cesium-137 half-life 30 years. 속도상수로 계산

The half-life of cesium-137 is 30 years.

Suppose we have a 100 mg sample.

a) Find the mass that remains after t years.

b) How much of the sample remains after 100 years?

c) After how long will only 1 mg remain?

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a) Find the mass that remains after t years.

[반감기의 정의로부터, 속도 상수(k) 계산]

> P_0 = 100 mg

> t = 30 years

---> 30 years 후 질량 = 50 mg = P_t

equation for decay

P_t = P_0 e^(kt)

( 참고 https://ywpop.tistory.com/26 )

( 참고: e^(kt) 대신 e^(–kt)를 사용해도 된다. )

50 = 100 × e^(30k)

e^(30k) = 50 / 100 = 0.5

30k × ln(e) = ln(0.5)

( 참고: ln(e) = 1 )

k = ln(0.5) / 30

= –0.023105

P_t = 100 × e^(–0.023105 × t)

답: 100 × e^(–0.023105 × t) mg

b) How much of the sample remains after 100 years?

P_t = 100 × e^(–0.023105 × t)

P_100 = 100 × e^(–0.023105 × 100)

= 9.92

답: 9.92 mg

c) After how long will only 1 mg remain?

P_t = 100 × e^(–0.023105 × t)

1 = 100 × e^(–0.023105 × t)

e^(–0.023105 × t) = 1 / 100 = 0.01

(–0.023105 × t) ln(e) = ln(0.01)

t = –ln(0.01) / 0.023105

= 199.3

답: 199.3 years

[키워드] 반감기 기준, 반감기 사전, cesium-137 기준, cesium-137 사전