Examine the graph showing the half-life of the radioactive isotope substance x. based on this graph, what might be the best use of this radioactive isotope?

 T_{1/2} = 5,776  10³ years

We see this life time as it is very close to the life time of carbon 14, so it could be used for dating ancient objects

Explanation:

The radioactive decay of described by an equation of the form

        N = N₀  

where N is the number of atoms present, N₀ is the number of initial atoms λ is the activity of the material.

The average life time is defined as the time for which the number of remainng atoms is N = N₀ / 2

          = ln 2 /λ

With these expressions, the best method to determine the average life time is to find the activity of the first equation.

For this we look for a point on the graph as accurate as possible,

  N₀ = 100,  N = 30 and  t = 10,000 years

we substitute in the equation

         30 = 100 e^{-\lambda 10000}

        ln 0.3 = - λ 10000

        λ = - (ln 0.3) / 10000

        λ = 1.2 10-4

now we can find the average life time

          = ln 2 / 1,2 10-4

         T_{1/2} = 5,776  10³ years

We see this life time as it is very close to the life time of carbon 14, so it could be used for dating ancient objects

vapour or gas assuming atmospheriv pressure

the temperature in kelvin of 2.0 dm3 of an ideal gas is doubled and its pressure is increased by a factor of four. what is the final volume of the gas