The average life of a radioactive element is ............. of the decay constant.

Answer

To find the relationship between the average life of a radioactive element and its decay constant, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Terms**:
- The average life (mean life) of a radioactive element is denoted by \( \tau \) (tau).
- The decay constant is denoted by \( \lambda \) (lambda).

2. **Definition of Decay Constant**:
- The decay constant \( \lambda \) represents the probability per unit time that a nucleus will decay. It is a measure of the rate of decay of the radioactive element.

3. **Relationship Between Average Life and Decay Constant**:
- The average life \( \tau \) is defined as the average time a nucleus exists before it decays. The relationship between average life and decay constant is given by the formula:
\[ \tau = \frac{1}{\lambda} \]
- This means that the average life of a radioactive element is the reciprocal (or inverse) of the decay constant.

4. **Understanding Half-Life**:
- The half-life \( t_{1/2} \) of a radioactive element is the time required for half of the radioactive nuclei to decay. The decay constant \( \lambda \) is also related to the half-life by the formula:
\[ \lambda = \frac{0.693}{t_{1/2}} \]
- This shows that the decay constant can be derived from the half-life.

5. **Final Statement**:
- Therefore, we conclude that the average life of a radioactive element is the reciprocal of the decay constant:
\[ \text{Average life} = \frac{1}{\lambda} \]

### Final Answer:
The average life of a radioactive element is the reciprocal (or inverse) of the decay constant.

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