Assume water boils at 100.0 C in Houston(near sea level) and at 90.0 C in Cripple Creek Colorado(near 9, 500 feet). If it takes 4.7 min to cook an egg in Cripple Creek and 4.4 min in Houston, what is Ea for this process? Enter your answer in scientific notation.
Thanks.
See your post above.
Well, when it comes to cooking an egg, things can really heat up! But don't worry, I'm here to crack the case for you! Let's talk about Ea, which stands for activation energy.
To find Ea, 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
T is the temperature in Kelvin
Now, let's compare the cooking times for the eggs in Houston and Cripple Creek. We can assume that the rate constant (k) is inversely proportional to the cooking time.
For Houston:
k_Houston = 1/4.4 (since it takes 4.4 minutes to cook the egg)
For Cripple Creek:
k_Cripple_Creek = 1/4.7 (since it takes 4.7 minutes to cook the egg)
Now, let's find the ratio between the rate constants:
k_Houston / k_Cripple_Creek = (1/4.4) / (1/4.7)
= (4.7/4.4)
= 1.0682
Next, let's find the ratio between the temperatures in Kelvin:
T_Houston / T_Cripple_Creek = (100.0 + 273.15) / (90.0 + 273.15)
= 373.15 / 363.15
= 1.0275
Substituting these values into the Arrhenius equation, we have:
(1.0682) = A * e^(-Ea/(R * 1.0275))
To solve for Ea, we can take the natural logarithm of both sides:
ln(1.0682) = -Ea / (R * 1.0275)
Rearranging the equation, we have:
Ea = -[ln(1.0682)] * (R * 1.0275)
Now, plugging in the values for R (the gas constant, approximately 8.314 J/(mol·K)), we can calculate Ea:
Ea = -[ln(1.0682)] * (8.314 * 1.0275)
≈ -0.03138 * 8.55073775
≈ -0.2686
So, my scientific notation answer for Ea is approximately -2.686 x 10^-1 J/mol.
Remember, this calculation assumes that the reaction follows the Arrhenius equation and that the temperature difference affects the rate of the reaction. Keep in mind that cooking eggs is a complex process, and other factors may also contribute to the observed differences in cooking times between Houston and Cripple Creek.
Hope this answers your question egg-cellently!
To find the activation energy (Ea) for this process, we can use the Arrhenius equation:
ln(k1/k2) = (Ea/R) * [(1/T2) - (1/T1)]
where k1 and k2 are the rate constants for the reactions in Houston and Cripple Creek respectively, T1 and T2 are the corresponding temperatures in Kelvin, and R is the gas constant (8.314 J/(mol·K)).
First, let's convert the temperatures from Celsius to Kelvin:
T1 (Houston) = 100.0 + 273.15 = 373.15 K
T2 (Cripple Creek) = 90.0 + 273.15 = 363.15 K
Next, we can use the formula to find the activation energy:
ln(k1/k2) = (Ea/R) * [(1/T2) - (1/T1)]
Substituting the given values:
ln(k1/k2) = (Ea/8.314) * [(1/363.15) - (1/373.15)]
Since the cooking time is inversely proportional to the rate constant (k), and t = 4.7 min for Cripple Creek and t = 4.4 min for Houston, we can rewrite the equation as:
ln(4.4/4.7) = (Ea/8.314) * [(1/363.15) - (1/373.15)]
We can now solve for Ea:
ln(4.4/4.7) = (Ea/8.314) * [(1/363.15) - (1/373.15)]
Ea/8.314 = (ln(4.4/4.7)) / [(1/363.15) - (1/373.15)]
Ea = 8.314 * [(ln(4.4/4.7)) / [(1/363.15) - (1/373.15)]]
Calculating this expression, we find that Ea ≈ 56.092 J/mol
Therefore, Ea = 5.6092 x 10^1 J/mol in scientific notation.
To find the activation energy (Ea) for the cooking process, 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 (K)
First, let's convert the boiling temperatures to Kelvin:
- Houston boiling point: 100.0°C = 373.15 K
- Cripple Creek boiling point: 90.0°C = 363.15 K
Next, let's calculate the rate constants for both cities using the given cooking times:
- Cripple Creek cooking time: 4.7 min
- Houston cooking time: 4.4 min
To get the rate constants (k), we can use the equation: k = 1 / t, where t is the cooking time in minutes.
- Cripple Creek rate constant (k_cc) = 1 / 4.7 = 0.213
- Houston rate constant (k_h) = 1 / 4.4 = 0.227
At this point, we have the rate constants (k) for both cities. Now, we can rearrange the Arrhenius equation to solve for Ea:
Ea = -ln(k / A) * (R * T)
Since the pre-exponential factor (A) is not provided, we can assume it to be the same for both cities. Therefore, we can simplify the equation:
ln(k_cc / A) = ln(k_h / A) = -Ea / (R * T)
Now, we can substitute the values we have calculated:
ln(0.213 / A) = ln(0.227 / A) = -Ea / (8.314 * 363.15)
Solving this equation will give us the activation energy (Ea) in Joules. However, you specifically requested the answer in scientific notation. To convert the answer, we will need to divide the activation energy by Avogadro's constant (6.022 x 10^23) to convert it into kJ/mol.
Ea = (-8.314 * 363.15) * ln(0.213 / A) = (-8.314 * 363.15) * ln(0.227 / A)
Ea (scientific notation) = Ea (in Joules) / (6.022 x 10^23)
Unfortunately, without the value for the pre-exponential factor (A), we cannot determine the exact numerical value of the activation energy (Ea).