JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Modern Physics

Assume that protons and neutrons have equal masses. Mass of a nucleon is

1.6 \times 1 0^{- 27}

kg and radius of nucleus is

1.5 \times 1 0^{- 15} A^{1/3}

m. The approximate ratio of the nuclear density and water density is

n \times 1 0^{13}

. The value of n is ______.

Answer: 11

Solution

Nuclear density is

ρ = \frac{A m}{\frac{4}{3} π r ^{3}}

with

r = 1.5 \times 1 0^{- 15} A^{1/3}

, so

r^{3} = 3.375 \times 1 0^{- 45} A

and the mass number cancels, leaving a value independent of

A

ρ = \frac{1.6 \times 1 0 ^{- 27}}{\frac{4}{3} π ( 3.375 \times 1 0 ^{- 45} )} \approx 1.13 \times 1 0^{17} kg/m^{3}

. Dividing by the density of water,

1 0^{3} kg/m^{3}

, gives

1.13 \times 1 0^{14} \approx 11 \times 1 0^{13}

. The value of

n

11

JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Integer answer

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