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.
Use the Arrhenius equation. I would use the time to determine a rate constant for each. 1 egg in 4.7 min = 1/4.7 = ? and 1/4.4 = ?
Substitute these for k1 and k2, use T1 and T2 and solve for Ea. Post your work if you get stuck.
To find the activation energy (Ea) for this process, we can use the Arrhenius equation:
ln(k2/k1) = (-Ea/R) * (1/T2 - 1/T1)
Where:
- k2 and k1 are the rate constants for the cooking process at different temperatures,
- Ea is the activation energy,
- R is the ideal gas constant (8.314 J/mol·K),
- T2 and T1 are the absolute temperatures in Kelvin.
First, let's convert the temperatures from Celsius to Kelvin:
- T1 (Houston) = 100 + 273.15 = 373.15 K
- T2 (Cripple Creek) = 90 + 273.15 = 363.15 K
Now, let's calculate the ratio of rate constants (k2/k1):
k2/k1 = (time2/time1)
Where:
- time1 (Houston) = 4.4 min = 4.4 * 60 s = 264 s
- time2 (Cripple Creek) = 4.7 min = 4.7 * 60 s = 282 s
k2/k1 = 282 s / 264 s = 1.068
Substituting the values into the Arrhenius equation, we have:
ln(1.068) = (-Ea/8.314) * (1/363.15 - 1/373.15)
Simplifying the equation:
0.0665 = (-Ea/8.314) * (-0.0276)
Multiplying both sides by -8.314:
-0.5519 = Ea * 0.0276
Dividing both sides by 0.0276:
Ea = -0.5519 / 0.0276 = -19.997
Therefore, the activation energy (Ea) for this process is approximately -19.997 J/mol.
Note: The activation energy is a positive value, but the negative sign in this case represents the energy release during the cooking process.
To find the activation energy (Ea) for this process, we can use the Arrhenius equation, which relates the rate constant (k) to the temperature (T) and Ea:
k = A * e^(-Ea/RT)
Where:
- k is the rate constant
- A is the pre-exponential factor (a constant)
- Ea is the activation energy
- R is the ideal gas constant
- T is the temperature in Kelvin (K)
To use this equation, we need to convert the temperatures from Celsius to Kelvin. The relationship between Kelvin (K) and Celsius (C) is given by the equation:
K = C + 273.15
Now, we can solve for Ea.
First, convert the boiling points to Kelvin:
- For Houston: 100.0 °C + 273.15 = 373.15 K
- For Cripple Creek: 90.0 °C + 273.15 = 363.15 K
Next, convert the cooking times to seconds:
- For Houston: 4.4 min * 60 sec = 264 sec
- For Cripple Creek: 4.7 min * 60 sec = 282 sec
Now, we can rearrange the Arrhenius equation to solve for Ea:
Ea = -ln(k2/k1) * R / (1/T2 - 1/T1)
Where:
- ln() is the natural logarithm
- k2 and k1 are the rate constants (cooking times) for Cripple Creek and Houston respectively
- R is the ideal gas constant (8.314 J/(mol·K))
- T2 and T1 are the temperatures (in Kelvin) for Cripple Creek and Houston respectively
Let's plug in the values and solve for Ea:
Ea = -ln(282/264) * 8.314 J/(mol·K) / (1/363.15 K - 1/373.15 K)
Evaluating this equation:
Ea ≈ (-0.063449) * 8.314 J/(mol·K) / (-9.5385043 × 10^-6 K^-1)
Ea ≈ 68,624.41 J/mol
To convert the answer to scientific notation, we can write it as:
Ea ≈ 6.862 × 10^4 J/mol
Therefore, the activation energy for this process is approximately 6.862 × 10^4 J/mol.