An industrial chemical reaction has a rate law: rate = k [C]. The activation energy for the reaction is 3.0000 x 10^4 J/mol and A = 2.000 x 10^-3. What must be the reaction temperature if a rate of 1.000 x 10^-9 M/s is required and [C] must be kept at 0.03000M. Hint: What must be the value of k to achieve the necessary rate?
I would do this.
Use rate = k[C], substitute from the problem and solve for k. Then use the Arrhenius equation, substitute k and the other variables and solve for T.
Here is a link for the Arrhenius equation.
http://en.wikipedia.org/wiki/Arrhenius_equation
To find the reaction temperature, we need to determine the value of the rate constant, k, using the given information.
The Arrhenius equation relates the rate constant, k, to the activation energy, Ea, and the temperature, T:
k = A * e^(-Ea/RT)
Given:
- Activation energy (Ea) = 3.0000 x 10^4 J/mol
- Rate constant pre-exponential factor (A) = 2.000 x 10^-3
- Required rate (rate) = 1.000 x 10^-9 M/s
- Concentration of [C] ([C]) = 0.03000 M
- Universal gas constant (R) = 8.314 J/(mol*K)
Substituting the given values into the Arrhenius equation, we get:
1.000 x 10^-9 = (2.000 x 10^-3) * e^(-3.0000 x 10^4 / (8.314 * T))
Now, let's solve the equation for T.
1. Divide both sides of the equation by (2.000 x 10^-3):
(1.000 x 10^-9) / (2.000 x 10^-3) = e^(-3.0000 x 10^4 / (8.314 * T))
2. Simplify the left side of the equation:
5.000 x 10^-7 = e^(-3.0000 x 10^4 / (8.314 * T))
3. Take the natural logarithm (ln) of both sides of the equation to eliminate the exponential:
ln(5.000 x 10^-7) = ln(e^(-3.0000 x 10^4 / (8.314 * T)))
4. Use the property of logarithms to bring down the exponent:
ln(5.000 x 10^-7) = (-3.0000 x 10^4 / (8.314 * T))
5. Multiply both sides of the equation by (8.314 * T):
(8.314 * T) * ln(5.000 x 10^-7) = -3.0000 x 10^4
6. Divide both sides of the equation by ln(5.000 x 10^-7):
T = (-3.0000 x 10^4) / (8.314 * ln(5.000 x 10^-7))
Now, calculate T using a calculator or programming tool:
T ≈ 1,299 K
Therefore, the reaction temperature must be approximately 1,299 Kelvin (K) to achieve a rate of 1.000 x 10^-9 M/s with a concentration of [C] at 0.03000 M.
To find the reaction temperature, you need to determine the value of k that will give you the necessary rate.
The rate constant (k) can be calculated using the Arrhenius equation:
k = A * e^(-Ea/RT)
Where:
k = rate constant
A = pre-exponential factor
Ea = activation energy
R = gas constant (8.314 J/(mol·K))
T = temperature in Kelvin
First, let's calculate the value of k using the given values of A and Ea.
k = 2.000 x 10^-3 * e^(-3.0000 x 10^4 J/mol / (8.314 J/(mol·K) * T))
Now, substitute the desired rate (1.000 x 10^-9 M/s) into the rate law equation:
1.000 x 10^-9 M/s = k * [C]
Substitute the given concentration ([C] = 0.03000 M) into the equation:
1.000 x 10^-9 M/s = k * 0.03000 M
Rearranging the equation to solve for k:
k = (1.000 x 10^-9 M/s) / (0.03000 M)
Now that you have the value of k, substitute it back into the Arrhenius equation:
k = 2.000 x 10^-3 * e^(-3.0000 x 10^4 J/mol / (8.314 J/(mol·K) * T))
Solve this equation for T to find the reaction temperature.
T = (-3.0000 x 10^4 J/mol) / (8.314 J/(mol·K) * ln(k / (2.000 x 10^-3)))
Plug in the value of k into the equation and calculate T using logarithms.