Calculate the rate constant at 200.°C for a reaction that has a rate constant of 9.20 × 10−4 s−1 at 90.°C and an activation energy of 59.7 kJ/mol.
To calculate the rate constant at 200.°C for a reaction, we can use the Arrhenius equation. The Arrhenius equation relates the rate constant (k) to the temperature (T) and the activation energy (Ea) of a reaction:
k = Ae^((-Ea)/(RT))
Where:
- k is the rate constant
- A is the pre-exponential factor or the frequency factor
- Ea is the activation energy
- R is the ideal gas constant (8.314 J/(mol·K))
- T is the temperature in Kelvin (K)
First, let's convert the temperatures to Kelvin:
- 90.°C + 273.15 = 363.15 K
- 200.°C + 273.15 = 473.15 K
Now, we need to calculate the pre-exponential factor (A) at 90.°C. Given that the rate constant is 9.20 × 10^(-4) s^(-1), we can rearrange the Arrhenius equation and solve for A:
A = k * e^((Ea)/(RT))
Substituting the known values:
A = (9.20 × 10^(-4) s^(-1)) * e^((-59.7 kJ/mol) / (8.314 J/(mol·K) * 363.15 K))
Note: We need to convert the activation energy from kJ/mol to J/mol by multiplying by 1000.
Simplifying the equation, we get:
A = (9.20 × 10^(-4) s^(-1)) * e^(-16.30)
Calculating the value of e^(-16.30):
e^(-16.30) ≈ 2.1 × 10^(-8)
Now, substituting the value of e^(-16.30) into the equation:
A = (9.20 × 10^(-4) s^(-1)) * (2.1 × 10^(-8))
Calculating A:
A ≈ 1.932 × 10^(-11) s^(-1)
Finally, we can calculate the rate constant (k) at 200.°C using the Arrhenius equation with the pre-exponential factor (A) we just obtained:
k = A * e^((-Ea)/(RT))
Substituting the values:
k = (1.932 × 10^(-11) s^(-1)) * e^((-59.7 kJ/mol) / (8.314 J/(mol·K) * 473.15 K))
Simplifying the equation, we get:
k = (1.932 × 10^(-11) s^(-1)) * e^(-12.70)
Calculating the value of e^(-12.70):
e^(-12.70) ≈ 4.2 × 10^(-6)
Now, substituting the value of e^(-12.70) into the equation:
k = (1.932 × 10^(-11) s^(-1)) * (4.2 × 10^(-6))
Calculating k:
k ≈ 8.11 × 10^(-17) s^(-1)
Therefore, the rate constant at 200.°C for the given reaction is approximately 8.11 × 10^(-17) s^(-1).