Silver Atom Properties and Calculations

Understanding Atomic Structure and Calculations for Silver

This section details the process of determining various properties of a silver atom, including its atomic and mass numbers, number of neutrons, and the number of atoms in a given sample, based on its nuclear mass and charge.

1. Determining Atomic Number (Z) and Mass Number (A)

The atomic number (Z) represents the number of protons in an atom's nucleus, which also equals the number of electrons in a neutral atom. The mass number (A) is the total number of protons and neutrons in the nucleus.

1.1. Atomic Number (Z) from Charge

The charge of the nucleus is solely due to its protons, as neutrons are neutral. Each proton carries an elementary charge of approximately $1.602 \times 10^{-19}$ C.

• Given Nuclear Charge (Q): $7.52 \times 10^{-18}$ C
• Elementary Charge (e): $1.602 \times 10^{-19}$ C
• Formula: $Z = \frac{Q}{e}$
• Calculation: $Z = \frac{7.52 \times 10^{-18} \text{ C}}{1.602 \times 10^{-19} \text{ C/proton}} \approx 46.94$
• Since the atomic number must be an integer, we round this to the nearest whole number.
• Result: $Z = 47$
• This atomic number corresponds to Silver (Ag) on the periodic table.

1.2. Mass Number (A) from Nuclear Mass

The mass of the nucleus is primarily due to its protons and neutrons. The mass of an electron is negligible in comparison. The approximate mass of a proton or neutron is about $1.67 \times 10^{-27}$ kg (or 1 atomic mass unit, amu).

• Given Mass of Nucleus (Y): $1.804 \times 10^{-25}$ kg
• Approximate Mass of a Nucleon (proton or neutron): $m_p \approx m_n \approx 1.67 \times 10^{-27}$ kg
• Formula: $A = \frac{\text{Mass of Nucleus}}{\text{Average Mass of a Nucleon}}$
• Calculation: $A = \frac{1.804 \times 10^{-25} \text{ kg}}{1.67 \times 10^{-27} \text{ kg/nucleon}} \approx 108.02$
• Since the mass number must be an integer, we round this to the nearest whole number.
• Result: $A = 108$

2. Calculating the Number of Neutrons (N)

The number of neutrons is found by subtracting the atomic number (number of protons) from the mass number (total number of protons and neutrons).

• Formula: $N = A - Z$
• Calculation: $N = 108 - 47 = 61$
• Result: The silver atom has 61 neutrons.

3. Number of Atoms in a 25g Sample of Silver

To find the number of atoms in a given mass of a substance, we use Avogadro's number and the molar mass of the element.

• Given Sample Mass: 25 g
• Molar Mass of Silver (Ag): The mass number (A) in grams per mole is approximately the molar mass. So, for Ag-108, the molar mass is approximately 108 g/mol.
• Avogadro's Number ($N_A$): $6.022 \times 10^{23}$ atoms/mol
• Formula: $\text{Number of Atoms} = \left( \frac{\text{Sample Mass}}{\text{Molar Mass}} \right) \times N_A$
• Calculation: $\text{Number of Atoms} = \left( \frac{25 \text{ g}}{108 \text{ g/mol}} \right) \times (6.022 \times 10^{23} \text{ atoms/mol})$
• $\text{Number of Atoms} \approx 0.23148 \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol}$
• Result: $\text{Number of Atoms} \approx 1.39 \times 10^{23}$ atoms

4. Symbolic Representation of the Atom

The symbolic representation of an atom follows the format: $^{A}_{Z}\text{X}$, where X is the element symbol, A is the mass number, and Z is the atomic number.

• Element Symbol: Ag (for Silver)
• Atomic Number (Z): 47
• Mass Number (A): 108
• Result: $^{108}_{47}\text{Ag}$

Key Takeaways

• The atomic number (Z) is determined by the nuclear charge and identifies the element.
• The mass number (A) is determined by the total nuclear mass and represents the sum of protons and neutrons.
• The number of neutrons (N) is calculated as $A - Z$.
• The number of atoms in a sample can be found using the sample's mass, the element's molar mass, and Avogadro's number.
• The symbolic representation $^{A}_{Z}\text{X}$ concisely summarizes an atom's composition.