How Does Alpha Decay Impact Water Temperature?

[shigg927]

Homework Statement

Thorium (with half-life T1/2 = 1.913 yr. and atomic mass 228.028715 u) undergoes alpha decay and produces radium (atomic mass 224.020186 u) as a daughter nucleus. (Assume the alpha particle has atomic mass 4.002603 u.)

The energy released from the decay of 0.6 g of thorium is used to heat 3.5 kg of water (assume all the energy released goes into the water).

(c) What is the change in temperature of the water after 2.6 hrs.?

Homework Equations

N=No(1/2)^n

Th --> Ra + He
[tex]\Delta[/tex]m= M(Th)-M(Ra)-M(He)
E=mc^2

The Attempt at a Solution

[tex]\Delta[/tex]m= .005926 u
So, I can find the energy, but in the "help," it says to find the number of atoms. How do I find the number of atoms?

 

[oedipa maas]

You know the atomic mass of Thorium - and you also know 1 atomic mass unit = 1.66053886 × 10-27 kilograms. You also know you have 0.6 g of Thorium -

N_o = total mass/ mass per particle.

 

[Moetasim]

I would first like to clarify that radioactive decay is a natural process in which an unstable atom emits particles or energy in order to become more stable. This process is not something to apologize for, as it is a fundamental aspect of the universe that has been studied and understood by scientists for decades.

Moving on to the problem at hand, we are given information about the decay of thorium and asked to find the change in temperature of water after a certain amount of time. To do this, we need to use the equations provided and some basic principles of thermodynamics.

Firstly, we can use the equation \Deltam= M(Th)-M(Ra)-M(He) to find the mass difference between thorium and its daughter nucleus, radium, and the alpha particle that is emitted. This mass difference represents the mass that is converted into energy during the decay process.

Next, we can use the equation E=mc^2 to calculate the total energy released from the decay of 0.6 g of thorium. This energy will be in the form of heat, which can be used to heat up the water.

To find the change in temperature of the water, we need to use the principle of conservation of energy. We know that all the energy released from the decay of thorium will go into heating the water, so we can set the energy released equal to the energy gained by the water.

Finally, we can use the specific heat capacity of water (4.186 J/g·K) to calculate the change in temperature of the water after 2.6 hours.

In order to find the number of atoms, we can use the equation N=No(1/2)^n, where N is the number of atoms at a given time, No is the initial number of atoms, and n is the number of half-lives that have passed. However, in this problem, we do not need to explicitly calculate the number of atoms.