The reaction
2NOCl(g) �¨ 2NO(g) + Cl2 (g)
has an activation energy of 100.0 kJ/mol and a rate constant at 350.0 K of 8.5 x 10-6 mol-1 L s-1. Determine the rate constant at 400 K.
Isn't this the Arrhenius equation?
Post your work if you get stuck.
.000008537
To determine the rate constant at 400 K, we can use the Arrhenius equation, which relates the rate constant (k) to the activation energy (Ea) and the temperature (T):
k = Ae^(-Ea/RT)
Where:
k = rate constant at the desired temperature
A = frequency factor (pre-exponential factor)
Ea = activation energy
R = gas constant (8.314 J/mol∙K)
T = temperature in Kelvin
First, let's convert the activation energy from kJ/mol to J/mol:
Ea = 100.0 kJ/mol * 1000 J/1 kJ = 100,000 J/mol
Next, we'll convert the activation energy from J/mol to J/particle by dividing by Avogadro's number (6.022 x 10^23 particles/mol):
Ea = 100,000 J/mol / (6.022 x 10^23 particles/mol) = 1.661 x 10^-19 J/particle
Now, let's calculate the rate constant at 400 K using the Arrhenius equation:
k2 = A * e^(-Ea/RT2)
Where:
k2 = rate constant at 400 K
A = frequency factor (pre-exponential factor)
Ea = activation energy (in J/particle)
R = gas constant (8.314 J/mol∙K)
T2 = temperature at 400 K
Substituting the known values:
k2 = (8.5 x 10^-6 mol^-1 L s^-1) * e^(-1.661 x 10^-19 J/particle / (8.314 J/mol∙K * 400 K))
Now, let's calculate the rate constant at 400 K:
k2 = (8.5 x 10^-6 mol^-1 L s^-1) * e^(-1.661 x 10^-19 J/particle / (3325.6 J/mol∙K))
k2 ≈ (8.5 x 10^-6) * e^-4.989 x 10^-23
Using a calculator, we find:
k2 ≈ (8.5 x 10^-6) * 0.99999999995
k2 ≈ 8.5000000175 x 10^-6 mol^-1 L s^-1
Therefore, the rate constant at 400 K is approximately 8.5000000175 x 10^-6 mol^-1 L s^-1.
To determine the rate constant at 400 K, we can use the Arrhenius equation, which relates the rate constant (k) to the activation energy (Ea) and the temperature (T):
k = A * e^(-Ea/RT)
Where:
- k is the rate constant
- A is the pre-exponential factor or the frequency factor
- e is the base of the natural logarithm (approximately 2.71828)
- Ea is the activation energy
- R is the ideal gas constant (8.314 J/(mol·K))
- T is the temperature in Kelvin
Given:
- Activation energy (Ea) = 100.0 kJ/mol = 100000 J/mol (converted to J by multiplying by 1000)
- Rate constant (k) at 350 K = 8.5 x 10^(-6) mol^(-1) L s^(-1)
- Temperature (T) at 400 K
First, we convert the activation energy (Ea) to J/mol:
Ea = 100000 J/mol
Next, we rearrange the equation to solve for the pre-exponential factor (A):
A = k / e^(-Ea/RT)
Substituting the given values:
A = (8.5 x 10^(-6) mol^(-1) L s^(-1)) / e^(-100000 J/mol / (8.314 J/(mol·K) * 350 K))
Now, we can calculate the pre-exponential factor (A):
A ≈ 1.08 x 10^13 mol^(-1) L s^(-1)
Finally, we can plug in the new temperature (T = 400 K) into the Arrhenius equation to find the rate constant (k):
k = A * e^(-Ea/RT)
k = (1.08 x 10^13 mol^(-1) L s^(-1)) * e^(-100000 J/mol / (8.314 J/(mol·K) * 400 K))
Calculating the rate constant (k) at 400 K will give you the answer.