Calculate the age for a rock that contains equal numbers of Potassium-40 and Argon-40 isotopes. The half-life of Potassium-40 is 1.25 Billion years.
The correct answer and explanation is :
To find the age of a rock that contains equal numbers of Potassium-40 (⁴⁰K) and Argon-40 (⁴⁰Ar), we can use the concept of radioactive decay. The key point is that radioactive decay follows an exponential decay process, and when half of the parent isotope (in this case, Potassium-40) has decayed into the daughter isotope (Argon-40), one half-life has passed.
🔢 Calculation:
If the number of ⁴⁰K atoms equals the number of ⁴⁰Ar atoms, that means half of the original ⁴⁰K has decayed into ⁴⁰Ar. By definition, this occurs after one half-life.
- Half-life of ⁴⁰K = 1.25 billion years
So,
👉 Age of the rock = 1.25 billion years
🧠 Explanation (300 words):
Radioactive dating is a powerful method used by geologists to determine the age of rocks and minerals. One of the commonly used isotopes for this purpose is Potassium-40 (⁴⁰K), a radioactive isotope that decays into Argon-40 (⁴⁰Ar), a stable gas. The decay process is predictable and occurs at a known rate, called the half-life — the time it takes for half of the original radioactive atoms to decay.
In this scenario, we’re told the rock has equal numbers of Potassium-40 and Argon-40 atoms. This indicates that half of the original ⁴⁰K has already decayed, because for every atom of Argon-40 present, one atom of Potassium-40 has transformed. That’s the definition of one half-life: 50% of the parent isotope has decayed into the daughter isotope.
Since the half-life of Potassium-40 is known to be 1.25 billion years, and we observe a 1:1 ratio of ⁴⁰K to ⁴⁰Ar, we can conclude that the rock has been undergoing decay for exactly one half-life.
This kind of radiometric dating is especially useful because Argon is a gas and doesn’t easily get trapped in rocks during formation — it accumulates only from radioactive decay. That makes the K-Ar dating method reliable for determining the absolute age of rocks that are hundreds of millions to billions of years old.
Thus, the age of the rock is:
👉 1.25 billion years.