Question

Over time, \(3.00 \mathrm{~g}\) of \({ }_{55}^{132} \mathrm{Cs}\) decays by \(\beta^{-}\) emission to produce stable xenon. If the half-life of \({ }_{55}^{132} \mathrm{Cs}\) is 55 days, how much \({ }_{55}^{132} \mathrm{Cs}\) will be left after 150 days?

Short answer

After 150 days, the remaining mass of Cesium-132 is given by the equation: \(m = 2 ^{-\frac{150}{55}} * 3.00 \) g.

Step-by-step solution

Calculate the number of half-lives in 150 days.

Divide the elapsed time (150 days) by the half-life (55 days). \(n = \frac{150}{55} \)

Use the exponent function to determine the number of half-lives more accurately.

The above calculation might not give a whole number. Therefore, we will use the following formula to determine the number of half-lives more accurately: \(n = 2 ^{-\frac{t}{T}} \) Where: n = remaining Cesium-132 amount fraction (Nb/N0) t = elapsed time, 150 days T = half-life, 55 days Plug in the values and solve for n: \(n = 2 ^{-\frac{150}{55}} \)

Calculate the remaining mass of Cesium-132.

Multiply the initial mass of Cesium-132 (3.00 g) by the remaining fraction n: \(m = n * m_{0}\) Where: m_0 = initial mass of Cesium-132, 3.00 g m = remaining mass of Cesium-132 Substitute the values: \(m = 2 ^{-\frac{150}{55}} * 3.00 \) g Now, calculate the final mass of Cesium-132 remaining after 150 days.

Key concepts

Radioactive Decay

Radioactive decay is a spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. This process can change one chemical element into another and is characterized by the type of radiation emitted, such as alpha particles, beta particles, or gamma rays.

The decay occurs because the configurations of particles within an atomic nucleus are not always stable, and, over time, the nucleus seeks a more stable arrangement. This instability is quantified by half-lives, which measure how long it takes for half of any given amount of a radioactive isotope to decay. Understanding radioactive decay is essential in fields ranging from medicine, where radioactive tracers are used for imaging, to archaeology, for dating artifacts.

Beta Emission

Beta emission is one of the common types of radioactive decay, where a beta particle, which can be either an electron or a positron, is emitted from the nucleus. In \( \beta^{-} \) emission, a neutron in the nucleus is transformed into a proton, an electron, and an electron antineutrino. The electron is ejected from the nucleus, which leads to an increase in the atomic number of the element while keeping the mass number the same.

This process is integral to nuclear equations in scientific calculations and is one of the paths for radioactive isotopes to reach stability. Cesium-132’s decay to stable xenon is an example of \( \beta^{-} \) emission, playing a critical role in understanding how elements evolve over time.

Exponential Decay

Exponential decay models the decrease of a quantity by a consistent percentage over time. Unlike linear decay, which removes a fixed amount at regular intervals, exponential decay accelerates the rate of decay as time advances. It's best described by the formula: \( N(t) = N_0 e^{-kt} \) where \( N_0 \) is the initial quantity, \( N(t) \) is the quantity at time \( t \) and \( k \) is the decay constant.

This model is ubiquitous in natural processes, including the rate of radioactive decay in elements. In the context of solving problems related to half-life calculation, understanding the concept of exponential decay allows us to predict the remaining amount of a substance after a given period.

Cesium-132 Decay

Cesium-132, denoted as \( {}_{55}^{132}\text{Cs} \), is a radioactive isotope that undergoes \( \beta^{-} \) emission to transform into stable xenon. The half-life parameter is a key aspect when discussing Cesium-132 since it dictates the time required for half of the substance to decay. With a half-life of 55 days, Cesium-132 provides an excellent example of exponential decay in action.

By applying the half-life formula, we can calculate how much Cesium-132 remains after any given period, as exemplified by the exercise which determines the remaining mass after 150 days. The calculation of such decay helps in various practical applications, including medical treatments and the management of nuclear materials.