Question

In the experiment, if the Ba- \(137 \mathrm{~m}\) sample were placed closer to the Geiger tube, the measured activity would be greater (inverse-square relationship). Would this affect the result of the half-life? Explain how this would affect the \(N\) -versus- \(t\) graph on Cartesian graph paper (or on semi- log paper).

Short answer

Moving the sample affects the measured activity but not the half-life; the graph would show higher counts initially but maintain the same decay rate.

Step-by-step solution

Understand the Question

The exercise is asking if moving a Ba-137m sample closer to the Geiger tube affects the measured activity and how this would influence the half-life calculation and the graph of count rate vs. time.

Inverse-Square Relationship

As the Ba-137m sample is moved closer to the Geiger tube, the activity will be measured as higher because of the inverse-square law, which states that intensity is inversely proportional to the square of the distance from the source.

Analyze Half-Life Calculation

Half-life is intrinsic to the radioisotope and is not dependent upon the distance between the sample and the detector. Therefore, moving the sample does not affect the half-life of Ba-137m.

Impact on N versus t Graph

The N-versus-t graph, where N denotes the number of counts and t is time, would have higher initial values if the sample is closer to the detector, but the general shape (the exponential decay curve) and the slope on a semi-log scale, which indicates half-life, remain unchanged.

Key concepts

Inverse-Square Law

The inverse-square law is a fundamental principle that describes how the intensity of a physical quantity, like light or radiation, diminishes with distance. When applied to a radioactive decay experiment, this law means that the intensity of radiation decreases as the distance from the radioactive source increases.

Specifically, the inverse-square law states that this intensity is inversely proportional to the square of the distance from the source. Mathematically, it can be expressed as:

• \( I \propto \frac{1}{d^2} \)

where \( I \) is the intensity and \( d \) is the distance from the source.

This means if you halve the distance between the Geiger tube and the Ba-137m sample, the intensity of detected radiation will increase by a factor of four. This principle is crucial in understanding how position affects the count rate measured by the Geiger tube and helps in ensuring accurate calibrations during experiments.

In summary, always remember that as the sample gets closer to the tube, the measured activity looks higher due to this law. It does not actually change the physical quantity being measured but changes how it's sensed due to proximity.

Half-Life Calculation

Half-life is a key concept in understanding radioactive decay. It refers to the time required for half of the radioactive atoms in a sample to decay. This intrinsic property depends solely on the nature of the isotopic element, not on external conditions or measurement techniques.

For example, in the case of Ba-137m, the half-life is specific to this isotope and remains constant irrespective of how far or close it is to the Geiger tube. Thus, when considering experimental setups or modifications such as changes in distance to the detector, the half-life value remains unaffected.

The reason for this invariance is because half-life is a measure of the probability of decay, which is an inherent characteristic of the isotope's nucleus. Calculations involving half-life can be performed using the decay formula:

• \( N(t) = N_0 \times e^{-\lambda t} \)

where \( N(t) \) is the remaining number of radioactive nuclei at time \( t \), \( N_0 \) is the initial number, and \( \lambda \) is the decay constant related to the half-life by \( \lambda = \frac{\ln(2)}{T_{1/2}} \).

Thus, in our experiment, while the count rate might increase due to a closer placement of the radioactive source, the half-life itself, derived from the slope of the decay curve on a semi-log scale, remains constant.

Geiger Tube Measurements

Geiger tubes are pivotal in detecting and measuring radioactive decay, making them essential in experiments involving radioactivity measurements. They work by ionizing radiation passing through a gas within the tube, which then triggers an electric pulse that is counted as an event.

When conducting a radioactive decay experiment, it's important to note that measured activity using a Geiger tube depends on several factors:

• Distance from the radioactive source (as the inverse-square law affects count rate)
• Efficiency of the tube
• Background radiation levels

The proximity of the source significantly affects the count rate due to the inverse-square law, hence careful positioning is crucial for consistent and repeatable measurements.

In relation to the N-versus-t graph, a closer proximity results in higher initial count rates. However, such measurements reflect only a part of the full scenario. They enable better counting statistics due to the higher number of events, leading to a more detailed graphical representation of decay over time.

Ultimately, while the Geiger tube readings provide immediate insights into activity, understanding their relation with distance, and careful interpretation, are critical in accurately assessing decay characteristics like half-life.