24.132 CP - Chemistry: Molecular Nature Of Matter And Change

Question

A metastable (excited) form of ⁵⁰Sc changes to its stable form by emitting $\gamma$ radiation with a wavelength of 8.73 pm. What is the change in mass of 1 mol of the isotope when it undergoes this change?

 

Step-by-Step Solution

Answer

The change of mass is $1.52\times {10}^{-7}\frac{kg}{mol}$.

Step 1: Definition of change of mass

In physics, mass is a “numerical representation” of inertia, which is a fundamental feature of all matter. It is in effect with a body's resistance to a change in speed or position caused by the application of a force.

 

The smaller the change caused by an applied force, the larger the mass of a body.

Step 2: Find the change of mass

Using the wavelength, calculate the gamma radiation's energy. The wavelength is 8.73 pm or $8.73\times {10}^{-12} m$. 

 

The formula is,

 $$\begin{gathered} E=\frac{hc}{\lambda } \\ =\frac{6.63\times {10}^{-34} J\times s\times 3.00\times {10}^8\frac{m}{s}}{8.73\times {10}^{-12} m} \\ =2.28\times {10}^{-14} J \end{gathered}$$

 

Calculate the mass defect using the equation:

$$\begin{gathered} E=\Delta m\times c^2 \\ \Delta m=\frac{E}{c^2} \\ =\frac{2.28\times {10}^{-14} J}{{\left(3.00\times {10}^8\right)}^2\frac{m^2}{s^2}} \\ =2.532\times {10}^{-31}\frac{kg}{atom} \end{gathered}$$

 

To get the mass defect per mole, multiply by Avogadro's number.

$$\begin{gathered} 2.532\times {10}^{-31}\frac{kg}{atoms}\times \frac{6.022\times {10}^{23}atoms}{1 mol} \\ =1.52\times {10}^{-7}\frac{kg}{mol} \end{gathered}$$

 

Therefore, the change of mass $1.52\times {10}^{-7}\frac{kg}{mol}$.