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Question: The decay constant of a radioactive element is \(1.5 \times {10^{ - 9}}{s^{ - 1}}\) . Its mean life ...

The decay constant of a radioactive element is 1.5×10−9s−11.5 \times {10^{ - 9}}{s^{ - 1}} . Its mean life in second will be.

Explanation

Solution

In this question, we will use the relation between the mean lifetime and the decay constant. Now, by substituting the given values, we will directly get the required result. Also, we will study about the wave-particle nature of an atom and basic of time, for our better understanding.

Formula used:
τ=1λ\tau = \dfrac{1}{\lambda }

Complete step by step solution:
We know that radioactive material is defined as any material containing unstable atoms that emit ionizing radiation as it decays.
Also, the decay constant of a radioactive element is its probability to decay per unit time.
As we know the mean life time of any radioactive element is given by:
τ=1λ\tau = \dfrac{1}{\lambda }
Now, by substituting the given values in the above equation, we get:
τ=11.5×10−9\tau = \dfrac{1}{{1.5 \times {{10}^{ - 9}}}}
∴τ=10915=6.67×10−8s⁡\therefore \tau = \dfrac{{{{10}^9}}}{{15}} = 6.67 \times {10^{ - 8}}\operatorname{s}
Therefore, we get the required answer, which gives us the mean life time of the radioactive element.

Additional information:
Wave- particle duality is a concept of quantum mechanics that according to this every particle may be described as either a particle or a wave. It expresses the inability of the classical concepts of particle and wave, to fully describe the behavior of quantum scale objects.
In general, an electron in a metal has a de-Broglie wavelength in order of ~10nm. So, we observe quantum-mechanical effects in the properties of a metal when the width of the sample is around this value. The S.I unit of this wavelength is meter (m).
De-Broglie won the Nobel Prize for physics in 1929, after the wave- like behavior of matter was experimentally demonstrated in 1927.
As we know, time is defined by its measurement. Time is what a clock reads. In classical and non- relativistic physics, it is a scalar quantity. We know that length, mass and charge is usually described as a fundamental quantity, similarly time is also a fundamental quantity. The S.I unit of time is second.

Note:
Should remember the S.I unit of time is second, which is represented s. One minute is equal to 60 seconds and 1 hour is equal to 3600 sec. If a particle is larger than its de-Broglie wavelength, or if it is interacting with other objects on a scale significantly larger than its de Broglie wavelength, then its wave- like properties are not acceptable.