### Problem Statement:We have an initial amount of \( H_3N_2Cl_4 \) of 72 milligrams, and a half-life of 30 days. Find the decay constant, \(k\), and determine how long it takes for 95% of the material to decay.### Solution:#### 1. Finding the Decay Constant, \(k\):The relationship between the half-life \(T_{1/2}\) and the decay constant \(k\) is given by the formula:\[T_{1/2} = \frac{\ln(2)}{k}\]Given:\[T_{1/2} = 30 \ \text{days}\]Rearrange the formula to solve for \(k\):\[k = \frac{\ln(2)}{T_{1/2}}\]Substitute the given half-life:\[k = \frac{\ln(2)}{30} \approx \frac{0.693}{30} \approx 0.0231 \ \text{days}^{-1}\]#### 2. Determining the Time for 95% Decay:We need to find the time \(t\) when 95% of the material has decayed, which means 5% of the original material remains.The decay formula is:\[N(t) = N_0 e^{-kt}\]Where:- \(N(t)\) is the amount of substance remaining at time \(t\),- \(N_0\) is the initial amount of the substance,- \(k\) is the decay constant,- \(t\) is the time.Since 5% of the original amount remains:\[0.05 N_0 = N_0 e^{-kt}\]Simplifying the equation by dividing both sides by \(N_0\):\[0.05 = e^{-kt}\]Take the natural logarithm of both sides:\[\ln(0.05) = -kt\]Solve for \(t\):\[t = \frac{\ln(0.05)}{-k}\]Substitute the value of \(k\):\[t = \frac{\ln(0.05)}{-0.0231}\]Calculate the value of \(\ln(0.05)\):\[

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Answered: We have an initial amount of H3N2C14 of…

We have an initial amount of H3N2C14 of 72 milligrams, and a half-life of 30 days. Find the decay constant, k, and determine how long it takes for 95% of the material to decay.

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...

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