How do you convert density to number density?

To convert density to number density, you need to relate the mass density to the number of particles in a given volume. Here's a step-by-step guide:

Understanding the Definitions

• Material Density (ρ): This is defined as the mass (m) of a substance per unit volume (V), expressed as ρ = m/V.
• Number Density (n): This represents the number of particles (N) per unit volume (V), expressed as n = N/V.

Conversion Process

Here's how to bridge the gap between these two:

1. Relate Mass to Number of Particles: The key is to relate the total mass m to the total number of particles N. This is often done using the molar mass (M) or the mass of a single particle (m₀).
   • Using Molar Mass: If you know the substance and its molar mass (M in g/mol), you can find the number of moles (n_mol) from the mass: n_mol = m/M. Then the total number of particles N can be found by multiplying the number of moles with Avogadro's number (N_A ≈ 6.022 x 10²³ particles/mol): N = n_mol * N_A
   • Using Mass of a Single Particle: If you know the mass of a single particle (m₀), you can calculate N directly by dividing the total mass by the single particle mass: N= m/m₀
2. Express Number Density:
   • Once you know the total number of particles (N), calculate the number density (n) using:
     • n = N/V
   • Substituting the expression for N based on the molar mass we get:
     • n = (m N_A) / (M V)
     • Where m/V is the material density (ρ), so we can simplify it to:
       • *n = (ρ N_A) / M**
   • Substituting the expression for N based on the mass of a single particle we get:
     • *n= m/(V m₀)**
     • Where m/V is the material density (ρ), so we can simplify it to:
       • n = ρ / m₀

Practical Example

Let’s say you have Helium gas.

• Helium's molar mass (M) is approximately 4 g/mol.
• The Avogadro's number (N_A) is approximately 6.022 x 10²³ particles/mol.
• Suppose you have Helium gas at a density (ρ) of 0.1785 kg/m³ (or 178.5 g/m³).

To find the number density (n) using molar mass we would apply the formula *n = (ρ N_A) / M**:

1. n = (178.5 g/m³ * 6.022 x 10²³ particles/mol ) / (4 g/mol)
2. n ≈ 2.68 x 10²⁵ particles/m³

To find number density (n) using the mass of a single atom. The mass of a single helium atom is 6.646 × 10<sup>−27</sup> kg.

To find the number density (n) using the mass of a single atom we would apply the formula n = ρ / m₀:

1. n = (0.1785 kg/m³ ) / (6.646 x 10^-27 kg)
2. n ≈ 2.68 x 10²⁵ particles/m³

Therefore, the number density of Helium is approximately 2.68 x 10²⁵ particles/m³.

Key Formula

Formula	Description
ρ = m/V	Material density is mass (m) per unit volume (V).
n = N/V	Number density is number of particles (N) per unit volume (V).
*n = (ρ NA) / M**	Number density calculated from material density (ρ), Avogadro's number (NA), and molar mass (M).
n = ρ / m0	Number density calculated from material density (ρ) and the mass of a single particle (m0).

Summary

Converting density to number density requires understanding the relationship between mass, volume, and the number of particles. Using molar mass or the mass of a single particle along with Avogadro's number enables you to determine the number density from the material density. In short, the material density is given by ρ=m/V, where m is mass and V is volume. Number density is given by n=N/V, where N is the total number of particle, as stated in the reference.