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Chapter 19: Problem 26

A freshly prepared radioactive source half-life \(2 \mathrm{hr}\) emits radiation of intensity which is 64 times the permissible safe level. The minimum time after which it would be possible to work safely with this source is (A) \(6 \mathrm{hr}\) (B) \(12 \mathrm{hr}\) (C) \(42 \mathrm{hr}\) (D) \(128 \mathrm{hr}\)

Short Answer

Expert verified
The minimum time after which it would be possible to work safely with this radioactive source is \(12 \mathrm{hr}\) (Option B).

Step by step solution

01

Understand the radioactive decay formula

The radioactive decay formula is given by: \(N(t) = N_0 \times 0.5^{(\frac{t}{T})}\) Where: - N(t): The amount of radioactive substance left after time 't'. - N_0: The initial amount of radioactive substance. - t: Time (in hours, in this exercise). - T: Half-life (2 hours in this problem). The safe level of radiation is when the intensity reduces to 1 (since the source is initially 64 times the safe level).
02

Set up the equation to find the minimum time

We need to find the minimum time 't' when the intensity reaches the safe level. Given that the initial intensity is 64 times the safe level, we can set up the radioactive decay equation as: \(N(t) = 64 \times 0.5^{(\frac{t}{2})}\) Our goal is to find the time 't' when N(t) = 1 (safe level).
03

Solve the equation for 't'

We can now solve the equation for the time 't': \(1 = 64 \times 0.5^{(\frac{t}{2})}\) To solve for 't', let's first take the logarithm of both sides: \(\ln(1) = \ln(64 \times 0.5^{(\frac{t}{2})})\) Since \(\ln(1) = 0\), we have: \(0 = \ln(64) + \ln(0.5^{(\frac{t}{2})})\) Now, use the logarithmic property for exponential expressions: \(0 = \ln(64) + \frac{t}{2} \ln(0.5)\) From here, we can solve for 't': \(-\ln(64) = \frac{t}{2} \ln(0.5)\) \(t = \frac{-2 \ln(64)}{\ln(0.5)}\) \(t \approx 12\)
04

Choose the correct answer

Using the radioactive decay formula, we found out that the minimum time after which it will be safe to work with this source is 12 hours. Thus, the correct answer is: (B) \(12 \mathrm{hr}\)

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Most popular questions from this chapter

Carbon, silicon and germanium have four valence electrons each. At room temperature which one of the following statements is most appropriate? (A) The number of free electrons for conduction is significant only in Si and Ge but small in \(C\) (B) The number of free conduction electrons is significant in \(C\) but small in Si and Ge (C) The number of free conduction electrons is negligibly small in all the three (D) The number of free electrons for conduction is significant in all the three.

A radioactive nucleus (initial mass number \(A\) and atomic number \(Z\) ) emits \(3 \alpha\)-particles and 2 positrons. The ratio of number of neutrons to that of protons in the final nucleus will be (A) \(\frac{A-Z-8}{Z-4}\) (B) \(\frac{A-Z-4}{Z-8}\) (C) \(\frac{A-Z-12}{Z-4}\) (D) \(\frac{A-Z-4}{Z-2}\)

In gamma ray emission from a nucleus (A) only the proton number changes. (B) both the neutron number and the proton number change. (C) there is no change in the proton number and the neutron number. (D) only the neutron number changes.

If \(M_{O}\) is the mass of an oxygen isotope \({ }_{8} \mathrm{O}^{17}, M_{P}\) and are the masses of a proton and a neutron respectively, the nuclear binding energy of the isotope is (A) \(\left(M_{O}-17 M_{N}\right) c^{2}\) (B) \(\left(M_{O}-8 M_{P}\right) c^{2}\) (C) \(\left(M_{O}-8 M_{P}-9 M_{N}\right) c^{2}\) (D) \(M_{O} c^{2}\)

A radioactive sample at any instant has its disintegration rate 5000 disintegration per minute. After 5 minute, the rate is 1250 disintegrations per minute. Then, the decay constant (per minute) is (A) \(0.4 \ln 2\) (B) \(0.2 \ln 2\) (C) \(0.1 \ln 2\) (D) \(0.8 \ln 2\)
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