The concentration of drug x in aqueous solution drops by 10% per month when stored at room temperature if the degradation occurs by first order what concentration will remain if a 5 mg/ml solution of the drug is stored under the same conditions for 3 months

To determine the remaining concentration of drug X after three months, given that it degrades by 10% per month and follows first-order kinetics, we can use the principles of first-order decay. In first-order reactions, the concentration of the substance decreases exponentially over time. Let's break this down step by step.

Understanding First-Order Kinetics

In a first-order reaction, the rate of degradation is proportional to the concentration of the drug at any given time. The general formula for the concentration of a substance undergoing first-order decay is:

[C] = [C₀] * e^-kt

Where:

• [C] = concentration at time t
• [C₀] = initial concentration
• k = rate constant
• t = time

Calculating the Rate Constant

Since the concentration drops by 10% each month, we can express this as:

[C] = [C₀] * (1 - 0.10) = [C₀] * 0.90

This means that after one month, the concentration is 90% of the initial concentration. To find the rate constant (k), we can use the relationship:

k = -ln(0.90)

Calculating this gives:

k ≈ 0.1054

Applying the Formula Over Three Months

Now that we have the rate constant, we can calculate the concentration after three months. The time (t) in this case is 3 months, and the initial concentration ([C₀]) is 5 mg/ml.

Using the first-order decay formula:

[C] = 5 mg/ml * e^{-0.1054 * 3}

Calculating the exponent:

-0.1054 * 3 ≈ -0.3162

Now, we find e^-0.3162:

e^-0.3162 ≈ 0.728

Now, substituting this back into our concentration formula:

[C] ≈ 5 mg/ml * 0.728 ≈ 3.64 mg/ml

Final Concentration After Three Months

Therefore, after storing the 5 mg/ml solution of drug X at room temperature for three months, the remaining concentration will be approximately 3.64 mg/ml.

This calculation illustrates how first-order kinetics can be applied to predict the concentration of a drug over time, which is crucial for understanding drug stability and efficacy in pharmaceutical applications.