Question

What mass of electrons is required to neutralize the charge of \(1.0 \mathrm{~g}\) of protons?

Short answer

The mass of electrons required to neutralize the charge of 1.0 g of protons is approximately 5.445 x 10^-7 kg or 0.5445 mg.

Step-by-step solution

Determine the number of protons in 1.0 g

To determine the number of protons in 1.0 g, use the mass of a single proton and Avogadro's number. The mass of a proton is approximately 1.6726 x 10^-27 kg. To convert 1.0 g of protons to kg, use the conversion factor of 1 g = 1 x 10^-3 kg. So, 1.0 g is 1.0 x 10^-3 kg. The number of protons, N_p, is the total mass divided by the mass of one proton, i.e., N_p = (1.0 x 10^-3) / (1.6726 x 10^-27).

Calculate the number of protons

Using the formula from the previous step, calculate the number of protons: N_p = (1.0 x 10^-3 kg) / (1.6726 x 10^-27 kg/proton). This calculation results in N_p being approximately equal to 5.978 x 10^23 protons.

Determine the mass of electrons required to neutralize the protons

Since each proton has the same magnitude of charge as an electron, but opposite sign, it will require an equal number of electrons to neutralize the charge of the protons. The mass of one electron is much smaller than that of a proton, approximately 9.1094 x 10^-31 kg. Therefore, the total mass of electrons, m_e, needed to neutralize the charge is the number of electrons (which is equal to the number of protons) times the mass of a single electron: m_e = N_p * 9.1094 x 10^-31 kg. Note that the number of electrons, N_e, is equal to N_p.

Calculate the mass of electrons

Using the formula from the previous step, m_e = N_p * 9.1094 x 10^-31 kg, and substituting the value of N_p from step 2, calculate the mass of electrons. The result is m_e = (5.978 x 10^23) * (9.1094 x 10^-31 kg/electron) which equals approximately 5.445 x 10^-7 kg.

Key concepts

Understanding Avogadro's Number

Avogadro's number, denoted as \(6.022 \times 10^{23}\), plays a fundamental role in chemistry and is one of the pillars of the mole concept. It represents the number of constituent particles, usually atoms or molecules, contained in one mole of a substance. This number is derived from the number of atoms found in exactly 12 grams of carbon-12, which is thought to be constant for any substance.

Counting the number of particles in even a tiny amount of substance is practically impossible. That's where Avogadro's number becomes useful. It provides a way to convert between a number of particles and an amount of substance in moles. So, when we have 1 mole of an element or compound, we know that we have \(6.022 \times 10^{23}\) particles of that element or compound, regardless of what substance it is.

Mass of a Proton

The mass of a proton is a critical feature in understanding the atomic scale and plays a role in calculations involving atomic particles. A proton is positively charged and found in the nucleus of an atom; its mass is approximately 1.6726 x 10^-27 kg.

It is this mass, along with Avogadro's number, that allows us to compute the total mass of protons in a given sample. When we know the mass of one proton, by dividing the total mass of a sample by the mass of a single proton, we can estimate the number of protons in that particular sample. This number is essential for solving problems in nuclear and atomic physics, as well as those in chemistry.

Mass of an Electron

An electron is a subatomic particle with a negative charge which, along with protons and neutrons, contributes to the structure of an atom. The mass of an electron is significantly lower than that of a proton, being approximately 9.1094 x 10^-31 kg.

The small mass of an electron is noteworthy because it allows us to ignore its contribution to the total mass of an atom when performing calculations concerning atomic mass. However, when dealing with electrical charge neutrality, we consider the number and not the mass of electrons, because each electron neutralizes the charge of a proton despite the massive difference in their mass.

Charge Neutralization

Charge neutralization is the process by which an equal number of positive and negative charges cancel each other out, resulting in a net charge of zero. This concept is critical when dealing with atomic and molecular systems.

At a subatomic level, charge neutralization typically involves the number of positively charged protons in the nucleus being balanced by an identical number of negatively charged electrons orbiting it. When an imbalance occurs, we have ions: positively charged if there are fewer electrons than protons (cations), or negatively charged if there are more electrons (anions). In the context of our textbook exercise, to neutralize the charge of a given mass of protons, you would require an equal number of electrons, thus maintaining the principle of charge neutrality.