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Question: How do you find the half -life of the uranium – 235?...

How do you find the half -life of the uranium – 235?

Explanation

Solution

Half-life of any radioactive substance can be defined as the time needed by the radioactive substance or we can say time needed by one half of the atoms to disintegrate or convert into a different substance.

Formula used:
N(t)=Noe−λtN\left( t \right)={{N}_{o}}{{e}^{-\lambda t}}

Complete step by step solution:
As we know to find the half-life of radioactive substance (in the case U235{{U}^{_{235}}} ) you need to begin with the initial quantity of the substance, N (o), then after a time interval, t, you have to measure the remaining quantity of the substance N (t)

Consider Formula to find the radioactive decay.
N(t)=Noe−λt....(1)N\left( t \right)={{N}_{o}}{{e}^{-\lambda t}}....\left( 1 \right)
Where, N(t)N\left( t \right)=is the quantity that left after decay.
N(o)N\left( o \right) = the initial quantity of the substance.
λ\lambda = decay constant.
t = time.

Now let’s discover half – life formula, now from definition when half substance decays and the time taken for that is t12{{t}_{\dfrac{1}{2}}} then,
t = ⇒t=t12\Rightarrow t={{t}_{\dfrac{1}{2}}}
⇒N(t12)=N(o)2\Rightarrow N\left( {{t}_{\dfrac{1}{2}}} \right)=\dfrac{N\left( o \right)}{2}

Substitute both the values in the equation (1)
⇒N(o)2=N(o)eλt12 ∴12=eλt12 \begin{aligned} & \Rightarrow \dfrac{N\left( o \right)}{2}=N\left( o \right){{e}^{\lambda {{t}_{\dfrac{1}{2}}}}} \\\ & \therefore \dfrac{1}{2}={{e}^{\lambda {{t}_{\dfrac{1}{2}}}}} \\\ \end{aligned}

Taking log to remove exponential,
⇒ln=(12)=λt12 ⇒−ln(2)=λt12 ∴t12=−ln(2)λ \begin{aligned} & \Rightarrow {{l}_{n}}=\left( \dfrac{1}{2} \right)=\lambda {{t}_{\dfrac{1}{2}}} \\\ & \Rightarrow -{{l}_{n}}\left( 2 \right)=\lambda {{t}_{\dfrac{1}{2}}} \\\ & \therefore {{t}_{\dfrac{1}{2}}}=\dfrac{-{{l}_{n}}\left( 2 \right)}{\lambda } \\\ \end{aligned}

From the above formula we can say that to find half- life we need λ\lambda or half-life to find which can be found by quantification that is decay in the time interval.

From the experiment half – life of U235{{U}^{_{235}}}is 7.04×1087.04\times {{10}^{8}} years.

Note:
To find half – life of U235{{U}^{_{235}}} we can’t divide the value of 235 by 12\dfrac{1}{2} because 235 is atomic mass number to find half – life we need time interval in which U235{{U}^{_{235}}} decaying it will give us value of the remaining U235{{U}^{_{235}}} at the time interval t with the value we can find but the life of U235{{U}^{_{235}}} using the equation (1).