A radioactive substance,with initial mass m_0 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let $m$ be the mass of the substance at time $t$. The rate of decay is given by $\frac{dm}{dt} = -km$,where $k > 0$ is the decay constant.Integrat

Vedclass · Accepted Answer

$-\frac{m_0}{h} \log 2$

Vedclass · Answer

(C) Let $m$ be the mass of the substance at time $t$. The rate of decay is given by $\frac{dm}{dt} = -km$,where $k > 0$ is the decay constant.Integrating the equation $\frac{dm}{m} = -k dt$,we get $\log m = -kt + c$.At $t = 0$,$m = m_0$,so $c = \log m_0$.Thus,$\log m = -kt + \log m_0$,which implies $\log(\frac{m}{m_0}) = -kt$.Given the half-life $h$,at $t = h$,$m = \frac{m_0}{2}$.Substituting these values: $\log(\frac{1}{2}) = -kh$,which gives $-\log 2 = -kh$,or $k = \frac{\log 2}{h}$.The initial decay rate is the value of $\frac{dm}{dt}$ at $t = 0$.$\frac{dm}{dt} = -km_0 = -(\frac{\log 2}{h})m_0 = -\frac{m_0}{h} \log 2$.

$A$ radioactive substance,with initial mass $m_0$,has a half-life of $h$ days. Then its initial decay rate is given by

Similar Questions

$A$ spherical balloon expands at a rate proportional to its surface area. Initially, its radius is $2 \text{ cm}$ and $5 \text{ minutes}$ later it increases to $7 \text{ cm}$. What will be the surface area of the spherical balloon after $12 \text{ minutes}$ (in $\text{ cm}^2$)?

The equation of the curve passing through $\left(2, \frac{9}{2}\right)$ and having the slope $\left(1-\frac{1}{x^2}\right)$ at $(x, y)$ is

The orthogonal trajectories of the family of curves $x^{2/3} + y^{2/3} = a^{2/3}$,where $a$ is an arbitrary constant,are given by:

The principal increases continuously in a newly opened bank at the rate of $10 \%$ per year. An amount of Rs. $2000$ is deposited with this bank. How much will it become after $5$ years? $(e^{0.5} = 1.648)$

$A$ radioactive substance has a half-life of $h \ days$. Its initial decay rate is given by (Note that at $t = 0, M = m_0$):

Vedclass Test Series

Exam Paper Generator

Online Exam Module