A radioactive element disintegrates at a rate proportional to the quantity of substance Q present at any time t . What is thhe differential equation of the disintegration ?

Options

A. `(dQ)/(dt) = -Q`
B. `(dQ)/(dt) = -kQ , k lt 0`
C. `(dQ)/(dt) = - kQ , k gt 0`
D. `(dQ)/(dt) = Q `

Answer

To derive the differential equation for the disintegration of a radioactive element, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
The problem states that a radioactive element disintegrates at a rate that is proportional to the quantity of the substance \( Q \) present at any time \( t \).

2. **Setting Up the Rate of Change**:
The rate of change of the quantity \( Q \) with respect to time \( t \) can be expressed as:
\[ \frac{dQ}{dt} \]
Since the element is disintegrating, this rate will be negative. Therefore, we can express this as:
\[ \frac{dQ}{dt} < 0 \]

3. **Expressing Proportionality**:
The problem states that the rate of disintegration is proportional to the quantity \( Q \). This can be mathematically expressed as:
\[ \frac{dQ}{dt} \propto -Q \]
Here, the negative sign indicates that as \( Q \) decreases, the rate of change is negative.

4. **Removing the Proportionality Sign**:
To convert the proportionality into an equation, we introduce a constant of proportionality \( k \):
\[ \frac{dQ}{dt} = -kQ \]
where \( k \) is a positive constant.

5. **Final Form of the Differential Equation**:
The final form of the differential equation representing the disintegration of the radioactive element is:
\[ \frac{dQ}{dt} = -kQ \]