Question

Using the semiempirical binding energy formula estimate the mass of a europium- $152$ atom.

Short answer

The mass of europium- $152$ atom is $151.90u$.

Step-by-step solution

Given data

The given atom is Europium- $152$.

Formula of Semi empirical binding Energy

Semi empirical binding Energy is given by the expression, $BE=c_{1}A-c_2{A}^{2/3}-c_3\frac{Z(Z-1)}{A^{1/3}}-c_4\frac{{(N-Z)}^2}{A}$.

Where,$c_1=15.8MeV,c_2=17.8MeV,c_3=0.71MeV,\text{\hspace{0.17em}}{c}_4=23.7MeV$.

$A\to$Mass Number,$Z\to$atomic Number and$N\to$ number of neutrons.

Calculation of binding energy by semi empirical binding energy formula

The binding energy calculated by the semi empirical binding energy formula for a europium-$152$atom is note europium has$Z=63$ and$A=152$ .

$\begin{array}{l}N=A-Z\\ N=152-63\\ N=89\end{array}$

Substitute $15.8\text{MeV}$ for $c_1,17.8\text{MeV}$ for $c_3,63$for$Z,152$ for$A$ for and 89 for $N$ in equation (1).

$\begin{array}{l}BE=c_{1}A-c_2{A}^{2/3}-c_3\frac{Z(Z-1)}{A^{1/3}}-c_4\frac{{(N-Z)}^2}{A}\\ BE=15.8\left(152\right)-17.8{\left(152\right)}^{2/3}-0.71\frac{63(63-1)}{{\left(152\right)}^{1/3}}-23.7\frac{{(89-63)}^2}{152}\\ BE=1269.58\text{MeV}\end{array}$

Calculate by binding energy formula

By the other model, obtain$BE=(Z{M}_H+N{m}_u-M_{Eu})c^2$ .

Rearrange the above equation for$M_{Eu}$.

$M_{Eu}=Z{M}_H+N{m}_u-\frac{BE}{c^2}$

Substitute $1.007825u$ for $M_H,1.008665u$ for $m_u$ and $1269.58\text{MeV}$ for $B.E$ in above equation.

$\begin{array}{c}{M}_{Eu}=\left(63\right)\left(1.007825u\right)+\left(99\right)\left(1.008665u\right)-\frac{\left(1269.58\text{MeV}\right)}{(931.5\text{MeV}/\text{u}u)}\\ M_{Eu}=151.90u\end{array}$