Samples of radioactive isotopes are decaying.

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Two different samples of radioactive isotopes are decaying. The isotopes have initial amounts of c1 and c2 and half-lives of k1 and k2, respectively. Find an expression for the time t required for the samples to decay to equal amounts.

 

Discussion 5.2
In a classroom designed for 30 students, the air conditioning system can move 450 cubic feet of air per minute.
(a) Determine the ventilation rate per child in a full classroom.
(b) Estimate the air space required per child.

Sample Solution

Radioactive Decay:

To find the time (t) when the two samples of radioactive isotopes decay to equal amounts, we need to consider the equations for their individual decays and set them equal to each other at the time t.

Sample 1:

  • Initial amount: c1
  • Half-life: k1
  • Amount remaining at time t: c1 * (1/2)^(t / k1)

Sample 2:

  • Initial amount: c2
  • Half-life: k2
  • Amount remaining at time t: c2 * (1/2)^(t / k2)

Setting the amounts remaining equal at time t:

c1 * (1/2)^(t / k1) = c2 * (1/2)^(t / k2)

Solving for t:

Taking the logarithm of both sides with base 2:

t / k1 = log2(c2 / c1) + t / k2

Combining like terms:

t (1/k1 – 1/k2) = log2(c2 / c1)

Therefore, the time t required for the samples to decay to equal amounts is:

t = (log2(c2 / c1) * k1 * k2) / (k2 – k1)

Air Conditioning in Classroom:

Part (a): Ventilation Rate per Child:

  • Total air moved per minute: 450 cubic feet
  • Number of students: 30

Ventilation rate per child:

= Total air moved / Number of students = 450 cubic feet/minute / 30 students = 15 cubic feet/minute/student

Part (b): Estimated Air Space per Child:

  • Ventilation rate per child: 15 cubic feet/minute/student

Assuming a standard breathing rate of 1 cubic foot per minute:

  • Fresh air needed per minute: 1 cubic foot/minute/student
  • Fresh air provided by ventilation: 15 cubic feet/minute/student

Therefore, the estimated air space required per child should be:

  • Total air space / (Ventilation air + Breathing air)
  • Air space / 16 cubic feet/minute/student

However, this calculation only considers the minimum fresh air requirement and doesn’t account for factors like building size, occupancy fluctuations, and carbon dioxide removal. Building codes and professional standards specify more comprehensive air space requirements based on these factors.

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