Problem 21 - Analytical Chemistry

Question

The steady state activity for \({ }^{14} \mathrm{C}\) in a sample is 13 cpm per gram of carbon. If counting is limited to \(1 \mathrm{hr}\), what mass of carbon is needed to give a percent relative standard deviation of \(1 \%\) for the sample's activity? How long must we monitor the radioactive decay from a 0.50 -g sample of carbon to give a percent relative standard deviation of \(1.0 \%\) for the activity?

Step-by-Step Solution

Answer

A mass of approximately 12.82 g is needed for 1% RSD in 1 hour. Monitor a 0.50-g sample for about 25.643 hours for 1% RSD.

Step 1: Understanding the Problem

We need to find the mass of carbon required to achieve a 1% relative standard deviation when counting for 1 hour. We also need to determine how long we must monitor a 0.50-g sample of carbon to achieve the same precision.

Step 2: Using the Formula for Relative Standard Deviation

The relative standard deviation (RSD) is given by the formula: \( \text{RSD} = \frac{1}{\sqrt{N}} \), where \(N\) is the total count. We set \( \text{RSD} = 0.01 \) for 1% precision.

Step 3: Finding Required Activity for Mass

For the first part of the problem, find \(N\) that achieves a 1% RSD in 1 hour. Since \(N = R \times t\), where \(R\) is the rate of activity in cpm, solve \( \frac{1}{\sqrt{N}} = 0.01 \) to find \(N = 10000\).

Step 4: Calculate Required Mass of Carbon for 1% RSD in 1 Hour

Given \(N = 10000\) and activity rate of 13 cpm per gram, we find \(Rm \cdot t = 10000\). So, \(m = \frac{10000}{13 \times 60} \approx 12.82\) grams.

Step 5: Determine Monitoring Time for 0.50-g Sample

For 0.50-g of carbon, \(R = 13 \times 0.5 = 6.5\) cpm. Find \(t\) such that \(R \cdot t = 10000\). So, \(t = \frac{10000}{6.5}/60 \approx 25.643\) hours.

Key Concepts

Relative Standard Deviation

The relative standard deviation (RSD) is an important concept in scientific measurements, particularly when precision is required. It gives the size of the standard deviation in relation to the mean of the data set. This makes it a handy way to gauge precision across different datasets. RSD is expressed as a percentage and calculated through:\[\text{RSD} = \frac{\text{Standard Deviation}}{\text{Mean}} \times 100\]In radioactive decay studies, such as this exercise with carbon-14, having a low RSD is critical to ensure measurements are accurate. If we require a 1% RSD in an experiment, it needs very precise counting and sometimes, larger sample sizes or longer counting times. This can be crucial when determining the starting mass of a radioactive sample or when calculating the time needed to monitor decay.

Carbon-14

Carbon-14, often symbolized as \({ }^{14} \mathrm{C} \), is a radioactive isotope of carbon. It is central to fields like archaeology and geology via radiocarbon dating. Radiocarbon dating uses the decay of carbon-14 to estimate the age of carbon-containing materials and is explored in this exercise. Carbon-14 is steadily created in the atmosphere and taken up by living organisms. Upon death, intake stops and carbon-14 decays with a half-life of about 5730 years, serving as a clock for determining elapsed time since the organism's death. The challenge for scientists is often ensuring accurate decay counting to make precise estimations, which is where concepts like counting statistics and RSD come into play.

Counting Statistics

Counting statistics is the method of measuring how often a random event occurs within a set time period. In radioactive decay studies, it's about how many decay events are counted in a sample over a duration. The key formula here is: \[ N = R \times t \]This formula links the number of decay events (\(N\)) to the rate of decay events per minute (\(R\)) and the time monitored (\(t\)). In our problem, a sample needed to reach an RSD of 1% in activity requires understanding these statistics. Accuracy in such statistics is vital, as seen in our exercise. More events or longer counting increases accuracy. Fewer events or short counting times lead to potential errors, underlining the importance of correctly selecting both sample size and duration.

Analytical Chemistry Problem Solving

Analytical chemistry problem solving involves a detailed understanding of the chemical and mathematical aspects of a problem. In our exercise on carbon-14 decay, to solve for precision requirements like 1% RSD, it's necessary to apply analytical problem-solving strategies. Breaking down the problem:

• Identify what's known: such as decay rate per gram of carbon.
• Use relevant formulas: in this case, relate RSD, sample mass, and monitoring time.
• Calculate precisely: derive the mass required or time needed by solving equations step-by-step.
• Check for consistency and accuracy: validate results with experimental expectations.

Solving these exercises hones skills in accurately implementing calculations and interpreting results applicable in diverse scenarios beyond just radioactive decay.