# The respective masses in amu of the proton, the neutron, and the nckel-60 atom are 1.00728, 1.00867, and 59.9308. What is the mass defect of the nickel-60 atom in g?

Jun 12, 2016

$\Delta m = 9.1409 \cdot {10}^{- 25} \text{g}$

#### Explanation:

You're looking for mass defect, $\Delta m$, which is defined as the difference that exists between the atomic mass of a nucleus and the total mass of its nucleons, i.e. of its protons and neutrons.

The idea here is that the energy that is released when the nucleus is formed will decrease its mass as described by Albert Einstein's famous equation $E = m \cdot {c}^{2}$.

In this regard, you can say that the actual mass of the nucleus will always be lower than the added mass of its nucleons.

Your goal here is to figure out the total mass of the protons and neutrons that make up a nickel-60 nucleus and subtract it from the known atomic mass of the nucleus.

Grab a periodic table and look for nickel, $\text{Ni}$. You'll find the element located in period 4, group 10. Nickel has an atomic number, $Z$, equal to $28$, which means that its nucleus contains $28$ protons.

The nickel-60 isotope has a mass number, $A$, equal to $60$, which means that its nucleus also contains

$A = Z + \text{no. of neutrons}$

$\text{no. of neutrons" = 60 - 28 = "32 neutrons}$

So, the total mass of the protons will be

${m}_{\text{protons" = 28 xx "1.00728 u" = "28.20384 u}}$

The total mass of the neutrons will be

${m}_{\text{neutrons" = 32 xx "1.00867 u" = "32.27744 u}}$

The total mass of the nucleuons will be

${m}_{\text{total" = m_"protons" + m_"neutrons}}$

${m}_{\text{total" = "28.20384 u" + "32.27744 u" = "60.48128 u}}$

The mass defect will be equal to

$\Delta m = {m}_{\text{total" - m_"actual}}$

$\Delta m = \text{60.48128 u" - "59.9308 u" = "0.55048 u}$

Now, to express this in grams, use the definition of the unified atomic mass unit, $\text{u}$, which is

$\textcolor{p u r p \le}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\text{1 u" = 1.660539 * 10^(-24)"g}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

In your case, you will have

0.55048 color(red)(cancel(color(black)("u"))) * (1.660539 * 10^(-24)"g")/(1color(red)(cancel(color(black)("u")))) = color(green)(|bar(ul(color(white)(a/a)color(black)(9.1409 * 10^(-25)"g")color(white)(a/a)|)))

Jun 12, 2016

The mass defect is $9.141 \times {10}^{- 25} \text{ g/atom}$.

#### Explanation:

Mass defect is the amount of mass that is lost when protons and neutrons combine to form a nucleus. The protons and neutrons become bound to each other, and the binding energy that's released shows up as mass being lost because of the relation $E = m {c}^{2}$. So when we talk about mass defect we really mean binding energy.

Nickel-60 has a mass number of 60 and an atomic number of 28, thus 28 protons and 32 neutrons are bound together. The mass of the geee particles is given by:

$\left(28 \times 1.00728\right) + \left(32 \times 1.00867\right) = 60.48128 \text{ g/mol}$

Compare that with the given atomic mass of nickel-60 $= 59.9308 \text{ g/mol}$. Take the didference and round to a multiple of $0.0001$ matching the given accuracy of the nuckel atomic mass:

Mass defect = $60.48128 - 59.9308 = 0.5505 \text{ g/mol}$

Note the units. To get grams per atom divide by Avogadro's Number:

{0.5505" g/mol"}/{6.022xx10^{23}" atoms/mol"}=9.141xx10^{-25}" g/atom".

Go back to the molar basis and see how much energy this is in $\text{J/mol}$. One joule is $1000 \text{ g}$ mass times $1 {\text{m/s}}^{2}$ acceleration times $1 \text{ m}$ distance:

E=mc^2={0.5505" g/mol"xx(299792458" m/s")^2xx1" J"}/{1000" g m"^2/"s"^2}=4.948xx10^{13}" J/mol"

This is tremendously larger than the energy changes associated in chemical reactions. This shows the potential power of the nuclear force and processes based on it.