The rate constant of a chemical reaction increased from 0.100 s-1 to 3.00 s-1 upon raising the temperature from 25.0 ∘C to 55.0 ∘C.
Part A: Calculate the value of ((1/T2)-(1/T1)) where T1 is the initial temperature and T2 is the final temperature. Express your answer numerically in K-1
Part B: Calculate the value of ln (k1/k2) where k1 and k2 correspond to the rate constants at the initial and the final temperatures as defined in part A. Express your answer numerically.
Part C: What is the activation energy of the reaction? Express your answer numerically in kilojoules per mole.
Answer:
Step 1: Explanation
The Arrhenius equation allows us to calculate activation energies if the rate constant is known as activation energy term Ea increases, the rate constant k decreases and therefore the rate of reaction decreases
K =Ae-Ea/RT
or,
when two different temperature and two different rate constant is given then we used the following given equation
ln ( k1/k2) = Ea/R(1/T2-1/T1)
where,
k represents the rate constant,
Ea is the activation energy,
R is the gas constant (8.3145 J/K mol),
and T is the temperature expressed in Kelvin
(A) Step 2: Calculation of temperature
Temperature (T1) =(25+273.15)K = 298.15 K
Temperature (T2) = ( 55+273.15)K = 328.15 K
hence, on substituting the value
=> 1/T2-1/T1
=> (1/328.15 K ) - (1/298.15 K )
=> -3.0663 × 10-4 K-1
(B) Step 3: Calculation of Rate constant
Rate constant (K1) = 0.100 s-1
Rate constant (K2) = 3.00 s-1
hence, on substituting the value
=> ln ( k1/k2)
=> ln ( 0.100 s-1 / 3.00 s-1)
=> -3.4012
(C) Step 5: Calculation of Activation energy
we got
ln ( k1/k2) = -3.4012
1/T2-1/T1 = -3.0663 × 10-4 K-1
R = 8.314 J/mol-1K-1
on substituting the value
ln ( k1/k2) = Ea/R(1/T2-1/T1)
=> -3.4012 = Ea / 8.314 J/ mol-1 K-1 ( -3.0663 × 10-4 K-1)
=> ( -3.4012 × 8.314 J/ mol-1 K-1 ) / ( -3.0663 × 10-4 K-1) = Ea
=> Ea = 92220 J/mol = 92.22 kJ/mol
[ note: 1000 J = 1 kJ ]
Hence, the activation energy = 92.22 kJ/mol
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