Math Snap

STEP 1

Assumptions1. There are two naturally occurring isotopes of boron ${ }^{10} \mathrm{~B}$ with a mass of $10.0129 \mathrm{~u}$ and ${ }^{11} \mathrm{~B}$ with a mass of $11.0093 \mathrm{~u}$.
. The average atomic mass of boron is $10.81 \mathrm{~u}$.
3. The sum of the abundances of the two isotopes is100%.

STEP 2

Let's denote the abundance of ${ }^{10} \mathrm{~B}$ as $x$ and the abundance of ${ }^{11} \mathrm{~B}$ as $y$. Since the sum of the abundances is100%, we can write the following equation$$x + y =1$$

STEP 3

The average atomic mass is calculated as the sum of the mass of each isotope times its relative abundance. We can write this as$$Average\, atomic\, mass = x \cdot Mass_{10} + y \cdot Mass_{11}$$

STEP 4

Substitute the given values into the equation from3$$10.81 = x \cdot10.0129 + y \cdot11.0093$$

STEP 5

We now have a system of two equations\begin{align} x + y &=1 \\ 10.81 &= x \cdot10.0129 + y \cdot11.0093\end{align}

STEP 6

We can solve this system of equations by substitution. First, solve the first equation for $y$$$y =1 - x$$

STEP 7

Substitute $y$ into the second equation$$10.81 = x \cdot10.0129 + (1 - x) \cdot11.0093$$

STEP 8

implify the equation$$10.81 =10.012x +11.0093 -11.0093x$$

STEP 9

Rearrange the equation to solve for $x$$$x = \frac{.81 -11.0093}{.0129 -11.0093}$$

STEP 10

Calculate the value of $x$$$x = \frac{10.81 -.0093}{10.0129 -.0093} \approx0.199$$

Substitute $x$ into the first equation to find $y$$$y = - x = -0.199 =0.801$$The abundance of ${ }^{10} \mathrm{~B}$ is approximately19.9% and the abundance of ${ }^{11} \mathrm{~B}$ is approximately80.%.