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Chapter 3: Problem 11

Chromium (atomic mass \(=51.9961\) amu) has four isotopes. Their masses are \(49.94605 \mathrm{amu}, 51.94051 \mathrm{amu}, 52.94065 \mathrm{amu}\), and \(53.93888\) amu. The first two isotopes have a total abundance of \(87.87 \%\), and the last isotope has an abundance of \(2.365 \%\). What is the abundance of the third isotope? Estimate the abundances of the first two isotopes.

Short Answer

Expert verified
Answer: The estimated abundances of the first and second isotopes are 81.68% and 6.19%, respectively. The calculated abundance of the third isotope is 9.765%.

Step by step solution

01

Calculate the abundance of the third isotope

To find the abundance of the third isotope, we will first calculate the total abundance of the first, second, and fourth isotopes, and then subtract it from 100%. Total abundance of first two isotopes = 87.87% Abundance of the fourth isotope = 2.365% Total abundance of first, second, and fourth isotopes = 87.87% + 2.365% = 90.235% Now we can calculate the abundance of the third isotope: Abundance of the third isotope = 100% - 90.235% = 9.765% The abundance of the third isotope is 9.765%. Now let's estimate the abundances of the first two isotopes.
02

Estimate the abundances of the first two isotopes

We are given that the total abundance of the first and second isotopes is 87.87%, and we need to estimate their individual abundances. We can use the given information about their masses and the atomic mass of chromium to estimate the abundances. Let the x represent the abundance of the first isotope as a decimal (i.e., 0.50 for 50%). Then, the abundance of the second isotope can be represented as (0.8787 - x). So, we can set up an equation using the given atomic mass of chromium: 51.9961 = (49.94605 * x) + (51.94051 * (0.8787 - x)) Now, we need to solve for x.
03

Solve for x

To solve for x, we will first simplify the equation: 51.9961 = 49.94605x + 45.6069 - 51.94051x 51.9961 = -1.99446x + 45.6069 Now, isolate the x term: 1.99446x = 51.9961 - 45.6069 x ≈ 0.8168 Now we can calculate the abundance of the first isotope: Abundance of the first isotope = x * 100% = 81.68% We can find the abundance of the second isotope by subtracting the abundance of the first isotope from the total abundance of the first two isotopes: Abundance of the second isotope = 87.87% - 81.68% = 6.19% So, the estimated abundances of the first and second isotopes are 81.68% and 6.19% respectively.

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