Problem 26 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

If the amount of radioactive substance is increases three times, the number of disintegration per unit time will be (a) doubled (b) one-third (c) triple (d) uncharged

Step-by-Step Solution

Answer

The number of disintegrations per unit time will be tripled.

Step 1: Identify the Law governing radioactive decay

The number of disintegrations per unit time, often referred to as the activity (A), of a radioactive substance is directly proportional to the amount of the substance present. This is described by the radioactive decay law which states that A is proportional to N, where N is the number of radioactive atoms. Mathematically, this can be expressed as A = λN, where λ is the decay constant.

Step 2: Analyze the change in the amount of radioactive substance

If the amount of the radioactive substance is increased three times, then the new number of radioactive atoms, labeled N', would be three times the initial amount, N. Hence, N' = 3N.

Step 3: Apply the decay law to find the new activity

Using the proportional relationship, the new activity A' can be found by substituting N' into the decay law equation. Hence, A' = λN' = λ(3N) = 3λN. Since λN is the initial activity A, the new activity A' is three times the initial activity, A' = 3A.

Key Concepts

Radioactive Substance

A radioactive substance is a material that contains unstable atoms which release energy in the form of radiation as they decay to become more stable. This process of decay occurs spontaneously and at a constant rate, unique to each radioactive isotope. The decay can result in the emission of alpha particles, beta particles, gamma rays, and other forms of radiation.

Each radioactive substance has a characteristic half-life, the time it takes for half of the radioactive atoms to decay. This property is fundamental to identifying the type of radioactive substance and predicting how its activity will change over time. In the context of the textbook exercise, we consider the relationship between the amount of radioactive substance present and the number of disintegrations per unit time, which corresponds to the substance's activity.

Disintegrations Per Unit Time

The term disintegrations per unit time refers to the number of radioactive decays occurring in a substance per unit time, often measured in seconds. This measure is also known as the activity of the radioactive substance. The activity is a crucial parameter because it represents the rate at which the substance undergoes radioactive decay.

By monitoring the disintegrations per unit time, scientists can deduce valuable information about the sample's current state and its future behavior. In our exercise, the direct proportionality between the amount of substance and its activity leads to a straightforward conclusion: tripling the amount of a radioactive substance results in tripling the activity, meaning the disintegrations per unit time also triple.

Decay Constant

The decay constant, symbolized by λ, represents the probability per unit time of a single radioactive nucleus decaying. It is a fundamental property of each radioactive isotope and plays a pivotal role in describing the kinetics of radioactive decay. The inverse of the decay constant is the mean lifetime of the atoms, which provides the average time an atom will exist before it decays.

The radioactive decay law is mathematically expressed as A = λN, where A denotes the activity and N the number of undecayed atoms. As evidenced by the exercise solution, when the amount of radioactive substance is increased, the decay constant remains unchanged. This is because it is an intrinsic property of the substance, not affected by the quantity present. Therefore, when the amount of substance increases threefold, the number of disintegrations per unit time is also tripled, as the decay rate per atom stays constant.