Assuming 1μg of trace radioactive element X with a half life of 30 years is absorbed by a growing tree. The amount of X remaining in the tree after 100 years is ______ × 10–1μg.
[Given : ln 10 = 2.303; log 2 = 0.30]
[Given : ln 10 = 2.303; log 2 = 0.30]
Correct Answer: 1
Approach Solution - 1
To solve the problem of how much of the radioactive element X remains after 100 years, we use the formula for radioactive decay: N(t) = N₀e–λt, where N(t) is the remaining quantity, N₀ is the initial quantity, λ is the decay constant, and t is time in years. First, compute the decay constant λ using the relationship λ = ln 2 / T₁/₂, where T₁/₂ is the half-life of the element.
Substituting known values, T₁/₂ = 30 years, we get:
λ = ln 2 / 30.
Given log 2 = 0.30, hence ln 2 = 2.303 × 0.30 = 0.6909.
Therefore, λ = 0.6909 / 30 ≈ 0.02303 per year.
Next, calculate the remaining quantity after 100 years:
N(100) = 1μg × e–0.02303 × 100 = 1μg × e–2.303.
To find e–2.303, recognize that e–2.303 = 1 / e2.303 = 1 / 10 = 0.1.
Therefore, N(100) = 1μg × 0.1 = 0.1μg.
The amount of X remaining is 1.0 × 10–1 μg, which is within the specified range of the solution, confirming its validity.
Approach Solution -2
\(kt=ln\frac {1}{1−x}\)
\(\frac {0.693}{30}(100)=ln\frac {1}{1−x}\)
\(2.303=2.303\ log\frac {1}{1−x}\)
\(⇒\frac {1}{1−x}=10\)
\(⇒1=10–10X\)
\(⇒X=\frac {9}{10}\)
\(⇒X =0.9\) μg
Amount of X remaining\(=1–X\)
\(=1–0.9\)
\(=0.1\) μg
\(=1×10^{−1}\) μg
So, the answer is \(1\).
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Concepts Used:
Structure of Atom
Atomic Structure:
The atomic structure of an element refers to the constitution of its nucleus and the arrangement of the electrons around it. Primarily, the atomic structure of matter is made up of protons, electrons and neutrons.
Dalton’s Atomic Theory
Dalton proposed that every matter is composed of atoms that are indivisible and indestructible.
The following are the postulates of his theory:
- Every matter is made up of atoms.
- Atoms are indivisible.
- Specific elements have only one type of atoms in them.
- Each atom has its own constant mass that varies from element to element.
- Atoms undergo rearrangement during a chemical reaction.
- Atoms can neither be created nor be destroyed but can be transformed from one form to another.
Cons of Dalton’s Atomic Theory
- The theory was unable to explain the existence of isotopes.
- Nothing about the structure of atom was appropriately explained.
- Later, the scientists discovered particles inside the atom that proved, the atoms are divisible.
Subatomic Particles
- Protons - are positively charged subatomic particles.
- Electron - are negatively charged subatomic particles.
- Neutrons - are electrically neutral particles and carry no charge
Atomic Structure of Isotopes
Several atomic structures of an element can exist, which differ in the total number of nucleons.These variants of elements having a different nucleon number (also known as the mass number) are called isotopes of the element. Therefore, the isotopes of an element have the same number of protons but differ in the number of neutrons. For example, there exist three known naturally occurring isotopes of hydrogen, namely, protium, deuterium, and tritium.