To determine the rupture time at a reduced temperature, you can use the Arrhenius equation, which relates the rate of a chemical reaction (or in this case, rupture) to temperature and activation energy. The Arrhenius equation is as follows:
k = Ae^(-Ea / RT)
Where:
k is the rate constant (reciprocal of the time it takes for rupture to occur).
A is the pre-exponential factor.
Ea is the activation energy (in this case, in calories per mole).
R is the universal gas constant (1.987 cal/(mol·K)).
T is the absolute temperature in Kelvin.
First, you need to calculate the rate constant (k) at the initial temperature of 900°C (1173 K) using the given information:
Stress (σ) = 20,000 psi
Activation energy (Ea) = 35,000 cal/mol
Rupture time (t) = 25,000 hours
Convert stress to force per unit area in Pascals (Pa):
1 psi = 6894.76 Pa
20,000 psi = 20,000 x 6894.76 Pa ≈ 137,895,200 Pa
Now, you can calculate the rate constant (k) at 900°C:
k = (σ / t) = (137,895,200 Pa) / (25,000 hours x 3600 seconds/hour) ≈ 2.5152 x 10^(-9) 1/s
Now, you want to determine the rupture time at the reduced temperature of 800°C (1073 K). You can rearrange the Arrhenius equation to solve for the new time (t2):
t2 = (1 / k2)
You need to calculate the rate constant (k2) at 800°C using the same activation energy:
T2 = 1073 K
Ea = 35,000 cal/mol
Now, calculate k2:
k2 = Ae^(-Ea / RT2) = (2.5152 x 10^(-9) 1/s) / e^(-35,000 cal/mol / (1.987 cal/(mol·K) x 1073 K))
Calculate the new rupture time (t2):
t2 = 1 / k2
Now, calculate t2:
t2 = 1 / k2 ≈ 2.65 x 10^8 seconds
To convert this to hours:
t2 ≈ 7,361,111 hours
So, if the temperature is reduced to 800°C, the rupture time is approximately 7,361,111 hours.