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Radioactivity and the Nucleus: Question 2

Syllabus 5.1.2, 5.2.4

Structured Core 6 marks

A water company suspects that an underground pipe is leaking somewhere along its length. To find the exact location without digging up the whole pipeline, engineers add a small amount of a radioactive tracer to the water flowing through the pipe. A detector is then moved slowly along the ground above the pipe; the count rate it records rises sharply above any point where leaked water (and therefore the tracer) has reached the soil close to the surface.

The tracer used is a nuclide represented as 57140X{}^{140}_{57}\text{X}.

(a) State the number of protons and the number of neutrons in a nucleus of this nuclide. [2]

(b) A different sample contains the nuclide 57142X{}^{142}_{57}\text{X}. Explain what is meant by saying that 57140X{}^{140}_{57}\text{X} and 57142X{}^{142}_{57}\text{X} are isotopes of the same element. [2]

(c) Before adding the tracer to the water, the engineers test a small fixed sample of it in the lab and record how its count rate changes with time.

Time / hours 0 1 2 3
Count rate / counts per minute 800 400 200 100

Use the table to determine the half-life of the tracer. [2]

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Worked solution

Part (a): Protons and neutrons from nuclide notation

In the nuclide notation ZAX{}^{A}_{Z}\text{X}, the bottom number ZZ is the proton number and the top number AA is the nucleon number (total protons + neutrons).

For 57140X{}^{140}_{57}\text{X}: Z=57Z = 57, so there are 57\boxed{57} protons.

Number of neutrons =A−Z= A - Z: 140−57=83140 - 57 = 83

So the nucleus contains 57 protons and 83 neutrons.

Part (b): Explaining isotopes

Comparing the two nuclides: Z(both)=57A1=140,A2=142Z(\text{both}) = 57 \qquad A_1 = 140, \quad A_2 = 142

Both nuclides have the same proton number (5757), so they are the same element. Since the atomic (proton) number is the same, they behave the same chemically.

However, they have different nucleon numbers, so the number of neutrons differs: 140−57=83 neutrons142−57=85 neutrons140 - 57 = 83 \text{ neutrons} \qquad 142 - 57 = 85 \text{ neutrons}

Nuclides of the same element that have different numbers of neutrons (and therefore different nucleon numbers) are called isotopes.

Part (c): Finding the half-life

Half-life is the time taken for the count rate (or activity) of a sample to fall to half its previous value.

Reading down the table: 800→400(1 hour:halved)800 \to 400 \quad (1\text{ hour}: \text{halved}) 400→200(1 hour:halved)400 \to 200 \quad (1\text{ hour}: \text{halved}) 200→100(1 hour:halved)200 \to 100 \quad (1\text{ hour}: \text{halved})

The count rate halves every 11 hour, consistently across the whole table, so: half-life=1 hour\text{half-life} = \boxed{1}\text{ hour}

Final answers

  • (a) 57 protons, 83 neutrons
  • (b) Same proton number (57) ⇒\Rightarrow same element; different nucleon numbers (140, 142) ⇒\Rightarrow different neutron numbers (83, 85) ⇒\Rightarrow isotopes
  • (c) Half-life == 1 hour