For a certain reaction, the activation energy is 52.1 kJ/mole. By what ratio will the rate constant change if the temperature is decreased from 175C to 75C?

Use the Arrhenius equation and remember to insert Ea in J/mol and not kJ/mol.

That answer is too basic... Could someone show the steps on how to solve this one? I keep on getting the 55.67 as an answer but the correct one is 0.0179.

@Eunji flip your numbers when finding for ratio

To solve this problem, we need to use the Arrhenius equation, which describes the dependence of the rate constant (k) on temperature (T) and activation energy (Ea):

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

where:
- k is the rate constant
- A is the pre-exponential factor or the frequency factor
- Ea is the activation energy
- R is the gas constant (8.314 J/(mol*K))
- T is the temperature in Kelvin (K)

First, we need to convert the temperatures from Celsius to Kelvin:
T1 = 175°C + 273.15 = 448.15 K
T2 = 75°C + 273.15 = 348.15 K

Next, we can rewrite the Arrhenius equation for the ratio of rate constants:

k1 / k2 = (A * exp(-Ea / (R * T1))) / (A * exp(-Ea / (R * T2)))

The pre-exponential factor (A) cancels out:

k1 / k2 = exp(-Ea / (R * T1)) / exp(-Ea / (R * T2))

Since exp(x) / exp(y) = exp(x - y), the equation simplifies to:

k1 / k2 = exp((-Ea / (R * T1)) - (-Ea / (R * T2)))

Simplifying further:

k1 / k2 = exp((-Ea / (R * T1) + Ea / (R * T2)))

To calculate the values inside the exponents, we substitute the given values:

k1 / k2 = exp(((-52.1 kJ/mol) / (8.314 J/(mol*K))) * ((1 / 448.15 K) - (1 / 348.15 K)))

Now, we can calculate the ratio of rate constants:

k1 / k2 ≈ exp(-0.7008 - (-0.8704))

k1 / k2 ≈ exp(0.1696)

Using the exponential rule, we can simplify further:

k1 / k2 ≈ 1.184

So, the rate constant will change by a ratio of approximately 1.184 when the temperature is decreased from 175°C to 75°C.