The energy released by αdecay in a 50.0-g sample of 94 ^239 Pu is to be used to heat 4.75 kg of

The energy released by αdecay in a 50.0-g sample of 94 ^239...

Key Concepts

Heat Transfer and Calorimetry

Heat transfer in calorimetry involves the conservation of energy principle, where the energy released from a reaction is assumed to be completely absorbed by another substance, causing a temperature change. This application of the conservation of energy in thermal systems is fundamental in experiments and calculations involving temperature changes induced by energy inputs.

Energy Conversion in Nuclear Reactions

In nuclear reactions, a small amount of mass is converted into a large amount of energy according to Einstein’s mass-energy equivalence principle. Calculating the energy released involves understanding the energy per decay event and converting it into usable energy units, which is a crucial step in analyzing processes like alpha decay.

Specific Heat Capacity

Specific heat capacity is a property of a material that indicates how much energy is needed to raise the temperature of a given mass of the substance by one degree Celsius. It is a key concept in calorimetry, allowing one to relate the energy input to the change in temperature of the substance being heated.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This process is a fundamental feature of nuclear physics and is characterized by a specific decay mode, a decay constant or half-life, and the release of energy associated with the transformation of the nucleus.

Alpha Decay

Alpha decay is a type of radioactive decay in which an unstable nucleus emits an alpha particle, which consists of two protons and two neutrons. This decay mode results in a decrease in both the atomic mass and atomic number of the original nucleus and typically releases a significant amount of energy.

Transcript

00:01 So this problem wants us to determine the increase in temperature of the water in 1r, given that all of the energy released by alpha decay goes into hitting the water, given that we have a plutonium sample.

00:23 So let's denote the changing temperature as delta t.

00:29 And if this is the sample, so we have 239, potonium 239, it goes in alpha decay, becomes uranium 235 and we have here the alpha particle which is same atomic mass in atomic number of helium atom so we can derive an expression for the changing temperature by the formula of heat which is q equals to product of the mass of the water the specific heat of the water and the changing temperature is the heat required to raise the temperature of the water and since you've stated that all of the energy release was converted to heat, then we can say that the net energy by the decay is equals to the...

01:22 And note that the total energy due to decay is the product of the total number of decays and the energy per one decay.

01:38 And the total number of decays is the difference of the initial number of nuclei and the remaining.

01:44 Nuclear after some time due to after some time as it decays.

01:51 And the energy per decay is the product of the change in mass from atom multiplied by the speed of light squared.

02:07 And the remaining nuclei is equals to this.

02:21 So we can further simplify our equation to this.

02:26 Note that lambda here is the decay constant of the plutonium in this example.

02:31 So we have ln of 2 over the half life.

02:35 And here is the elapsed time, which is also given.

02:47 And the change in mass of the atom here is this.

02:55 The sum of the mass of uranium atom and the helium atom minus the mass of the plutonium atom.

03:16 And the initial number of nuclei is the quotient of the given mass and the mass of the atom.

03:43 Now, let us solve this.

03:52 So the given mass of the plutonium, which is 50 grams.

03:55 So let's convert it to kilograms by this factor.

04:00 And the mass of protonium atom is 239 .052 -158u...

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