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Answer :
the calculated activation energy is approximately [tex]\( 83.26 \, \text{kJ/mol} \)[/tex]
The correct option is (a).
To calculate the activation energy (Ea) using the Arrhenius equation:
[tex]\[ k = A \times e^{\frac{{-Ea}}{{RT}}} \][/tex]
Where:
- ( k ) is the rate constant
- ( A ) is the frequency factor
- ( Ea ) is the activation energy
- ( R ) is the gas constant (8.314 J/(mol·K))
- ( T ) is the temperature in Kelvin
First, let's convert the rate constant to J/mol·s to match the units of the gas constant:
[tex]\[ k = 13.0 \, \text{s}^{-1} = 13.0 \, \text{mol}^{-1}\text{s}^{-1} \][/tex]
Given[tex]\( A = 6.10 \times 10^{14} \, \text{s}^{-1} \) and \( T = 320 \, \text{K} \),[/tex]we can rearrange the Arrhenius equation to solve for ( Ea ):
[tex]\[ Ea = -\frac{{\ln{\left(\frac{k}{A}\right)}}}{{\frac{1}{RT}}} \][/tex]
[tex]\[ Ea = -\frac{{\ln{\left(\frac{13.0}{6.10 \times 10^{14}}\right)}}}{{\frac{1}{(8.314 \, \text{J/mol·K}) \times 320 \, \text{K}}}} \][/tex]
Now, let's plug in the values and calculate:
[tex]\[ Ea = -\frac{{\ln{\left(\frac{13.0}{6.10 \times 10^{14}}\right)}}}{{\frac{1}{(8.314 \times 320)}}} \][/tex]
[tex]\[ Ea =-\frac{{\ln{\left(2.13 \times 10^{-14}\right)}}}{{\frac{1}{2651.52}}} \][/tex]
[tex]\[ Ea =\frac{{-31.34}}{{0.0003767}} \][/tex]
[tex]\[ Ea = 83.26 \, \text{kJ/mol} \][/tex]
So, the calculated activation energy is approximately [tex]\( 83.26 \, \text{kJ/mol} \)[/tex]
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Rewritten by : Barada
Activation energy (Ea) is calculated using the Arrhenius equation by arranging it to solve for Ea and substituting the given values of the frequency factor (A), the rate constant (k), and temperature (T) in Kelvin. The calculated Ea in Joules/mol can then be converted to kJ/mol.
To calculate the activation energy (Ea) for a reaction, we can use the Arrhenius equation:
k = Ae^(-Ea/RT)
Where:
k is the reaction rate constant.
A is the frequency factor.
Ea is the activation energy in joules per mole.
R is the ideal gas constant, 8.314 J/mol/K.
T is the temperature in Kelvin.
Given: k = 13.0 s-1, A = 6.10 imes 1014 s-1, and T = 320 K. We rearrange the Arrhenius equation to solve for Ea:
Ea = -R × T × ln(k/A)
Plug in the values:
Ea = -(8.314 J/mol/K) × (320 K) × ln(13.0 s⁻¹ / 6.10 × 10¹⁴ s⁻¹)
After calculating, the value for Ea will be found in joules per mole, which can then be converted into kilojoules per mole (kJ/mol) by dividing by 1000.
Upon calculation:
Ea = - (8.314 J/mol/K) × (320 K) × ln(13.0 / 6.10 × 10¹⁴)
= - (8.314 J/mol/K) × (320 K)× (-32.2452)
= 84,960 J/mol or approximately 85.0 kJ/mol. (Since none of the provided options exactly matches this result, the closest option should be chosen, which would need to be further checked for accuracy.)
