Calculate the temperature at which k = 2.0 s-1 if k = 10.0 s-1 at 371 K. The activation energy for the reaction is 30.0 kJ mol-1.
To calculate the temperature at which k = 2.0 s-1, we can make use of the Arrhenius equation. The Arrhenius equation relates the rate constant (k) of a reaction to the temperature (T) and the activation energy (Ea) using the equation:
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.
We are given that k = 10.0 s-1 at 371 K and the activation energy Ea = 30.0 kJ mol-1.
First, let's convert the activation energy from kJ/mol to J/mol.
Ea = 30.0 kJ mol-1 * 1000 J/1 kJ = 30000 J/mol
Now, let's rearrange the Arrhenius equation to solve for T:
T = (-Ea/(R * ln(k/A)))
Given k1 = 10.0 s-1 and k2 = 2.0 s-1, we can calculate T2 using the above equation.
For k1 = 10.0 s-1:
T1 = (-Ea/(R * ln(k1/A)))
T1 = (-30000 J/mol / (8.314 J/(mol*K) * ln(10.0 s-1 / A)))
Now, let's solve for A using T1 and k1:
T1 = (-Ea/(R * ln(k1/A)))
(ln(k1/A)) = -Ea/(R * T1)
A = k1 / (e^(-Ea/(R * T1)))
Given that T1 = 371 K, we can substitute the values into the equation to find A.
A = 10.0 s-1 / (e^(-30000 J/mol/(8.314 J/(mol*K) * 371 K)))
Calculate the value of A using the formula provided above.
Once A is determined, substitute the values of k2 = 2.0 s-1, Ea = 30000 J/mol, and A into the equation for T2.
T2 = (-Ea/(R * ln(k2/A)))
Calculate T2 to find the desired temperature.
After following these steps, we will obtain the temperature (T2) at which k = 2.0 s-1.