Consider an exponential decay function P(t)=P0 e^-λt, and let T denote its time constant. Show t

step 1 Problem number 54, consider exponential decay function. P of T is going to P0E to land of the T.

Consider an exponential decay function P(t)=P0 e^-λt, and...

Consider an exponential decay function $P(t)=P_{0} e^{-\lambda t}$, and let $T$ denote its time constant. Show that, at $t=T$, the function $P(t)$ decays to about one-third of its initial size. Conclude that the time constant is always larger than the half-life.

Key Concepts

Relationship Between Time Constant and Half-Life

Since the time constant (T = 1/?) corresponds to the time taken for the quantity to decay to about 37% of its initial value, and the half-life (t?/? = (ln 2)/?) corresponds to a decay to 50% of the initial value, the time constant is always larger than the half-life. This is mathematically evident because ln 2 (approximately 0.693) is less than 1, meaning t?/? < T.

Exponential Decay Function

An exponential decay function describes a process in which a quantity decreases continuously at a rate proportional to its current value. It is typically expressed as P(t) = P?e^(–?t), where P? is the initial value and ? is the decay constant that governs how fast the decay occurs.

Decay Constant

The decay constant, denoted by ?, is a parameter that quantifies the exponential rate at which a quantity decays. A larger value of ? indicates a faster decay. It directly appears in the exponent of the decay function, determining the time scale of the process.

Time Constant

The time constant T is defined as the reciprocal of the decay constant (T = 1/?). It represents the time required for the initial quantity to decay to approximately 1/e (about 37%) of its original value. The time constant is a useful measure for understanding the speed of the decay process.

Half-Life

The half-life is the period of time required for the quantity to decrease to half of its initial value. For an exponential decay, the half-life is calculated as t?/? = (ln 2)/?. This concept is widely used to compare the rates of decay in various processes, such as radioactive decay or chemical reactions.

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