Question

You have developed a grand unified theory that predicts the following things about the decay of protons: (1) protons never get any older, in the sense that their probability of decay per unit time never changes, and (2) half the protons in any given collection of protons will have decayed in \(1.80 \cdot 10^{29}\) yr. You are given experimental facilities to test your theory: a tank containing \(1.00 \cdot 10^{4}\) metric tons of water and sensors to record proton decays. You will be allowed access to this facility for 2 years. How many proton decays will occur in this period if your theory is correct?

Short answer

Answer: According to the given grand unified theory about proton decay, approximately 5 proton decays will be observed in the tank during the 2-year period.

Step-by-step solution

Calculate the total number of protons in the given amount of water

To find the total number of protons in the given amount of water (10,000 metric tons), we first need to determine the number of moles of water. We know that 1 mole of water (H\(_2\)O) weighs 18 grams (1 mole H = 1 gram, 1 mole O = 16 grams). The given amount of water is \(1.00 \cdot 10^{4}\) metric tons, which is equal to \(1.00\cdot10^{4}\cdot 10^3 kg = 1.00\cdot10^{7} kg\). Converting this to grams, we have \(1.00\cdot10^{10} g\). So the number of moles of water is: $$n_{H_2O} = \frac{1.00\cdot10^{10} \text{ g}}{18 \text{ g/mol}} = 5.56\cdot10^8 \text{ moles}$$ Considering 1 mole of water contains 2 moles of hydrogen atoms (protons), we have: $$n_{H} = 2 \cdot n_{H_2O} = 2 \cdot 5.56\cdot10^8\text{ moles} = 1.11\cdot10^9 \text{ moles}$$ Finally, we need to convert moles to the total number of protons, using Avogadro's constant (\(N_A = 6.022\cdot10^{23}\) atoms/mol): $$N_{p} = n_{H}\cdot N_A = 1.11\cdot10^9 \text{ moles} \cdot 6.022\cdot10^{23} \text{ atoms/mol} = 6.68\cdot10^{32} \text{ protons}$$

Find the decay constant of the proton

The decay constant (\(\lambda\)) represents the probability of decay per unit time. Since half the protons decay in \(1.80 \cdot 10^{29}\) years, this is the half-life (T\(_{1/2}\)) of the proton. We can find the decay constant using the half-life formula: $$\lambda = \frac{ln2}{T_{1/2}}$$ Substituting the given half-life value: $$\lambda = \frac{ln(2)}{1.80\cdot10^{29} \text{ years}} = 3.86 \cdot 10^{-30} \text{ yr}^{-1}$$

Calculate the number of proton decays in the given period

To find the number of proton decays in a given period (2 years), we will use the decay formula: $$N(t) = N_{0}e^{-\lambda t}$$ Where \(N(t)\) is the number of remaining protons, \(N_{0}\) is the initial number of protons, \(\lambda\) is the decay constant, and \(t\) is the time (in years). Rearranging the equation to find the number of decayed protons: $$N_{decayed}(t) = N_{0} - N(t) = N_{0}(1-e^{-\lambda t})$$ Substitute the values: $$N_{decayed}(2\text{ years}) = 6.68\cdot10^{32}\text{ protons} (1 - e^{-3.86\cdot 10^{-30} \text{ yr}^{-1} \cdot 2 \text{ years}})$$ After calculating, we get: $$N_{decayed}(2\text{ years}) \approx 5.15 \text{ proton decays}$$ Therefore, if the given grand unified theory about proton decay is correct, approximately 5 proton decays will be observed in the tank during the 2-year period.

Key concepts

Understanding Half-life

Half-life is a crucial concept in nuclear physics and chemistry that describes the time required for half of the unstable nuclei or particles in a sample to undergo radioactive decay. More simply, it's the period in which the quantity of a substance inside a decaying sample is reduced by 50%.

In the context of the exercise provided, where protons are the subject of decay, it is stated that half of the protons will have decayed after a span of approximately \(1.80 \cdot 10^{29}\) years, which is an extraordinarily long half-life. In practical terms, this suggests that proton decay is an extremely rare event.

The calculation of how many protons decay over a certain time period, say 2 years as per the exercise, inherently depends on knowing this half-life. This is because the half-life is directly related to the decay constant, a value that gives us the rate at which the protons decay over time. By understanding the half-life, we're equipped to predict the behavior of the sample over time and can calculate expected decay events.

Decay Constant and Its Role in Radioactive Decay

The decay constant, represented by the symbol \(\lambda\), is a probability factor that signifies the likelihood of a single particle or nucleus to decay per time unit. It is a vital parameter in the exponential decay law, characterizing the intrinsic stability or instability of particles.

The decay constant can be derived from the half-life. As the exercise solutions illustrate, you use the formula \(\lambda = \frac{\ln(2)}{T_{1/2}}\), where \(T_{1/2}\) is the half-life. For protons with a half-life of \(1.80 \cdot 10^{29}\) years, the decay constant must be exceptionally small, reflecting protons' long-standing stability against decay.

This decay constant enables us to model the behavior of radioactive decay. With it, we can not only estimate the number of protons that will decay over any given period but also understand the mechanism driving that decay.

Avogadro's Constant: The Bridge Between the Macroscopic and Atomic Worlds

Avogadro's Constant, denoted by \(N_A\), is a fundamental figure in chemistry and physics, anchoring the macroscopic and atomic scales together. It is defined as the number of units (e.g., atoms, molecules, ions) in one mole of substance, and its approximate value is \(6.022 \cdot 10^{23}\) moles. This constant enables us to translate a macroscopic mass, like a kilogram of water, into a mole quantity and down further to the number of constituent particles.

In our exercise, Avogadro's Constant is pivotal for converting moles of water into an actual number of protons. This is critical for calculating the expected number of decay events: we first need to determine the number of protons in our water sample before applying decay-related calculations. The provided solution expertly demonstrates this by determining the total number of protons, starting with the mass of water and ultimately using Avogadro's Constant to reach the desired microscopic level of understanding.