I have two questions:
1. A reaction has a first order rate constant of 3.83×10−5 s−1 at 25 °C and 0.00174 s−1 at 75 °C. What is the value of the rate constant at 53 °C?
2. A catalyst decreases the activation energy of a given reaction by exactly 9 kJ mol−1. By what factor does the rate constant increase at 25 °C when the catalyst is used? Assume that the Arrhenius pre-exponential factor, A, is the same for the catalyzed and uncatalyzed reactions.
1. Use the Arrhenius equation. Plug in k1 and k2 at T1 and T2 and solve for Ea, the activation energy. Knowing Ea, use the new T and either of the other two k values to calculate the new k. Post your work you get stuck.
I would start by making up a convenient number for the Ea of some hypothetical reaction. I would use something like 150 or 200 kJ (convert that to J) and calculate k1/k2. Then reduce Ea by 9 kJ and recalculate k1/k2.
for the first one, do i just use the new t value? or do i need to use one of the t values in addition to the new one since the equation is 1/t2-1/t1?
so what i did was calc Ea which i got as 65.81kj/mol. then i used only the new t value with the Ea i found to get k, which i got as 261.73. but that number looks way too big
No. Use T1 and T2 with corresponding values of k1 and k2 and calculate Ea. Then use k1 with T1 and Ea you found with T3 to calculate the new k3. Alternatively you can use k2 with T2 and Ea you found with T3 and calculate k3.
A reaction has a first order rate constant of 3.83×10−5 s−1 at 25 °C and 0.00174 s−1 at 75 °C. What is the value of the rate constant at 53 °C?
The problem tells you that k is 3.83E-5 @ 25 C and 1.74E-3 @ 75 C so the k @ 53 C I would think would be between these two values? Right?
To solve these questions, we need to understand the relationship between temperature and rate constant for a chemical reaction. The Arrhenius equation describes this relationship:
k = A * e^(-Ea/RT)
Where:
- k is the rate constant
- A is the pre-exponential factor
- Ea is the activation energy
- R is the gas constant (8.314 J/(mol·K))
- T is the temperature in Kelvin (K)
Let's use this equation to find the answers to your questions.
1. To find the rate constant at 53 °C, we need to use the Arrhenius equation with the given rate constants at 25 °C and 75 °C.
Given:
- k1 = 3.83×10^(-5) s^(-1) at 25 °C (298 K)
- k2 = 0.00174 s^(-1) at 75 °C (348 K)
- T1 = 25 °C (298 K)
- T2 = 75 °C (348 K)
- T3 = 53 °C (?) - we need to find this rate constant
We can rewrite the Arrhenius equation for the two given temperatures as follows:
ln(k1) = ln(A * e^(-Ea/(R * T1)))
ln(k2) = ln(A * e^(-Ea/(R * T2)))
We can divide one equation by the other to eliminate the pre-exponential factor A:
ln(k1/k2) = ln(A * e^(-Ea/(R * T1)) / (A * e^(-Ea/(R * T2))))
ln(k1/k2) = ln(e^(-Ea/(R * T1)) / e^(-Ea/(R * T2)))
ln(k1/k2) = -Ea/(R * T1) + Ea/(R * T2)
ln(k1/k2) = Ea/R * (1/T2 - 1/T1)
Now we can solve for Ea:
Ea = (R * ln(k1/k2)) / (1/T2 - 1/T1)
Substituting the given values:
Ea = (8.314 J/(mol·K) * ln(3.83×10^(-5) s^(-1) / 0.00174 s^(-1))) / (1/348 K - 1/298 K)
Ea = 52969.2 J/mol
Now we can find the rate constant at 53 °C (326 K) using the obtained Ea:
k3 = A * e^(-Ea/(R * T3))
k3 = A * e^(-52969.2 J/mol / (8.314 J/(mol·K) * 326 K))
This will give us the answer for the rate constant at 53 °C.
2. To find the factor by which the rate constant increases at 25 °C (298 K) when the catalyst is used, we can use the Arrhenius equation again.
Given:
- Ea = 9 kJ/mol
- T = 25 °C (298 K)
We need to compare the rate constant with and without the catalyst at the same temperature.
Without catalyst:
k1 = A1 * e^(-Ea/(R * T))
With catalyst:
k2 = A2 * e^(-Ea'/(R * T))
We can divide the equation for the rate constants with and without the catalyst to eliminate the pre-exponential factor A:
k2/k1 = (A2 * e^(-Ea'/(R * T))) / (A1 * e^(-Ea/(R * T)))
k2/k1 = (A2 / A1) * (e^(-Ea'/(R * T)) / e^(-Ea/(R * T)))
k2/k1 = (A2 / A1) * e^(-Ea'/(R * T) + Ea/(R * T))
k2/k1 = (A2 / A1) * e^((Ea - Ea')/(R * T))
The pre-exponential factor A is assumed to be the same for both reactions, so A2/A1 = 1.
k2/k1 = e^((Ea - Ea')/(R * T))
Now we can solve for the factor by which the rate constant increases:
k2/k1 = e^((9,000 J/mol) / (8.314 J/(mol·K) * 298 K))
This will give us the answer for the factor by which the rate constant increases at 25 °C when the catalyst is used.
Remember to substitute the appropriate temperature values and units when performing the calculations.