[tex]390 \; \text{K}[/tex]
Explanation
The rate of a chemical reaction is directly related to its rate constant [tex]k[/tex] if the concentration of all reactants in its rate-determining step is held constant. The rate "constant" is dependent on both the temperature and the activation energy of this particular reaction, as seen in the Arrhenius equation:
[tex]k = A \cdot e^{-E_A/( R\cdot T)[/tex]
where
Taking natural logarithms of both sides of the expression yields:
[tex]\ln k = \ln A - {E_a}/ ({R \cdot T})[/tex]
[tex]k_2 = 4.50 \; k_1[/tex], such that
[tex]\ln k_2 = \ln 4.5 + \ln k_1[/tex]
[tex]\ln A- {E_a}/ ({R \cdot T_2}) = \ln k_2 \\\phantom{\ln A-{E_a}/ ({R \cdot T_2})} = \ln 4.5 + \ln k_1\\ \phantom{\ln A- {E_a}/ ({R \cdot T_2})} = \ln 4.5 +\ln A- {E_a}/ ({R \cdot T_1})[/tex]
Rearranging gives
[tex]-{E_a}/ ({R \cdot T_2}) = \ln 4.5- {E_a}/ ({R \cdot T_1})[/tex]
Given the initial temperature [tex]T_1 = 355 \; \text{K}[/tex] and activation energy [tex]E_A = 49.40 \; \text{kJ} \cdot \text{mol}^{-1}[/tex]- assumed to be independent of temperature variations,
[tex]- {49.40 \; \text{kJ} \cdot \text{mol}^{-1}}/ ({8.314 \times 10^{-3} \; \text{kJ} \cdot \text{mol}^{-1} \cdot \text{K}^{-1} \cdot T_2}) \\= \ln 4.5- {49.40 \; \text{kJ} \cdot \text{mol}^{-1}}/ ({355 \; \text{K}\cdot 8.314 \times 10^{-3} \; \text{kJ} \cdot \text{mol}^{-1} \cdot \text{K}^{-1}})[/tex]
Solve for [tex]T_2[/tex]:
[tex]-(8.314 \times 10^{-3}/ 49.40) \; T_2 = 1/ (\ln 4.5 - 49.40 / (355 \times 8.314 \times 10^{-3})[/tex]
[tex]T_2 = 390 \; \text{K}[/tex]
Given:
Ea(Activationenergy):49.4kJ/molecule
k1: the rate constant of the first reaction
k2 : rate constant of the second reaction.
T2: Temperature of the second reaction.
T1: Temperature of the first reaction.
k2/k1=4.55
Now by Arrhenius equation we get
log(k2/k1)=[Ea/(2.303xR)] x[(1/T1)-(1/T2)]
Where k1 is the rate constant of the first reaction.
k2 is the rate constant of the second equation.
T2 is the temperature of the second reaction measured in K
T1 is the temperature of the first reaction measured in K
Ea is the activation energy kJ/mol
R is the gas constant measured in J/mol.K
Now substituting the given values in the Arrhenius equation we get:
log(k2/k1)=[Ea/(2.303xR)] x[(1/T1)-(1/T2)]
log(4.55)=[Ea/(2.303xR)] x[(1/T1)-(1/T2)]
0.66=[49.4/(2.303x8.314x10^-3)]x[(1/355)-(1/T2)]
0.66= 2579.75x [(1/355)-(1/T2)]
0.000256= (T2-355)/355T2
0.0908T2-T2= -355
0.9092T2=355
T2=390.46K
