Question

A painting supposedly by Rembrandt \((1609-\) 1669 was found to contain \(96.1 \%\) of the amount of \({ }^{14} \mathrm{C}\) found in a living plant. Could this painting have been done by Rembrandt? [The half-life of \({ }^{14} \mathrm{C}\) is 5715 years. ] Justify your answer.

Short answer

The approximate age of the painting is 438 years, which means it was likely created in 1584. Since Rembrandt was born in 1609, the painting could not have been done by him based on its calculated age using the carbon-14 content.

Step-by-step solution

Determine the variables in the decay formula

To calculate the age of the painting, we will use the decay formula for carbon-14: \[ N(t) = N_0 * 0.5^{\frac{t}{t_{1/2}}} \] Where: - \(N(t)\) is the remaining amount of carbon-14 at time \(t\), - \(N_0\) is the initial amount of carbon-14, - \(t\) is the age of the painting (in years), - \(t_{1/2}\) is the half-life of the isotope (in this case, 5715 years). We know that the painting has 96.1% of the carbon-14 content found in a living plant, which means: \[ N(t) = 0.961 * N_0\] Our task is to find the value of \(t\), the age of the painting.

Calculate the age of the painting

Rearrange the decay formula to solve for \(t\): \[ 0.961 = 0.5^{\frac{t}{5715}}\] Take the natural logarithm on both sides of the equation: \[ \ln(0.961) = \frac{t}{5715} * \ln(0.5)\] Divide by the natural logarithm of 0.5: \[ t = \frac{\ln(0.961)}{\ln(0.5)} * 5715\] Now, calculate the value of \(t\): \[ t \approx 437.66\] Therefore, the approximate age of the painting is 438 years.

Check if the age of the painting matches Rembrandt's lifespan

Rembrandt was born in 1609 and died in 1669. To determine if this painting could have been done by Rembrandt, we should compare the age of the painting with his lifespan. Calculate the year when the painting was made by subtracting its age from the current year: \[ 2022 - 438 = 1584\] The painting was supposedly made in 1584, which is before Rembrandt's birth (1609).

Conclusion

Based on the carbon-14 content found in the painting and its calculated age, we conclude that this painting could not have been done by Rembrandt, as it predates his birth.

Key concepts

Rembrandt

Rembrandt van Rijn was a renowned Dutch painter and etcher born in 1609. Known for his profound influence on the Dutch Golden Age of Art, Rembrandt's works are revered for their emotional depth and innovative use of shadow and light.

During his lifetime, Rembrandt produced a vast number of paintings, drawings, and etchings, making a significant mark on European art history. The scientific practice of verifying artworks allegedly belonging to such famous artists involves identification techniques like provenance research and, sometimes, technical methods like carbon dating.

When a painting recently claimed to be by Rembrandt was analyzed for its carbon-14 content, it became evident upon calculation that the painting's age did not coincide with Rembrandt's lifespan. Though technical methods cannot solely determine authorship, they can provide critical data to support historical art investigations.

Carbon-14 Decay

Carbon-14 decay is a fundamental concept in carbon dating, widely used to date organic materials. Carbon-14, an isotope of carbon, is radioactive and decays over time. In living organisms, carbon-14 levels remain constant through carbon exchange with the environment.

Once the organism dies, the exchange stops, and carbon-14 begins to decay at a known rate. This decay is critical for determining the age of the material. In the exercise, determining the age of a painting using its carbon-14 content helped conclude whether it was from Rembrandt's era.

The decay of carbon-14 is a predictable process, and its rate is quantified by its half-life, making it a powerful tool for archaeologists, paleontologists, and art experts alike.

Half-life Calculation

Half-life is the period required for half of the radioactive nuclei in a sample to decay. In the context of carbon dating, half-life is crucial because it allows us to calculate the age of an object containing the isotope carbon-14.

The half-life of carbon-14 is 5715 years, a relatively long period that enables dating of materials up to about 50,000 years old. In our example, the painting's carbon-14 content indicated it was older than Rembrandt himself, which presents a fascinating insight into historical artifacts.

Through half-life calculation, the portion remaining in the sample can be used to extrapolate back to determine the time since the organism's death, giving us an age estimate of the item in question.

Isotope Decay Formula

The isotope decay formula is at the core of carbon dating calculations. This formula helps in tracing the decay of radioactive isotopes like carbon-14 over time. It is mathematically represented as: \[ N(t) = N_0 \times 0.5^{\frac{t}{t_{1/2}}} \]where:

• \(N(t)\) is the amount of the isotope remaining
• \(N_0\) is the initial quantity of the isotope
• \(t\) is the elapsed time
• \(t_{1/2}\) is the half-life of the isotope

This formula was pivotal in the exercise where determining the painting's age involved plugging in known values, such as the half-life of carbon-14 and the percentage of carbon-14 remaining.

By manipulating this decay formula, students arrived at the conclusion that the painting was created before Rembrandt's birth, demonstrating the formula's utility in validating historical claims.