Answer

Verified

340.2k+ views

**Hint:**Here in this question we have to find the exponential decay rate. That is, the rate at which a population of something decays is directly proportional to the negative of the current population at time t. So we can introduce a proportionality constant. Then further applying an integration on both sides and on simplification we get the required result.

**Complete step by step answer:**

Exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time. It can be expressed by the formula \[y = a{\left( {1 - b} \right)^x}\] where y is the final amount, a is the original amount, b is the decay factor, and x is the amount of time that has passed. Exponential decays typically start with a differential equation of the form:

\[ \Rightarrow \,\,\dfrac{{dN}}{{dt}}\alpha \, - N(t)\]

That is, the rate at which a population of something decays is directly proportional to the negative of the current population at time t. So we can introduce a proportionality constant \[\alpha \]:

\[ \Rightarrow \,\,\dfrac{{dN}}{{dt}} = - \alpha \,N(t)\]

We will now solve the equation to find a function of \[N(t)\]:

\[ \Rightarrow \,\,\dfrac{{dN}}{{N(t)}} = - \alpha \,dt\]

\[ \Rightarrow \,\,\dfrac{{dN}}{N} = - \alpha \,dt\]

Apply integration both sides, then

\[ \Rightarrow \,\,\int {\dfrac{{dN}}{N}} = \int { - \alpha } \,dt\]

\[ \Rightarrow \,\,\int {\dfrac{{dN}}{N}} = - \alpha \int {dt} \,\]

Using the integration formula \[\int {\dfrac{1}{x}dx = \ln x + c} \] and \[\int {dx = x + c} \], where c is an integrating constant.

\[ \Rightarrow \,\,\ln N = - \alpha t + c\,\]

As we know the logarithm function is the inverse form of exponential function, then

\[ \Rightarrow \,\,N = {e^{ - \alpha \,\,t + c}}\,\]

Or it can be written as:

\[ \therefore \,\,N = A{e^{ - \alpha \,t}}\]

Where A is a constant.

**Hence, the general form of exponential decay rate is \[N = A{e^{ - \alpha \,t}}\].**

**Note:**Exponential decay describes the process of reducing an amount by a constant percentage rate over a period of time. The integration is inverse of the differentiation so to cancel differentiation the integration is applied. Likewise the logarithm is inverse of exponential. Hence by using these concepts we obtain answers.

Recently Updated Pages

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

What happens when entropy reaches maximum class 11 chemistry JEE_Main

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Describe the poetic devices used in the poem Aunt Jennifers class 12 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Change the following sentences into negative and interrogative class 10 english CBSE

State the laws of reflection of light

State and prove Bernoullis theorem class 11 physics CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE