At time `t=0 N_(1)` nuclei of decay constant `lambda_(1)& N_(2)` nuclei of decay constant `lambda_(2)` are mixed. The decay rate of the mixture at time 't' is:
At time `t=0 N_(1)` nuclei of decay constant `lambda_(1)& N_(2)` nuclei of decay constant `lambda_(2)` are mixed. The decay rate of the mixture at time 't' is:
A
`N_(1)N_(2)e^((lambda_(1)+lambda_(2))t)`
B
`+((N_(1))/(N_(2)))e^(-(lambda_(1)-lambda_(2))t)`
C
`+(N_(2)lambda_(1)e^(lambda_(1)t) + N_(2)lambda_(2)e^(-lambda_(2)t))`
D
`+N_(1)lambda_(1)N_(2)lambda_(2)e^(-(lambda_(1)+lambda_(2))t)`
Text Solution
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The correct Answer is:
C
`(dN)/(dt)= lambda_(1)N_(1)+lambda_(2)N_(2) implies lambda_(1)N_(10)e^(-lambda_(1)t)+lambda_(2)N_(20)e^(-lambda_(2)t)`
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The decay constant (lambda) for radioactive decay is independent of
A radionuclide A_(1) with decay constant lambda_(1) transforms into a radionuclide A_(2) with decay constant lambda_(2) . Assuming that at the initial moment the preparation contained only the radionuclide A_(1) , find: (a) the equation describing accumulation of the radionuclide A_(2) with time, (b) the time interval after which the activity value.
Knowledge Check
The decay constant lambda of a radioactive sample:
The decay constant lambda of a radioactive sample:
A
Decreases with increase of external pressure and temperature
B
Increase with increase of external pressure and temperature
C
Decreases with increase of temperature and increase of pressure
D
Is independent of temperature and pressure.
Uranium ._(92)^(238)U is an ustable nucleus. It decays to Thorium ._(92)^(238)Th , which is again an unstable nucleus which further decays to ._(91)^(234)Pa . Let ._(92)^(238)U be called A of decay constant lambda_(1) and ._(90)^(234)Th is called as B of decay constant lambda_(2) and stable nuclei ._(91)^(234)Pa be called as C . Here A is called parent nucleus and B is called daughter nucleus of A . Any two adjacent nuclei may be consider parent or daughter nuclei A, B and C respectively at time 't' . Then we can write Aoverset(lambda_(1))rarrBoverset(lambda_(2))rarrC Rate of disintergration of A=(dN_(1))/(dt)=lambda_(1)N_(1) Rate of disintergration of B=(dN_(2))/(dt)=lambda_(1)N_(1)-lambda_(2)N_(2) Rate of formation of nuclei C is equal to (dN_(3))/(dt)=lambda_(2)N_(2) If at t=0 , there are N_(0) number of nuclei of A where as nuclei B and C are absent in the sample Answer the following questions ltbgt Numeber of the nuclei of nuclei C at time t is
Uranium ._(92)^(238)U is an ustable nucleus. It decays to Thorium ._(92)^(238)Th , which is again an unstable nucleus which further decays to ._(91)^(234)Pa . Let ._(92)^(238)U be called A of decay constant lambda_(1) and ._(90)^(234)Th is called as B of decay constant lambda_(2) and stable nuclei ._(91)^(234)Pa be called as C . Here A is called parent nucleus and B is called daughter nucleus of A . Any two adjacent nuclei may be consider parent or daughter nuclei A, B and C respectively at time 't' . Then we can write Aoverset(lambda_(1))rarrBoverset(lambda_(2))rarrC Rate of disintergration of A=(dN_(1))/(dt)=lambda_(1)N_(1) Rate of disintergration of B=(dN_(2))/(dt)=lambda_(1)N_(1)-lambda_(2)N_(2) Rate of formation of nuclei C is equal to (dN_(3))/(dt)=lambda_(2)N_(2) If at t=0 , there are N_(0) number of nuclei of A where as nuclei B and C are absent in the sample Answer the following questions ltbgt Numeber of the nuclei of nuclei C at time t is
A
`N_(0)[1+(lambda_(1))/(lambda_(2)-lambda_(1))e^(-lambda_(2)t)-(lambda_(2))/(lambda_(2)-lambda_(1))e^(-lambda_(1)t)]`
B
`N_(0)[(lambda_(1))/(lambda_(2)-lambda_(1))e^(-lambda_(2)t)-(lambda_(2))/(lambda_(2)-lambda_(1))e^(-lambda_(1)t)]`
C
`N_(0)(lambda_(1))/(lambda_(2)-lambda_(1))(e^(-lambda_(2)t)-e^(-lambda_(1)t))`
D
`N_(0)[(lambda_(1))/(lambda_(2)-lambda_(1))e^(-lambda_(2)t)-(lambda_(2))/(lambda_(2)-lambda_(1))e^(-lambda_(1)t)-1]`
Uranium ._(92)^(238)U is an ustable nucleus. It decays to Thorium ._(92)^(238)Th , which is again an unstable nucleus which further decays to ._(91)^(234)Pa . Let ._(92)^(238)U be called A of decay constant lambda_(1) and ._(90)^(234)Th is called as B of decay constant lambda_(2) and stable nuclei ._(91)^(234)Pa be called as C . Here A is called parent nucleus and B is called daughter nucleus of A . Any two adjacent nuclei may be consider parent or daughter nuclei A, B and C respectively at time 't' . Then we can write Aoverset(lambda_(1))rarrBoverset(lambda_(2))rarrC Rate of disintergration of A=(dN_(1))/(dt)=lambda_(1)N_(1) Rate of disintergration of B=(dN_(2))/(dt)=lambda_(1)N_(1)-lambda_(2)N_(2) Rate of formation of nuclei C is equal to (dN_(3))/(dt)=lambda_(2)N_(2) If at t=0 , there are N_(0) number of nuclei of A where as nuclei B and C are absent in the sample Answer the following questions Number of nuclei of B at any time t is
Uranium ._(92)^(238)U is an ustable nucleus. It decays to Thorium ._(92)^(238)Th , which is again an unstable nucleus which further decays to ._(91)^(234)Pa . Let ._(92)^(238)U be called A of decay constant lambda_(1) and ._(90)^(234)Th is called as B of decay constant lambda_(2) and stable nuclei ._(91)^(234)Pa be called as C . Here A is called parent nucleus and B is called daughter nucleus of A . Any two adjacent nuclei may be consider parent or daughter nuclei A, B and C respectively at time 't' . Then we can write Aoverset(lambda_(1))rarrBoverset(lambda_(2))rarrC Rate of disintergration of A=(dN_(1))/(dt)=lambda_(1)N_(1) Rate of disintergration of B=(dN_(2))/(dt)=lambda_(1)N_(1)-lambda_(2)N_(2) Rate of formation of nuclei C is equal to (dN_(3))/(dt)=lambda_(2)N_(2) If at t=0 , there are N_(0) number of nuclei of A where as nuclei B and C are absent in the sample Answer the following questions Number of nuclei of B at any time t is
A
`(N_(0)lambda_(1))/(lambda_(2)-lambda_(1))(e^(lambda_(1)t)-e^(lambda_(2)t))`
B
`(N_(0)lambda_(1))/(lambda_(2)-lambda_(1))(e^(-lambda_(1)t)-e^(-lambda_(2)t))`
C
`(N_(0)lambda_(1))/(lambda_(2)-lambda_(1))(e^(-lambda_(1)t)-e^(-lambda_(2)t))`
D
`(N_(0)lambda_(2))/(lambda_(2)-lambda_(1))(e^(-lambda_(1)t)+e^(-lambda_(2)t))`
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Uranium ._(92)^(238)U is an ustable nucleus. It decays to Thorium ._(92)^(238)Th , which is again an unstable nucleus which further decays to ._(91)^(234)Pa . Let ._(92)^(238)U be called A of decay constant lambda_(1) and ._(90)^(234)Th is called as B of decay constant lambda_(2) and stable nuclei ._(91)^(234)Pa be called as C . Here A is called parent nucleus and B is called daughter nucleus of A . Any two adjacent nuclei may be consider parent or daughter nuclei A, B and C respectively at time 't' . Then we can write Aoverset(lambda_(1))rarrBoverset(lambda_(2))rarrC Rate of disintergration of A=(dN_(1))/(dt)=lambda_(1)N_(1) Rate of disintergration of B=(dN_(2))/(dt)=lambda_(1)N_(1)-lambda_(2)N_(2) Rate of formation of nuclei C is equal to (dN_(3))/(dt)=lambda_(2)N_(2) If at t=0 , there are N_(0) number of nuclei of A where as nuclei B and C are absent in the sample Answer the following questions The graph N_(1), N_(2) and N_(3) with time can be best represent by
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