Question

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

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

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.

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.

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.

Key concepts

Atomic Mass

When we talk about atomic mass, we are referring to the weighted average mass of all isotopes of an element as they appear in nature. The atomic mass is expressed in atomic mass units (amu) and allows chemists to understand the average contribution of each isotope to an element's mass. In the case of chromium, this is expressed as 51.9961 amu, a value that reflects the presence and proportions of its naturally occurring isotopes.

The atomic mass is crucial because it helps in calculating the mass of an atom when considering multiple isotopes of the same element. To find this value, multiply each isotope's mass by its relative abundance (often in decimal form), then sum up all these values. This provides a description of chromium's isotopic composition as a whole, which is significant in understanding its properties and behavior.
Understanding the atomic mass is essential as it underlies the calculation of molecular mass and the formulation of compounds in chemistry.

Chromium Isotopes

Isotopes are basically variants of a particular chemical element that have the same number of protons, but different numbers of neutrons. This results in different masses for each isotope of the element. Chromium has four notable isotopes with masses of 49.94605 amu, 51.94051 amu, 52.94065 amu, and 53.93888 amu.

Each isotope of chromium naturally contributes to the overall atomic mass of chromium based on its abundance and mass. Their abundance in nature can vary, which is what makes the atomic mass a weighted average. More abundant isotopes will have a larger effect on the atomic mass than less abundant ones. It is this relationship that is exploited to calculate individual isotope abundances when given the combined atomic mass of chromium.
Knowing the isotopic composition is critical for applications that rely on specific isotopes, including industrial and medical uses. In research settings, certain chromium isotopes could be targeted to study specific chemical reactions or properties.

Mass Percentage Calculations

In chemistry, mass percentage calculations help determine how much of a certain isotope contributes to the overall atomic mass of an element. This is often presented as a percentage that reflects how much of each isotope exists relative to the others.

To solve for mass percentages, especially when dealing with isotopes like those of chromium, follow these steps:

• Add up the known abundances of the isotopes that are given.
• Subtract these from 100% to find the abundance of the isotope with an unknown abundance.
• Use the mass of each isotope and its abundance to calculate the atomic mass, and solve the equations to find individual isotope abundances.

A definite understanding of mass percentage calculations can allow you to estimate the isotopic composition of elements more accurately, which is vital for both theoretical and practical chemistry applications. When done methodically, it provides a deeper insight into the different roles and occurrences of isotopes within an element, greatly assisting scientific applications and calculations.