Questions about the Differential Equation (Dacay)

In summary: So, if x(t) is the amount of C-14 present after t years, then x0=x(0) is the amount present initially, and x(t) is the amount present after t years. Does that make sense?In summary, the half-life of a radioactive isotope is the amount of time it takes for a quantity of radioactive material to decay to one-half of its original amount. To determine the decay-rate parameter for C-14 and I-131, we can use the equation dx/dt = -bx where b is a constant. By separating the variables and integrating, we can find the decay constant \lambda for each isotope. It is important to note that x represents the quantity remaining, not the quantity
  • #1
mybingbinghk
5
0
The half-life of a radioactive isotope is the amount of time it takes for a quantity
of radioactive material to decay to one-half of its original amount.
i) The half-life of Carbon 14 (C-14) is 5230 years. Determine the decay-rate
parameter  for C − 14.
ii)The half-life of Iodine 131 (I-131) is 8 days. Determine the decay-rate param-
eter for I − 131.

Homework Equations



dx/dt = -bx where b is a constant


The Attempt at a Solution



i) let x(t) be the quantity of radionactuve meterial at time t in year and dx/dt be the rate of change of the quantity.

dx/dt = -bx where b is a constant

I dun no how to process after doing this assumption. Please help to give some hints on doing these questions. Many thanks!
 
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  • #2
Let's let [itex]t_{1/2}=5230\,\text{yr}[/itex] be the half-life of C-14. What does this mean? Well, if at some initial time [itex]t_0[/itex] the total quantity of C-14 is [itex]x_0=x(t_0)[/itex], then after an additional time [itex]t_{1/2}[/itex] the amount of carbon remaining is [itex]x_0/2[/itex]; algebraically, [itex]x(t_0+t_{1/2})=x_0/2[/itex].

The decay constant [itex]\lambda[/itex], instead of specifying how long it takes for the quantity to halve, tells you the rate of change of the percent change in quantity. That's a mouthful. So, what does that mean? Well, consider the same quantities [itex]x_0[/itex] and [itex]t_0[/itex] as before. How much carbon is left after an infinitesimal time [itex]dt[/itex]? Let's go take a look at our (verbal) definition for the decay constant: "percent change in quantity" means [itex]dx/x(t)[/itex] (remember that [itex]dx=x(t+dt)-x(t)[/itex]), and the "rate of change" of this is [itex][dx/x(t)]/dt[/itex]. Thus, the decay constant is given algebraically by
[tex]\lambda=-\frac{dx}{dt}\frac{1}{x(t)}.[/tex]​

Can you think of a way of incorporating these to find [itex]\lambda[/itex] for C-14?
 
  • #3
t0 =0, x0 = 0
When t = 5230, then I dun no how to find the X=?
then can't find the lambda?
 
  • #4
Starting with dx/dt = -bx, "separate" the variables by separating the derivative into differentials: dx= -bx dt so dx/x= -bdt. Integrate both sides: [itex]\int dx/x= -b\int dt[/itex].
 
  • #5
that means lambda is dx/dt, right?

As u said, I do the calculation and find the follow
ln|x| = -bt +C where C is a constant

then what should I do?

Can u explain more details?
 
  • #6
mybingbinghk said:
t0 =0, x0 = 0
When t = 5230, then I dun no how to find the X=?
then can't find the lambda?

Why do you think that x0=0?
 
  • #7
I think if the time is zero, there is no decay so x is equal to ZERO
 
  • #8
mybingbinghk said:
I think if the time is zero, there is no decay so x is equal to ZERO

Ahah! There's your problem. The value x measures the quantity remaining, not the quantity which has been removed.
 

Related to Questions about the Differential Equation (Dacay)

What is a differential equation?

A differential equation is a mathematical equation that relates the rate of change of a dependent variable to the values of independent variables and the dependent variable itself. It is commonly used to model and describe a wide range of physical, biological, and social phenomena.

What is decay in a differential equation?

In a differential equation, decay refers to the decrease in the value of a dependent variable over time or with respect to another independent variable. This can occur in various systems, such as radioactive decay, population growth, and electrical circuits.

What are the different types of decay in a differential equation?

The most common types of decay in a differential equation are exponential decay, power law decay, and logarithmic decay. Each type has a unique mathematical form and can be used to model different physical processes.

How is decay represented in a differential equation?

Decay is typically represented as a negative rate of change in a differential equation, with the value of the dependent variable decreasing over time or with respect to another independent variable. In some cases, decay may also be represented by a negative exponent or a negative coefficient in the equation.

What are some real-life examples of decay modeled by differential equations?

Differential equations can be used to model a wide range of decay phenomena in the natural and social sciences. Some examples include radioactive decay of isotopes, decay of biological populations, degradation of environmental pollutants, and depreciation of assets in economics.

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