Nuclei

Step 1: Understanding the Concept:
The half-life ( ) of a radioactive substance is the time required for half of the initial number of radioactive nuclei to decay. It is inversely related to the decay constant ( ), which represents the probability of decay per nucleus per unit time.
Step 2: Key Formula or Approach:
The relationship between half-life ( ) and the decay constant ( ) is given by:
Using the approximation , the formula becomes:
Step 3: Detailed Explanation:
We are given the decay constant per year.
Substitute this value into the half-life formula:
Now, we perform the calculation:
Expressing this in standard scientific notation:
Step 4: Final Answer:
The half-life of the radioactive substance is approximately years.

Step 1: Understanding the Concept:
Carbon dating, also known as radiocarbon dating, is a scientific technique that uses the properties of a naturally occurring radioactive isotope of carbon ( C) to determine the age of carbon-based materials up to about 50,000 to 60,000 years old.
Step 2: Detailed Explanation:
The principle behind carbon dating is as follows:
1. Formation of Carbon-14:
Cosmic rays from space strike the Earth's upper atmosphere, producing neutrons. These neutrons collide with Nitrogen-14 ( N) atoms, converting them into Carbon-14 ( C), which is radioactive.
2. Incorporation into Living Organisms:
This radioactive C combines with oxygen to form carbon dioxide ( ). This radioactive carbon dioxide mixes with regular carbon dioxide in the atmosphere and is absorbed by plants during photosynthesis. Animals then eat these plants. As a result, all living organisms continuously incorporate both stable carbon ( C) and radioactive carbon ( C) into their bodies, maintaining a relatively constant ratio of C to C, the same as in the atmosphere.
3. Decay After Death:
When an organism dies, it stops taking in carbon from the environment. The stable C in its tissues remains, but the radioactive C begins to decay back into Nitrogen-14 ( N) with a known half-life of approximately 5,730 years.
4. Age Determination:
By measuring the ratio of the remaining C to the stable C in a sample of organic material (like wood, bone, or cloth) and comparing it to the atmospheric ratio, scientists can calculate how many half-lives have passed since the organism died. This allows them to determine the age of the sample.
Step 3: Final Answer:
Carbon dating is a method of age determination for organic artifacts by measuring the decay of the radioactive isotope C they contain since the time of their death.

Step 1: Understanding the Concept:
The half-life ( ) of a radioactive isotope is the time it takes for half of the initial number of nuclei to decay. After each half-life, the amount of the remaining substance is halved.
Step 2: Key Formula or Approach:
The fraction of nuclei remaining undecayed ( ) after a certain time can be calculated using the number of half-lives, . The fraction remaining is then given by: Step 3: Detailed Explanation:
Given values:
Half-life, days.
Total time, days.
First, calculate the number of half-lives ( ) that have passed in 20 days: So, 4 half-lives have occurred.
Now, calculate the fraction of undecayed nuclei: Step 4: Final Answer:
After 20 days (4 half-lives), the fraction of the nuclei remaining undecayed will be . Option (D) is correct.

Step 1: Understanding the Concept:
Radioactivity (or simply activity) of a radioactive sample is defined as the rate at which the nuclei of its constituent atoms decay. It is a measure of the number of disintegrations per unit time.
Step 2: Detailed Explanation:
Let's analyze the units given in the options:
(A) MeV (Mega-electron Volt): This is a unit of energy, commonly used in nuclear physics. Joules. It is not a unit of decay rate.
(B) Curie (Ci): This is a traditional unit of radioactivity. It is defined as decays per second. It is a valid unit for radioactivity.
(C) a.m.u. (atomic mass unit): This is a unit of mass, used for atomic and subatomic particles. kg. It is not a unit of decay rate.
(D) Joule (J): This is the SI unit of energy or work. It is not a unit of decay rate.
The SI unit of radioactivity is the Becquerel (Bq), where 1 Bq = 1 decay per second. The Curie is a larger, non-SI unit that is still widely used.
Step 3: Final Answer:
Among the given options, only the curie is a unit used to measure radioactivity. Therefore, option (B) is correct.

Step 1: Understanding the Concept:
Penetrating power is the ability of a form of radiation to pass through matter. The more a particle or ray interacts with the atoms of the material it passes through, the more quickly it loses its energy and the lower its penetrating power.
Step 2: Detailed Explanation:
Let's compare the penetrating power of the given radiations:
(A) -rays (Alpha particles): These are helium nuclei ( ). They have a large mass and a double positive charge (+2e). Due to their strong charge and large size, they interact very strongly with matter through ionization. They lose their energy over a very short distance and have the \textit{lowest} penetrating power. They can be stopped by a sheet of paper or even the outer layer of skin.
(B) -rays (Beta particles): These are high-speed electrons or positrons. They are much lighter than alpha particles and have a single charge (-e or +e). They interact less strongly with matter than alpha particles, so they can penetrate further. They have a \textit{medium} penetrating power and can be stopped by a few millimeters of aluminum.
(C) -rays (Gamma rays): These are high-energy photons (electromagnetic radiation). They have no mass and no charge. Because they are uncharged, they do not interact strongly via electrostatic forces and penetrate deeply into matter. They have the \textit{highest} penetrating power and require thick layers of dense materials like lead or concrete to be significantly attenuated.
(D) Cathode rays: These are streams of electrons, essentially the same as beta particles but typically with lower energies found in cathode ray tubes. Their penetrating power is less than that of gamma rays and is comparable to beta particles.
Step 3: Final Answer:
Comparing the four options, gamma rays have the least interaction with matter due to their lack of charge and mass, giving them the highest penetrating power. Therefore, option (C) is correct.