# What are the percentages of isotopes? [duplicate]

The average atomic mass of a sample of an element 'X' is 16.2u. What are the percentages of isotopes $\ce{^16_8X}$(atomic number = 8, atomic mass = 16) and $\ce{^18_8X}$(atomic number = 8, atomic mass = 18) in the sample?

## marked as duplicate by Martin - マーチン♦Dec 8 '17 at 7:01

• – Martin - マーチン Dec 7 '17 at 12:55
• Hey Moderators, I tried something new on this obvious candidate for closed as OT, homework. – TAR86 Dec 7 '17 at 12:56
• Hey @TAR86, that's nice of you. – Martin - マーチン Dec 7 '17 at 12:58

We have two isotopes, $A,B$ with atomic masses $m_A, m_B$. We thus have two unknowns $x_A, x_B$ representing the percentage amounts of each isotope (i.e. the mole fraction), which are trivially connected as

$$x_A + x_B = 1$$

We also know something about the average atomic mass $\overline{m}$, namely

$$x_A \cdot m_A + x_B \cdot m_B = \overline{m}$$

So we now have a system of two linear equations to plug in values and solve. Steps:

$$x_A = 1 - x_B$$

.

$$(1 - x_B ) \cdot m_A + x_B \cdot m_B = \overline{m}$$

.

$$x_B \cdot (m_B - m_A) = \overline{m} - m_A$$

.

$$x_B = \frac{\overline{m} - m_A}{m_B - m_A}$$

And plug back in the first equation.

• $x_A$ representing the percentage amounts of each isotope = mole fraction? – Martin - マーチン Dec 7 '17 at 13:03