1)a radioactive material has mean life 1620 yrs and 540 yrs for alpha-emission and beta- emission respectively.The time in which half of the material remains undecayed emitting both these radiations is________2)mass of 2 curie uranium(234)is_______3)Half life of radium is 1620yrs &its atomic weight =226kg/kilomole.No. of atoms that will decay from 1 gm. of sample per second will be _________4)Ratio of momenta of an electron of mass me and an alpha particle of mass ma which are accelerated from rest by a potential difference of 250V is_______

Let's break down each of your questions step by step, ensuring we understand the concepts involved and how to arrive at the answers.

1. Time for Half of the Material to Remain Undecayed

To find the time in which half of the radioactive material remains undecayed while emitting both alpha and beta radiation, we need to calculate the effective half-life. The formula for the effective half-life (T_eff) when two decay processes are involved is given by:

1/T_eff = 1/T_alpha + 1/T_beta

Here, T_alpha is the half-life for alpha emission, and T_beta is the half-life for beta emission. We can derive the half-lives from the mean life (τ) using the relationship:

T_half = τ × ln(2)

Given:

• Mean life for alpha emission (τ_alpha) = 1620 years
• Mean life for beta emission (τ_beta) = 540 years

Calculating the half-lives:

• T_alpha = 1620 × ln(2) ≈ 1620 × 0.693 ≈ 1125.54 years
• T_beta = 540 × ln(2) ≈ 540 × 0.693 ≈ 374.82 years

Now, substituting these values into the effective half-life formula:

1/T_eff = 1/1125.54 + 1/374.82

Calculating this gives:

• 1/T_eff ≈ 0.000889 + 0.00267 ≈ 0.00356
• T_eff ≈ 281.5 years

Thus, the time in which half of the material remains undecayed is approximately 281.5 years.

2. Mass of 2 Curie Uranium-234

To find the mass of 2 curies of Uranium-234, we first need to know the activity in terms of disintegrations per second. One curie (Ci) is defined as 3.7 × 10¹⁰ disintegrations per second. Therefore, 2 curies is:

Activity = 2 × 3.7 × 10¹⁰ = 7.4 × 10¹⁰ disintegrations per second

Next, we use the decay constant (λ) to relate activity to the number of atoms (N) present:

Activity (A) = λN

For Uranium-234, the half-life (T_half) is about 245,500 years. The decay constant can be calculated as:

λ = ln(2) / T_half

Substituting the half-life:

λ = 0.693 / (245500 × 365 × 24 × 60 × 60) ≈ 2.82 × 10^-10 s^-1

Now, rearranging the activity formula to find N:

N = A / λ = 7.4 × 10¹⁰ / 2.82 × 10^-10 ≈ 2.62 × 10²⁰ atoms

Finally, to find the mass, we use the molar mass of Uranium-234, which is approximately 234 g/mol:

Mass = (N / Avogadro's number) × Molar mass

Mass = (2.62 × 10²⁰ / 6.022 × 10²³) × 234 g/mol ≈ 9.89 × 10^-3 g

Thus, the mass of 2 curies of Uranium-234 is approximately 0.00989 grams.

3. Number of Atoms Decaying from 1 g of Radium

To find the number of atoms that will decay from 1 gram of radium per second, we first need to calculate the number of moles in 1 gram of radium:

Number of moles = Mass / Molar mass

Given the atomic weight of radium is 226 kg/kilomole, we convert this to grams:

226 kg/kilomole = 226 g/mol

Now, calculating the number of moles in 1 g:

Number of moles = 1 g / 226 g/mol ≈ 0.00442 mol

Next, we find the number of atoms:

Number of atoms = Number of moles × Avogadro's number

Number of atoms ≈ 0.00442 mol × 6.022 × 10²³ ≈ 2.66 × 10²¹ atoms

Now, we need the decay constant (λ) for radium, which can be calculated from its half-life:

λ = ln(2) / T_half

Given T_half = 1620 years:

λ ≈ 0.693 / (1620 × 365 × 24 × 60 × 60) ≈ 2.68 × 10^-10 s^-1

Finally, the activity (A) gives us the number of decays per second:

A = λN ≈ 2.68 × 10^-10 s^-1 × 2.66 × 10²¹ atoms ≈ 7.13 × 10¹¹ decays/second

Thus, the number of atoms that will decay from 1 g of radium per second is approximately 7.13 × 10¹¹.

4. Momentum Ratio of an Electron

Let's tackle each of your questions step by step, breaking them down to ensure clarity and understanding. We'll explore the concepts of radioactive decay, half-life, and momentum in a way that makes sense.

1. Time for Half of the Material to Remain Undecayed

To find the time in which half of the radioactive material remains undecayed while emitting both alpha and beta radiation, we need to calculate the effective half-life. The formula for the effective half-life (T_eff) when two decay processes are involved is given by:

1/T_eff = 1/T_alpha + 1/T_beta

Here, T_alpha is the half-life for alpha emission, and T_beta is the half-life for beta emission. Given:

• T_alpha = 540 years
• T_beta = 1620 years

Substituting these values into the formula:

1/T_eff = 1/540 + 1/1620

Calculating the right side:

1/T_eff = (3 + 1) / 1620 = 4 / 1620

Thus, T_eff = 1620 / 4 = 405 years.

So, the time in which half of the material remains undecayed is 405 years.

2. Mass of 2 Curie Uranium-234

To find the mass of 2 curies of Uranium-234, we first need to understand what a curie represents. One curie (Ci) is defined as the amount of radioactive material that undergoes 3.7 x 10¹⁰ disintegrations per second. The activity (A) in curies can be related to the number of atoms (N) and the decay constant (λ) by:

A = λN

For Uranium-234, the half-life (T_1/2) is approximately 245,500 years. The decay constant can be calculated using:

λ = ln(2) / T_1/2

Substituting the half-life:

λ = 0.693 / (245500 * 365 * 24 * 3600) ≈ 2.82 x 10^-9 s^-1

Now, for 2 curies:

A = 2 * 3.7 x 10¹⁰ = 7.4 x 10¹⁰ disintegrations/second

Now, rearranging the activity formula:

N = A / λ = (7.4 x 10¹⁰) / (2.82 x 10^-9) ≈ 2.62 x 10¹⁹ atoms

To find the mass, we use the molar mass of Uranium-234, which is approximately 234 g/mol:

Mass = (N / Avogadro's number) * Molar mass

Mass = (2.62 x 10¹⁹ / 6.022 x 10²³) * 234 g ≈ 0.1 g

Thus, the mass of 2 curies of Uranium-234 is approximately 0.1 grams.

3. Number of Atoms Decaying from 1 g of Radium

To find the number of atoms that decay from 1 gram of radium per second, we start with the half-life and atomic weight. Given:

• Half-life of radium = 1620 years
• Atomic weight = 226 kg/kilomole

First, convert the atomic weight to grams:

226 kg/kilomole = 226 g/mole

Now, calculate the number of moles in 1 gram of radium:

Number of moles = mass / molar mass = 1 g / 226 g/mol ≈ 0.00442 moles

Using Avogadro's number (6.022 x 10²³ atoms/mole), we find the total number of atoms:

Total atoms = 0.00442 moles * 6.022 x 10²³ ≈ 2.66 x 10²¹ atoms

Next, we calculate the decay constant:

λ = ln(2) / T_1/2 = 0.693 / (1620 * 365 * 24 * 3600) ≈ 4.3 x 10^-10 s^-1

Now, the number of decays per second (activity) is:

Activity = N * λ = (2.66 x 10²¹) * (4.3 x 10^-10) ≈ 1.14 x 10¹² decays/second

Therefore, the number of atoms that will decay from 1 gram of radium per second is approximately 1.14 x 10¹² atoms.

4. Ratio of Momenta of an Electron and an Alpha Particle

To find the ratio of momenta of an electron (mass = m_e) and an alpha particle (mass = m_a) when both are accelerated from rest by a potential difference of 250 V, we start with the kinetic energy gained by each particle:

K.E. = qV

Where q is the charge of the particle. For an electron, q = -e (approximately -1.6 x 10^-19 C), and for an alpha particle, q = +2e (since it has two protons).

The kinetic energy for each particle can be expressed as:

• K.E._electron = e *