Answer: A. Therefore, there are approximately 4.01 × 10^22 nuclides in the sample.
B. The decay constant for the element is approximately 9.21 × 10^12 s⁻¹.
C. The half-life of the element is approximately 7.54 × 10⁻¹³ seconds.
Explanation: Given:
Mass of the element = 7.00 g
Mass number = 105
Decay rate = 6.14 × 10^11 Bq (becquerels)
A. To calculate the number of nuclides in the sample, we need to determine Avogadro's number (Na) and the molar mass of the element. Since the mass number is not the same as the atomic mass, we need to determine the actual molar mass.
The molar mass of the element (M) can be calculated as:
M = mass / N₀
N₀ = Avogadro's number / molar mass
Using the known values:
Mass of the element (m) = 7.00 g
Atomic mass of the element (A) = 105 g/mol
M = m / N₀
105 g/mol = 7.00 g / N₀
Solving for N₀:
N₀ = 7.00 g / (105 g/mol) = 0.0667 mol
To determine the number of nuclides (n) in the sample:
n = N₀ * Na
Na is Avogadro's number (approximately 6.022 × 10^23)
n = 0.0667 mol * (6.022 × 10^23) ≈ 4.01 × 10^22 nuclides
Therefore, there are approximately 4.01 × 10^22 nuclides in the sample.
B. The decay constant (λ) can be determined using the decay rate (λ = decay rate / N₀). Given the decay rate as 6.14 × 10^11 Bq and N₀ as 0.0667 mol:
λ = (6.14 × 10^11 Bq) / (0.0667 mol)
Calculating:
λ ≈ 9.21 × 10^12 s⁻¹
The decay constant for the element is approximately 9.21 × 10^12 s⁻¹.
C. The half-life (T₁/₂) can be calculated using the formula:
T₁/₂ = ln(2) / λ
Given:
λ ≈ 9.21 × 10^12 s⁻¹
T₁/₂ = ln(2) / (9.21 × 10^12 s⁻¹)
Calculating:
T₁/₂ ≈ 7.54 × 10⁻¹³ s
The half-life of the element is approximately 7.54 × 10⁻¹³ seconds.