Question

From the information in this chapter on the mass of the proton, the mass of the electron, and the sizes of the nucleus and the atom, calculate the densities of a hydrogen nucleus and a hydrogen atom.

Short answer

The density of a hydrogen nucleus is \(\rho_n = \frac{m_p}{V_n}\), where the mass of a proton is \(m_p = 1.67 × 10^{-27} kg\) and the volume is calculated as \(V_n = \frac{4}{3}π({1.2 × 10^{-15}})^3\). The density of a hydrogen atom is \(\rho_a = \frac{m_{H}}{V_a}\), where the mass of a hydrogen atom is approximately equal to the mass of a proton and the volume is calculated as \(V_a = \frac{4}{3}π({5.3 × 10^{-11}})^3\).

Step-by-step solution

Gather information

From the chapter, we have the following information: - Mass of a proton: \(m_p = 1.67 × 10^{-27} kg\) - Mass of an electron: \(m_e = 9.11 × 10^{-31} kg\) - Mass of a hydrogen atom \((1 \, proton + 1\, electron)\): \(m_{H} = m_p + m_e\) - Radius of a hydrogen nucleus: \(r_n = 1.2 × 10^{-15} m\) - Radius of a hydrogen atom: \(r_a ≈ 5.3 × 10^{-11} m\)

Calculate the volume of a hydrogen nucleus

In order to find the volume of a hydrogen nucleus, we will consider it as a sphere and use the formula for the volume of a sphere: \(V = \frac{4}{3}πr^3\). Calculating the volume of the hydrogen nucleus using the given radius: \[V_n = \frac{4}{3}π({1.2 × 10^{-15}})^3\]

Calculate the volume of a hydrogen atom

Similarly, we will find the volume of a hydrogen atom. Using the given radius of a hydrogen atom, we will use the same sphere volume formula: \[V_a = \frac{4}{3}π({5.3 × 10^{-11}})^3\]

Calculate the mass of a hydrogen atom

To find the mass of a hydrogen atom, we will sum the masses of its constituents: one proton and one electron. \[m_{H} = m_p + m_e\] \[m_{H} = 1.67 × 10^{-27} kg + 9.11 × 10^{-31} kg\] It's important to note that the mass of an electron is much smaller than the mass of a proton, so the mass of a hydrogen atom is approximately equal to the mass of the proton.

Calculate the density of a hydrogen nucleus

Now we will calculate the density of a hydrogen nucleus using the formula: \(\text{density} = \frac{\text{mass}}{\text{volume}}\). Using the mass of a proton, the volume of the hydrogen nucleus and the radius that we have calculated previously: \[\rho_n = \frac{m_p}{V_n}\]

Calculate the density of a hydrogen atom

Similarly, we will calculate the density of a hydrogen atom using the formula: \(\text{density} = \frac{\text{mass}}{\text{volume}}\). Using the mass of a hydrogen atom, the volume of the hydrogen atom and the radius that we have calculated previously: \[\rho_a = \frac{m_{H}}{V_a}\] Now you have calculated the densities of a hydrogen nucleus and a hydrogen atom.