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 tackle these questions one by one, breaking down each concept to ensure clarity and understanding. We'll explore the principles of radioactivity, half-life, and momentum in a structured manner.

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 occur 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

Now, substituting these values into the formula:

1/T_eff = 1/540 + 1/1620

Finding a common denominator (which is 1620), we can rewrite this as:

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

Thus, T_eff = 1620/4 = 405 years.

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

2. Mass of 2 Curie Uranium-234

To determine the mass of 2 curies of Uranium-234, we first need to understand the relationship between curies and the number of radioactive decays. One curie (Ci) is defined as 3.7 x 10¹⁰ decays per second.

For 2 curies, the decay rate is:

2 Ci = 2 x 3.7 x 10¹⁰ = 7.4 x 10¹⁰ decays/second

Next, we need the decay constant (λ) for Uranium-234, which can be calculated from its half-life (T_1/2). The decay constant is given by:

λ = ln(2) / T_1/2

Assuming the half-life of Uranium-234 is about 245,500 years, we convert this to seconds:

T_1/2 = 245,500 years x 365 days/year x 24 hours/day x 3600 seconds/hour

Calculating this gives us approximately 7.74 x 10¹² seconds. Now substituting into the decay constant formula:

λ = ln(2) / 7.74 x 10¹² ≈ 8.97 x 10^-14 s^-1

Using the relationship between activity (A), decay constant, and number of atoms (N):

A = λN

We can rearrange this to find N:

N = A / λ = (7.4 x 10¹⁰) / (8.97 x 10^-14) ≈ 8.25 x 10²³ atoms

Now, to find the mass, we use the atomic weight of Uranium-234, which is approximately 234 g/mol:

Mass = (N / Avogadro's number) x Atomic weight

Substituting Avogadro's number (6.022 x 10²³ atoms/mol):

Mass = (8.25 x 10²³ / 6.022 x 10²³) x 234 g ≈ 33.5 g

Thus, the mass of 2 curies of Uranium-234 is approximately 33.5 grams.

3. Number of Atoms Decaying from 1 gm of Radium

To find the number of atoms decaying 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/mol

Next, we calculate the number of moles in 1 gram of radium:

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

Now, using Avogadro's number to find the number of atoms:

Number of atoms = moles x Avogadro's number = 0.00442 mol x 6.022 x 10²³ atoms/mol ≈ 2.66 x 10²¹ atoms

Next, we find the decay constant (λ) using the half-life:

λ = ln(2) / T_1/2 = ln(2) / (1620 years x 365 days/year x 24 hours/day x 3600 seconds/hour)

Calculating this gives:

λ ≈ 4.27 x 10^-14 s^-1

Finally, the activity (A) can be calculated as:

A = λN = (4.27 x 10^-14) x (2.66 x 10²¹) ≈ 1.14 x 10⁸ decays/second

Therefore, the number of atoms that will decay from 1 gram of radium per second is approximately 1.14 x 10⁸ decays/second.

4. Momentum Ratio of an Electron and an Alpha Particle

To find the ratio of momenta of an electron and an alpha particle