Question

What is the ratio of the nuclear densities of two nuclie having mass numbers in ratio 1:4

Answer 1

`1:1`

Step-by-step explanation:

As the nuclear density is independent of mass number, so the ratio of nuclear densities of the two given nuclei is 1:1.

Answer 2

Given:-

• Two nuclei have mass number in the ratio of 1:4.

To find:-

• The ratio of nuclear densities of the two nuclei.

Answer:-

The nuclear densities for all atoms is same and it doesn't depends on mass number.

So the ratio of the nuclear densities will be 1:1 .

`\implies \underline{\underline{\green{ \rm Ratio= 1:1}}}`

_________________________________________

Additional information:-

Here I would be providing you the proof for the same .

Firstly we know that the volume of nucleus is directly proportional to its mass number ( Why ? because as the mass number increases , nucleons increases so volume increases) . Therefore;

`\implies V = (4)/(3)\pi r^3 \\`

And ,

`\implies V \propto A \\`

So ,

`\implies (4)/(3)\pi r^3 \propto A \\`

`\implies r^3 \propto A \\`

`\implies r\propto A^{(1)/(3)} \\`

`\implies r = r_0 A^{(1)/(3)} \dots\rm{(1)} \\`

where,

• 
  `r_0` is a constant and its value is 1.2 fm .
• 
  `A` = Mass number
• 
  `r` = radius of nucleus.

Now in order to find out the nuclear density , say
`\rho` ; we have ;

`\implies \rho =(Mass_(\rm nucleus))/(Volume_(\rm nucleus)) \\`

Now let's assume that there are m nucleons and their average mass number is A . So the average mass of m nucleons would be mA .

`\implies \rho = (mA)/( (4)/(3)\pi r^3)\\`

From equation (1) , we have ;

`\implies \rho = \frac{mA}{(4)/(3)\pi (r_0 A^{(1)/(3)})^3} \\`

`\implies \rho = (mA)/((4)/(3)\pi r_0^3 A) \\`

`\implies \underline{\underline{\green{\rho = (3m)/(4\pi r_0^3)}}} \\`

Now we can see that all the terms in the RHS of the equation are constants. This implies that nuclear density is also constant for all atoms independent of their mass numbers .

And we are done!