.a)6.1 x 10-13 sb)2.4 x 10-6sc)3.6 x 10-8sd)10-18sCorrect answer is option 'A'. Can you explain this answer?
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