The disintegration constant, also known as the decay constant, is a measure of how quickly a radioactive material decays. It represents the probability of decay per unit time. To calculate the disintegration constant, we need to use the formula:

N = N0 * e^(-λt)

Where:
N = Final number of moles
N0 = Initial number of moles
λ = Disintegration constant
t = Time

Given that four moles of a radioactive material decays into 0.5 moles in 90 seconds, we can use this information to calculate the disintegration constant.

Step 1: Convert the number of moles into fractions of moles
Since the disintegration constant represents a probability, we need to convert the number of moles into fractions of moles. In this case, we have:
N0 = 4 moles
N = 0.5 moles

Step 2: Rearrange the formula to solve for the disintegration constant
Rearranging the formula, we have:
λ = -ln(N/N0) / t

Step 3: Substitute the given values into the formula
Substituting the given values into the formula, we have:
λ = -ln(0.5/4) / 90

Step 4: Calculate the disintegration constant
Using a calculator, we can evaluate the natural logarithm and divide it by 90 to find the disintegration constant. The correct answer is between 2.30 and 2.32 x 10^-2 s^-1.

In conclusion, the disintegration constant is calculated by using the formula N = N0 * e^(-λt), where N and N0 represent the final and initial number of moles, λ is the disintegration constant, and t is the time. By rearranging the formula and substituting the given values, we can calculate the disintegration constant. In this case, four moles of a radioactive material decaying into 0.5 moles in 90 seconds gives a disintegration constant between 2.30 and 2.32 x 10^-2 s^-1.