Understanding the high-temperature behavior of nitrogen oxides is essential for controlling pollution generated in automobile engines. The decomposition of nitric oxide (NO) to N2 and O2 is second order with a rate constant of 0.0796 M−1*s−1 at 737∘C and 0.0815 M−1*s−1 at 947∘C. Calculate the activation energy in kJ/mol.

I'm not exactly sure which equation to use out of the almost 50 equations I have...

Use the following equation:

Ln(k2/k1)=Ea/R[(1/T1)-(1/T2)]

Where

k1=0.0796 M−1*s−1
k2=0.0815 M−1*s−1
R=8.314 J/mol K.
T1=273+737=1010K
T2=273+947=1220K
and
Ea=????

Solve for Ea:

Ea={R*[Ln(k1/k2)]}/[(1/T1)-(1/T2)]

To calculate the activation energy (Ea) for the decomposition of nitric oxide (NO) to N2 and O2, you can use the Arrhenius equation:

k = A * exp(-Ea / (R * T))

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

You have two sets of rate constant values at different temperatures, so you can use the Arrhenius equation to find the activation energy.

Let's use the first set of rate constant values at 737∘C (convert to Kelvin: 737 + 273 = 1010 K).

Rearranging the equation, we can write it as:

ln(k1) = ln(A) - (Ea / (R * T1))

Taking natural logarithm on both sides, we have:

ln(k1) = ln(0.0796 M−1*s−1) - (Ea / (8.314 J/(mol·K) * 1010 K))

Now we can use the second set of rate constant values at 947∘C (convert to Kelvin: 947 + 273 = 1220 K) to find the activation energy.

ln(k2) = ln(A) - (Ea / (R * T2))

ln(k2) = ln(0.0815 M−1*s−1) - (Ea / (8.314 J/(mol·K) * 1220 K))

Now we have a system of two equations with two unknowns (ln(A) and Ea). We can solve these equations simultaneously to find the activation energy.

Take the ratio of the two equations:

ln(k1/k2) = ln(0.0796 M−1*s−1 / 0.0815 M−1*s−1) - (Ea / (8.314 J/(mol·K))) * (1/1010 K - 1/1220 K)

ln(k1/k2) = -Ea / (8.314 J/(mol·K)) * (1/1010 K - 1/1220 K)

Simplifying further:

Ea = -8.314 J/(mol·K) * ln(k1/k2) * (1/1010 K - 1/1220 K)

Now you can substitute the given values for k1 and k2 to calculate the activation energy (Ea) in joules. To convert it to kJ/mol, divide by 1000 and multiply by the molar mass of NO.

To solve for the activation energy, we can use the Arrhenius equation, which relates the rate constant (k) of a reaction to the temperature (T) and the activation energy (Ea). The Arrhenius equation is:

k = A * e^(-Ea/RT)

Where:
- k is the rate constant of the reaction
- A is the pre-exponential factor or the frequency factor
- Ea is the activation energy
- R is the gas constant in units of energy per temperature per mole (8.314 J/mol*K or 0.008314 kJ/mol*K)
- T is the temperature in Kelvin

We have two sets of data points, one at 737∘C and the other at 947∘C. We need to convert these temperatures to Kelvin by adding 273.15 to each value.

Let's assume the rate constant at 737∘C is k1 and at 947∘C is k2. We can rewrite the Arrhenius equation for each temperature as follows:

k1 = A * e^(-Ea/RT1)
k2 = A * e^(-Ea/RT2)

Dividing the second equation by the first equation gives:

k2/k1 = e^(-Ea/RT2) / e^(-Ea/RT1)

If we simplify the equation, we get:

k2/k1 = e^(-Ea/RT2 + Ea/RT1)

Taking the natural logarithm (ln) of both sides of the equation, we can solve for Ea:

ln(k2/k1) = -Ea/R(1/T2 - 1/T1)

Plugging in the values for T1, T2, R, and the rate constants k1 and k2, we can solve for Ea.

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