An excited state of an atom has a 25 ns lifetime. What is the probability that an excited atom will emit a photon during a 0.50 ns interval?

검증된 단계별 안내

Understand the concept: The problem involves exponential decay, which describes the probability of an excited atom emitting a photon over time. The lifetime (τ) of the excited state is given as 25 ns, and we are tasked with finding the probability of emission during a 0.50 ns interval.

Write the formula for the probability of decay in a small time interval Δt: The probability is given by \( P = \frac{\Delta t}{\tau} \), where \( \tau \) is the lifetime of the excited state and \( \Delta t \) is the time interval of interest.

Substitute the given values into the formula: \( \Delta t = 0.50 \; \text{ns} \) and \( \tau = 25 \; \text{ns} \). The formula becomes \( P = \frac{0.50}{25} \).

Simplify the fraction to determine the probability: Perform the division \( P = \frac{0.50}{25} \) to find the probability of emission during the 0.50 ns interval.

Interpret the result: The calculated probability represents the likelihood that an excited atom will emit a photon within the specified 0.50 ns time interval. Ensure the result is expressed as a decimal or percentage for clarity.

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주요 개념

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Excited State

An excited state of an atom refers to a condition where one or more electrons have absorbed energy and moved to a higher energy level than their ground state. This state is unstable and will eventually return to the ground state, often resulting in the emission of a photon. The lifetime of the excited state indicates how long the atom typically remains in this higher energy configuration before transitioning back.

Lifetime of an Excited State

The lifetime of an excited state is the average time an atom remains in that state before emitting a photon and returning to a lower energy level. It is a crucial parameter in quantum mechanics and is often denoted by the symbol τ (tau). A shorter lifetime indicates a higher probability of photon emission in a given time interval, while a longer lifetime suggests a lower probability.

Probability of Emission

The probability of emission during a specific time interval can be calculated using the ratio of the time interval to the lifetime of the excited state. This is based on the assumption that the emission process follows an exponential decay model, where the likelihood of emission increases with time spent in the excited state. For a lifetime τ and a time interval Δt, the probability P of emitting a photon is given by P = Δt/τ.