For the decomposition of dinitrogen oxide, the rate constant is 0.043 L/mol-min at 665 oC and 12.5 L/mol-min at 828 oC. What is the activation energy of the decomposition reaction?

ln K = -Ea/RT

100 To find the activation energy of the decomposition reaction, we can use the Arrhenius equation:

k = A * e^(-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

We are given two sets of data:
- At 665 °C (938 K), the rate constant is 0.043 L/mol-min.
- At 828 °C (1101 K), the rate constant is 12.5 L/mol-min.

We can use both sets of data to set up a system of equations:

0.043 = A * e^(-Ea/(8.314 * 938))
12.5 = A * e^(-Ea/(8.314 * 1101))

First, we can divide the first equation by the second equation to get rid of A:

0.043/12.5 = e^(-Ea/(8.314 * 938))/e^(-Ea/(8.314 * 1101))

Next, simplify the left side:

0.00344 = e^(-Ea/(8.314 * 938 - 1101))

Then, take the natural logarithm of both sides to eliminate the exponential:

ln(0.00344) = -Ea/(8.314 * 938 - 1101)

Now, solve for Ea:

Ea = -ln(0.00344) * (8.314 * 938 - 1101)

Calculating this expression will give you the activation energy for the decomposition reaction.