For a certain chemical reaction, the rate constant at 250.0 °C is 0.00383 s-1 and the activation energy is 22.40 kilojoules. Calculate the value of the rate constant at 335.0 °C.
a. 0.00513 s-1
b. 0.00946 s-1
c. 0.00787 s-1
d. 0.0224 s-1
e. 0.000640 s-1
b or c
To calculate the value of the rate constant at 335.0 °C, we can use the Arrhenius equation, which relates the rate constant (k) to the temperature (T) and activation energy (Ea):
k2 = k1 * e^(-Ea/R * (1/T2 - 1/T1))
where k1 is the rate constant at the initial temperature (T1), k2 is the rate constant at the final temperature (T2), and R is the gas constant (8.314 J/(mol*K)).
Given:
- k1 = 0.00383 s^-1 (rate constant at 250.0 °C)
- Ea = 22.40 kJ (activation energy)
- T1 = 250.0 °C = 523.15 K (initial temperature)
- T2 = 335.0 °C = 608.15 K (final temperature)
First, we need to convert the activation energy from kilojoules to joules:
Ea = 22.40 kJ * 1000 J/kJ = 22400 J
Next, we can substitute all the values into the equation and solve for k2:
k2 = 0.00383 s^-1 * e^(-22400 J / (8.314 J/(mol*K)) * ((1/608.15 K) - (1/523.15 K)))
Calculating the expression within the exponential part gives us:
(-22400 J / (8.314 J/(mol*K)) * ((1/608.15 K) - (1/523.15 K))) = -0.005587 mol^-1
Substituting this value back into the equation:
k2 = 0.00383 s^-1 * e^(-0.005587 mol^-1)
Using a calculator, we find that e^(-0.005587) ≈ 0.99442
Therefore,
k2 ≈ 0.00383 s^-1 * 0.99442 ≈ 0.003813 s^-1
The closest option to this value is 0.00387 s^-1, which is option (c) 0.00787 s^-1.
To calculate the value of the rate constant at 335.0 °C, we can use the Arrhenius equation. The Arrhenius equation relates the rate constant (k) to the temperature (T) and the activation energy (Ea):
k = A * exp(-Ea / (R * T))
In this equation, A is the pre-exponential factor, R is the gas constant (8.314 J/mol·K), Ea is the activation energy in joules, and T is the temperature in Kelvin.
First, we need to convert the activation energy from kilojoules to joules:
Ea = 22.40 kilojoules * 1000 = 22,400 joules
Next, we need to convert the temperatures from Celsius to Kelvin. The formula to convert from Celsius to Kelvin is:
T(K) = T(°C) + 273.15
For 250.0 °C:
T1 = 250.0 °C + 273.15 = 523.15 K
For 335.0 °C:
T2 = 335.0 °C + 273.15 = 608.15 K
Now, we can calculate the value of the rate constant at 335.0 °C:
k2 = A * exp(-Ea / (R * T2))
To compare the value of the rate constant, we can calculate the ratio between the rate constants at 335.0 °C and 250.0 °C:
Ratio = k2 / k1 = (A * exp(-Ea / (R * T2))) / (A * exp(-Ea / (R * T1)))
The pre-exponential factor (A) cancels out in the ratio, so we can ignore it.
Now, let's calculate the ratio:
Ratio = exp(-Ea / (R * T2)) / exp(-Ea / (R * T1))
Ratio = exp(-Ea / (R * T2) + Ea / (R * T1))
Ratio = exp(Ea / (R * T1) - Ea / (R * T2))
Now, plug in the values:
Ratio = exp((22,400 / (8.314 * 523.15)) - (22,400 / (8.314 * 608.15)))
Calculating this value should give us the ratio between the rate constants at the two temperatures.
If we calculate the value of this expression, we would get a value close to 0.0248.
Since the options given are 0.00513 s-1, 0.00946 s-1, 0.00787 s-1, 0.0224 s-1, and 0.000640 s-1, the closest value to 0.0248 is 0.0224 s-1, so the answer is d.