Question

A counter rate meter is used to measure the activity of a radioactive sample . At a certain instant, the count rate was recorded as 475 counters per minute . Five minutes later , the count rate recorded was 270 count per minute. calculate the decay constant and half life period of the sample .

Answer 1

There is a relation between decay constant and half-life period of sample .A radioactive sample is those made up of atoms whose nuclei are unstable and give off atomic radiation as part of a process of attaining stability.

Complete step by step answer:
Radioactivity refers to the particles which are emitted from nuclei as a result of nuclear instability. Because the nucleus experiences the intense conflict between the two strongest forces in nature , it should not be surprising that there are many nuclear isotopes which are unstable and emit some kind of radiation.
The most common types of radiation are alpha , gamma radiation .Radioactivity decay rates are normally stated in terms of their half lives . The different types of radioactivity leads to different decay paths.
Decay constant: it is the probability of decay per unit time. The number of parent nuclides P therefore decreases with time t as $\dfrac{{dP}}{P}dt = - \lambda$ . the decay constant related to the half life of the nuclide ${T_{1/2}}$ through ${t_{1/2}} = \ln 2/\lambda$ .
Half life , in radioactivity , the interval of time required for one half of the atomic nuclei of a radioactive sample to decay .
The half life of a first order reaction is a constant that is related to the rate constant for the reaction:
${t_{1/2}} = 0.693/\lambda$
Now comes to the solution part,
Initially, rate = $475$ counters per minute =$\lambda {N_0}$
After 5 minutes , rate = $270$ counts per minute= $\lambda N$
Then, $\lambda {N_0}/\lambda N$ $\Rightarrow {N_0}/N$ = $475/270$
$\Rightarrow 1.76$
Now, $\lambda = 1/t\left( {\log {N_0}/N} \right)$
$\Rightarrow 1/5\left( {\log 1.76} \right) = 0.113$ ${\min ^{ - 1}}$
Now, ${t_{1/2}} = 0.693/\lambda$
$\Rightarrow {t_{1/2}} = 0.693/.113$
$\Rightarrow 6.1mi{n^{ - 1}}$

Hence, the required answer is $6.1{\min ^{ - 1}}$ .

Note:
There are some factors on which rate of reaction depends, rate may be defined as the change in concentration of reactant or product per unit time .
Factors are , 1. Temperature
On increasing the temperature normally the rate of reaction increases but in exothermic temperature should decrease.
2. concentration
In addition to the reaction , reaction goes in a forward direction hence the rate of reaction increases .
3. catalyst.
In addition, the catalyst rate of reaction increases by decreasing the activation energy .But not all catalysts are used for increasing the rate. There are some negative catalysts which are used to slow down the rate.