Incandescent light bulbs operate at a temperature of 2500 C when bright and 2200 C when dim. The creep rate of the filament when bright is 1.5 times that when dim. Assuming that the activation energy is constant within this temperature range, and that a single creep diffusion mechanism operates, calculate the activation energy in kJ/mol. You can assume that the stress is the same when the light is bright or dim.
Q (in kJ/mol)
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To calculate the activation energy in kJ/mol, we can use the Arrhenius equation:
k = A * exp (-Ea/RT)
where:
k is the creep rate
A is the pre-exponential factor
Ea is the activation energy (in kJ/mol)
R is the gas constant (8.314 J/(mol*K))
T is the absolute temperature (in Kelvin)
We are given that the creep rate when bright is 1.5 times that when dim. Let's assume the creep rate when dim is k_dim. So the creep rate when bright is 1.5 * k_dim.
Also, we know the temperature when the light is bright (T_bright = 2500 °C) and when it is dim (T_dim = 2200 °C).
We need to convert these temperatures to Kelvin by adding 273.15.
First, let's calculate the ratio of the two creep rates:
1.5 * k_dim / k_dim = 1.5
Now, we can write two Arrhenius equations, one for the bright light condition and the other for the dim light condition, and set up a ratio:
k_bright = A * exp (-Ea/RT_bright)
k_dim = A * exp (-Ea/RT_dim)
Dividing the first equation by the second equation:
(1.5 * k_dim) / k_dim = (A * exp (-Ea/RT_bright)) / (A * exp (-Ea/RT_dim))
Simplifying:
1.5 = exp ((Ea/RT_dim) - (Ea/RT_bright))
Now, we need to plug in the temperature values and solve for the activation energy.
Ea = ((RT_bright * RT_dim) / (RT_dim - RT_bright)) * ln(1.5)
Substituting the temperature values and the gas constant (R = 8.314 J/(mol*K)):
Ea = ((8.314 J/(mol*K)) * (273.15 + 2500) * (273.15 + 2200)) / ((273.15 + 2200) - (273.15 + 2500)) * ln(1.5)
Ea ≈ 178.2 kJ/mol
Therefore, the activation energy is approximately 178.2 kJ/mol.