The rate constant of a certain reaction is known to obey the Arrhenius equation, and to have an activation energy E = 5.0 kJ/mol. If the rate constant of this reaction is 1.9 × 107 M¹-s Round your answer to 2 significant digits. -1 at 89.0 °C, what will the rate constant be at 121.0 °C?
The Arrhenius equation is given by:
k = A * exp(-E/(R*T))
where k is the rate constant, A is the pre-exponential factor, E is the activation energy, R is the gas constant, and T is the temperature in Kelvin.
To find the rate constant k at a different temperature, we can use the ratio of the Arrhenius equation at the two temperatures:
k2 / k1 = exp(E/(R*T1) - E/(R*T2))
We are given k1 = 1.9 × 10^7 M^-1 s^-1, E = 5.0 kJ/mol = 5000 J/mol, T1 = 89.0 °C = 362.15 K, and T2 = 121.0 °C = 394.15 K. The gas constant R = 8.314 J/mol*K.
Now we can solve for k2:
k2 = k1 * exp(E/(R*T1) - E/(R*T2))
k2 = (1.9 × 10^7 M^-1 s^-1) * exp(5000/(8.314*362.15) - 5000/(8.314*394.15))
k2 = (1.9 × 10^7 M^-1 s^-1) * exp(1.7083 - 1.5468)
k2 = (1.9 × 10^7 M^-1 s^-1) * exp(0.1615)
k2 ≈ 2.3 × 10^7 M^-1 s^-1
So, the rate constant at 121.0 °C is approximately 2.3 × 10^7 M^-1 s^-1.
To determine the rate constant at a different temperature using the Arrhenius equation, we can use the following equation:
k₂ = k₁ * e^(-E/RT₂ + E/RT₁)
Where:
- k₂ is the rate constant at the new temperature (in this case, at 121.0°C)
- k₁ is the rate constant at the initial temperature (in this case, at 89.0°C)
- E is the activation energy of the reaction (given as 5.0 kJ/mol)
- R is the gas constant (8.314 J/(K·mol))
- T₂ is the new temperature (in Kelvin)
- T₁ is the initial temperature (in Kelvin)
First, we need to convert the temperatures from Celsius to Kelvin:
T₂ = 121.0°C + 273.15 = 394.15 K
T₁ = 89.0°C + 273.15 = 362.15 K
Now we can substitute the values into the Arrhenius equation:
k₂ = (1.9 * 10^7 M¹-s) * e^(-5.0 kJ/mol / (8.314 J/(K·mol)) * (1/394.15 K - 1/362.15 K)
Calculating this expression will give us the value of the rate constant at 121.0°C in M¹-s. Remember to round the final answer to 2 significant digits.
To determine the rate constant at 121.0 °C using the Arrhenius equation, we need to calculate the ratio of rate constants at the two temperatures. The Arrhenius equation is given by:
k = A * e^(-Ea/RT)
Where:
k is the rate constant,
A is the pre-exponential factor or frequency factor,
Ea is the activation energy,
R is the gas constant (8.314 J/mol·K), and
T is the temperature in Kelvin.
Let's begin by converting the temperatures to Kelvin:
T1 = 89.0 °C + 273.15 = 362.15 K
T2 = 121.0 °C + 273.15 = 394.15 K
Next, we need to calculate the ratio of the rate constants:
k2 / k1 = (A * e^(-Ea/RT2)) / (A * e^(-Ea/RT1))
The pre-exponential factor (A) cancels out, and we can simplify the equation to:
k2 / k1 = e^((Ea/R) * ((1/T1) - (1/T2)))
Now, substitute the values:
k2 / (1.9 × 10^7 M^-1 s^-1) = e^((5.0 kJ/mol / (8.314 J/mol·K)) * ((1/362.15 K) - (1/394.15 K)))
Simplifying this expression will give us the ratio of the rate constants. We can then multiply this ratio by the known rate constant at 89.0 °C to find the rate constant at 121.0 °C.