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Question: The rate constant for decomposition of hydrocarbons is \(2.418 \times {10^{ - 5}}{s^{ - 1}}\) at \(5...

The rate constant for decomposition of hydrocarbons is 2.418×10−5s−12.418 \times {10^{ - 5}}{s^{ - 1}} at 546K546K . If the energy of activation is 179.9kJ/mol179.9kJ/mol , what will be the value of pre – exponential factor?

Explanation

Solution

The Arrhenius equation is a formula that depicts the relationship between the rate constant (of a chemical process), absolute temperature, and the AA factor depicts the relationship between reaction rates and absolute temperature.
Formula Used:
k=Ae−EaRTk = A{e^{\dfrac{{ - {E_a}}}{{RT}}}}
k=k = Rate of chemical reaction
A=A = Pre-exponential factor
Ea={E_a} = Activation energy
R=R = Gas constant
T=T = Temperature

Complete answer:
Given:
k=2.418×10−5s−1k = 2.418 \times {10^{ - 5}}{s^{ - 1}}
T=546KT = 546K
Ea=179kJmol−1=179.9×103Jmol−1{E_a} = 179kJmo{l^{ - 1}} = 179.9 \times {10^3}Jmo{l^{ - 1}}
To find: Pre-exponential factor
According to the Arrhenius equation,s
k=Ae−EaRTk = A{e^{\dfrac{{ - {E_a}}}{{RT}}}} .
Ink=InA−EaRTInk = InA - \dfrac{{{E_a}}}{{RT}}
log⁡k=log⁡A−Ea2.303RT\log k = \log A - \dfrac{{{E_a}}}{{2.303RT}}
log⁡A=log⁡k+Ea2.303RT\log A = \log k + \dfrac{{{E_a}}}{{2.303RT}}
Substituting the given values in above formula ,
log⁡A=log⁡(2.418×10−5s−1)+179.9×103Jmol−12.303×8.314JK−1×546K\log A = \log (2.418 \times {10^{ - 5}}{s^{ - 1}}) + \dfrac{{179.9 \times {{10}^3}Jmo{l^{ - 1}}}}{{2.303 \times 8.314J{K^{ - 1}} \times 546K}}
On Solving the above equation we get,
(0.3835−5)+17.082(0.3835 - 5) + 17.082 == 12.591712.5917
Therefore,
A=A = Antilog( 12.591712.5917)
A=3.9×1012s−1A = 3.9 \times {10^{12}}{s^{ - 1}}
Therefore, the value of the pre-exponential factor is 3.9×1012s−13.9 \times {10^{12}}{s^{ - 1}} .

Additional Information: J.J. Hood developed the Arrhenius equation after studying the fluctuation in rate constants of several processes as a function of temperature. The equation is named after Swedish chemist Svante Arrhenius, who demonstrated that it may be applied to practically any reaction. Both the Arrhenius activation energy and the rate constant kk have been measured experimentally, and they represent macroscopic reaction-specific characteristics that are not simply connected to threshold energies and the success of individual molecule collisions.

Note:
The pre-exponential component, also known as the frequency factor, is represented by the letter ‘AA' in the Arrhenius equation. This factor is concerned with molecule collisions and is defined as the frequency of correctly oriented collisions between molecules with sufficient energy to initiate a chemical reaction.