A particular chemical reaction occurs at room temperature (293 K) at half the rate that it does at 300 K. What is the activation energy for this reaction?
Question
A particular chemical reaction occurs at room temperature (293 K) at half the rate that it does at 300 K. What is the activation energy for this reaction?
Solution 1
To solve this problem, we can use the Arrhenius equation which describes the temperature dependence of reaction rates:
k = Ae^(-Ea/RT)
where:
- k is the rate constant,
- A is the pre-exponential factor,
- Ea is the activation energy,
- R is the gas constant (8.314 J/(mol·K)),
- T is the temperature in Kelvin.
Given that the reaction rate at 293 K is half of that at 300 K, we can set up the following equation:
k1/k2 = e^((Ea/R)(1/T1 - 1/T2))
where:
- k1 is the rate constant at T1 (293 K),
- k2 is the rate constant at T2 (300 K).
Since k1 is half of k2, we can substitute k1/k2 with 0.5:
0.5 = e^((Ea/R)(1/293 - 1/300))
To solve for Ea, we first take the natural logarithm of both sides:
ln(0.5) = (Ea/R)(1/293 - 1/300)
Then we can solve for Ea:
Ea = R * ln(0.5) / (1/293 - 1/300)
Substituting R with 8.314 J/(mol·K), we get:
Ea = 8.314 * ln(0.5) / (1/293 - 1/300)
Calculating the above expression will give us the activation energy in Joules per mole.
Solution 2
To solve this problem, we can use the Arrhenius equation which describes the temperature dependence of reaction rates:
k = Ae^(-Ea/RT)
where:
- k is the rate constant,
- A is the pre-exponential factor,
- Ea is the activation energy,
- R is the gas constant (8.314 J/(mol·K)),
- T is the temperature (in Kelvin).
Given that the reaction rate at 293 K is half of that at 300 K, we can set up the following equation:
k1/k2 = e^((Ea/R)(1/T1 - 1/T2))
where:
- k1 is the rate constant at T1 (293 K),
- k2 is the rate constant at T2 (300 K).
Since k1 is half of k2, we can substitute k1/k2 with 0.5:
0.5 = e^((Ea/R)(1/293 - 1/300))
To solve for Ea, we first take the natural logarithm of both sides:
ln(0.5) = (Ea/R)(1/293 - 1/300)
Then, we can solve for Ea:
Ea = R * ln(0.5) / (1/293 - 1/300)
Substituting R with 8.314 J/(mol·K), we get:
Ea = 8.314 * ln(0.5) / (1/293 - 1/300)
Calculating the above expression will give us the activation energy in Joules per mole.
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