Question

Using the mass of the proton 1.0073 amu and assuming its diameter is 1.0×10−15m, calculate the density of a proton in g/cm3.

Answer 1

The density of a proton in g / cm₃ = 3.19.10⁻¹⁵

Further explanation

Atoms are composed of 3 types of basic particles namely protons, electrons, and neutrons

the mass of the basic particle is expressed in units of atomic mass

This atomic unit uses the standard atomic mass, that is, the C-12 isotope

1 atom C-12 = 12 atomic mass units

1 atomic mass unit = 1/12 x mass 1 C-12 atom

1 unit of atomic mass = 1.66.10 ⁻²⁴ g

the basic particle mass in atomic mass units is:

electron mass = 9.11.10⁻²⁸ g

proton mass = 1.6726.10⁻²⁴ g

neutron mass = 1,675.10⁻²⁴ g

The proton mass is considered to be equal to 1 atomic mass unit and is also considered to be equal to 1 atomic mass unit

1 unit of atomic mass = 1.6726.10⁻²⁴ g = 1.6726.10⁻²⁷kg

Density is a quantity derived from the mass and volume

density is the ratio of mass per unit volume

The unit of density can be expressed in g / cm³ or kg / m3³

Density formula:

$\large {\boxed {\bold {\rho ~ = ~ \frac {m} {V}}}}$

ρ = density

m = mass

v = volume

Assume protons as spheres

with a diameter of 1.0 × 10⁻¹⁵m = 1.0.10⁻¹³ cm

then the volume = 4/3 π r³ or 1/6πd³

proton volume = 1 / 6π(1.0 × 10⁻¹³)³

proton volume = 5.23.10⁻⁴⁰

density becomes:

density = mass: volume

density = 1.6726.10⁻²⁴: 5.23.10⁻⁴⁰

density = 3.19.10⁻¹⁵ g/cm³

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Answer 2

Answer : $3.2 X 10^{15} g/cm^{3}$

Explanation :  To convert amu i.e. atomic mass unit in grams we have the conversion factor as 1 amu = $1.66054 X 10^{-24} g$

we know the mass of the proton is 1.0073 amu

So converting it into grams we have to multiply;

1.0073 amu X  $1.66054 X 10^{-24} g/amu$ = $1.673 X 10^{-24} g$

Now, Volume = 1/6πd³ as diameter is given as $1.0 X 10^{-15} m$ converting it to cm will require to multiply with 100

∴ Volume  = 1/6π $(1.0 X 10^{-15}mX 100 cm / 1 m)^{3}$

Hence, volume =  $5.236 X 10^{-40} cm^{3}$

Therefore, Density = mass / volume

∴ Density =  $1.673 X 10^{-24} g / 5.236 X 10^{-40} cm^{3}$

Therefore, Density will be $3.2 X 10^{15} g/cm^{3}$.