At room temperature (about 20C) milk turns sour in about 64 hours. In a refrigerator at 3C, milk can be stored for about three times as long before turning sour. (i) Determine the approximate activation energy for the reaction that causes milk to sour, and (ii) estimate how long it would take for milk to sour at 37C.
Use the Arrhenius equation and solve for Ea. You might use for k something like 1 L sours/64 hours etc.
Use the same equation again but change the T to 37. Post your work if you get stuck.
I guess I just don't know what value to use for A in the Arrhenius equation.
Usually when one doesn't know the value of A we substitute 1 for it. These are estimates anyway.
To determine the approximate activation energy for the reaction that causes milk to sour, we can use the Arrhenius equation. The Arrhenius equation relates the rate constant (k) of a reaction to the activation energy (Ea), the gas constant (R), and the temperature in Kelvin (T).
The Arrhenius equation is given by:
k = A * e^(-Ea / (R * T))
Where:
k: rate constant
A: pre-exponential factor, a constant
e: base of the natural logarithm
Ea: activation energy
R: gas constant (8.314 J/(mol K))
T: temperature in Kelvin
(i) To determine the activation energy, we can use the information given:
At room temperature (20°C = 293K), milk turns sour in about 64 hours.
In a refrigerator at 3°C = 276K, milk can be stored for about three times as long before turning sour.
Let's assume the initial rate constant at room temperature is k1, and the rate constant at the refrigerator temperature is k2, and the activation energy is Ea.
Using the Arrhenius equation, we can set up the following equations:
k1 = A * e^(-Ea / (R * 293))
k2 = A * e^(-Ea / (R * 276))
Since we are given that milk can be stored for three times as long at the refrigerator temperature compared to room temperature, we can write:
64 hours at room temperature = 3 * 64 hours = 192 hours at the refrigerator temperature.
This implies that the rate constant at the refrigerator temperature is one-third of the rate constant at room temperature. Therefore, we can write:
k2 = (1/3) * k1
Substituting the equations for k1 and k2 in terms of A and Ea, we have:
A * e^(-Ea / (R * 293)) = (1/3) * (A * e^(-Ea / (R * 276)))
Canceling out the pre-exponential factor A, we get:
e^(-Ea / (R * 293)) = (1/3) * e^(-Ea / (R * 276))
Taking the natural logarithm of both sides, we have:
-Ea / (R * 293) = ln(1/3) - Ea / (R * 276)
Simplifying further, we can isolate Ea:
Ea / (R * 276) - Ea / (R * 293) = ln(1/3)
Ea * (1/ (R * 276) - 1/ (R * 293)) = ln(1/3)
Ea = ln(1/3) / ((1/ (R * 276) - 1/ (R * 293)))
Using the given value for R (8.314 J/(mol K)), we can calculate the approximate activation energy.
(ii) To estimate how long it would take for milk to sour at 37°C, we can use the rate constant calculated from part (i) to determine the time required for the reaction.
Using the Arrhenius equation again, we can write:
k3 = A * e^(-Ea / (R * 310))
Here, k3 represents the rate constant at 37°C = 310K.
To estimate the time taken for the reaction to sour the milk, we can use the equation:
time = 1 / k3
Substituting the calculated value of k3, we can determine the approximate time it would take for milk to sour at 37°C.
Please note that the values used in the calculations might vary based on experimental data and the specific reaction involved.