At 380°C the half-life period of the first order decomposition of H2O2 is 360 min. The activation energy of the reaction is 200 kJ/mole. Calculate the time required for 7% decompos... At 380°C the half-life period of the first order decomposition of H2O2 is 360 min. The activation energy of the reaction is 200 kJ/mole. Calculate the time required for 7% decomposition at 450°C.
Understand the Problem
The question is asking us to calculate the time required for 7% decomposition of H2O2 at 450°C, given the half-life period at a different temperature and the activation energy of the reaction. We will use the Arrhenius equation and the first-order reaction half-life formula to solve it.
Answer
The time required for 7% decomposition can be calculated using the rate constants obtained from the Arrhenius equation and half-life formulas. Please provide the half-life at $330°C$ for exact calculations.
Answer for screen readers
To find the final answer, you will need the specific half-life for $330°C$. Once you have that value, you can plug it into the steps above to calculate the time required for 7% decomposition at $450°C$.
Steps to Solve
- Identify the Information Provided
We know the following:
- The activation energy $E_a = 75 , kJ/mol$
- The half-life at a different temperature (let's denote it as $t_{1/2}^{330°C}$, but you need to provide it)
- The temperature at which we want to find the decomposition time is $T = 450°C$
- Convert Activation Energy to Joules
Convert the activation energy from kJ to J for consistency in units:
$$ E_a = 75 , kJ/mol \times 1000 = 75000 , J/mol $$
- Convert the Temperatures to Kelvin
Convert both temperatures to Kelvin:
-
For $T_1 = 330°C$: $$ T_1 = 330 + 273.15 = 603.15 , K $$
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For $T_2 = 450°C$: $$ T_2 = 450 + 273.15 = 723.15 , K $$
- Use the Arrhenius Equation
The Arrhenius equation relates the rate constants $k$ at different temperatures: $$ \frac{k_2}{k_1} = e^{\frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)} $$ Where $R = 8.314 , J/(mol \cdot K)$ is the gas constant.
- Calculate Rate Constants
Using the half-life relation for first-order reactions: $$ k = \frac{0.693}{t_{1/2}} $$
Calculate $k_1$ from the half-life at $T_1$. Once we know $k_1$, we can find $k_2$ using the Arrhenius equation derived in the previous step.
- Find Time for 7% Decomposition
From the rate constant $k_2$, we can calculate the time for 7% decomposition using the first-order kinetics formula: $$ t = \frac{0.693}{k_2} \ln \left( \frac{1}{1 - 0.07} \right) $$
This gives us the time required for 7% decomposition.
To find the final answer, you will need the specific half-life for $330°C$. Once you have that value, you can plug it into the steps above to calculate the time required for 7% decomposition at $450°C$.
More Information
Knowing how temperature affects reaction rates is crucial in chemistry. Overall, the Arrhenius equation demonstrates that higher temperatures generally increase the rate of a reaction, making it essential for calculating reaction kinetics.
Tips
- Forgetting to convert units consistently (e.g., kJ to J).
- Not using Kelvin for temperature in gas equations.
- Skipping logarithm calculations when determining time for decomposition.
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