Samples of two different isotopes, X and Y. Both contain the same number of radioactive atoms. Sample X has a half life twice than that of Y. How do their activities be compared?

Options

A. X has a smaller activity than Y
B. X has a greater activity than Y
C. Activities of X and Y are equal
D. Need to know number of neutrons in nucleus

Answer

To compare the activities of two radioactive isotopes, X and Y, given that sample X has a half-life that is twice that of sample Y, we can follow these steps:

### Step 1: Understand the relationship between half-life and decay constant
The half-life (T_half) of a radioactive substance is related to its decay constant (λ) by the formula:
\[ T_{1/2} = \frac{\ln(2)}{\lambda} \]
From this, we can express the decay constant in terms of half-life:
\[ \lambda = \frac{\ln(2)}{T_{1/2}} \]

### Step 2: Define the half-lives of isotopes X and Y
Let the half-life of isotope Y be \( T_{1/2} \). Then, according to the problem, the half-life of isotope X is:
\[ T_{1/2(X)} = 2T_{1/2(Y)} \]

### Step 3: Calculate the decay constants for both isotopes
Using the relationship from Step 1:
- For isotope Y:
\[ \lambda_Y = \frac{\ln(2)}{T_{1/2(Y)}} \]
- For isotope X:
\[ \lambda_X = \frac{\ln(2)}{T_{1/2(X)}} = \frac{\ln(2)}{2T_{1/2(Y)}} = \frac{1}{2} \lambda_Y \]

### Step 4: Relate the activities of the isotopes
The activity (R) of a radioactive sample is given by:
\[ R = \lambda N \]
where \( N \) is the number of radioactive atoms present.

Since both samples X and Y contain the same number of radioactive atoms (let's denote it as \( N_0 \)):
- Activity of isotope X:
\[ R_X = \lambda_X N_0 \]
- Activity of isotope Y:
\[ R_Y = \lambda_Y N_0 \]

### Step 5: Substitute the decay constants into the activity equations
Substituting \( \lambda_X = \frac{1}{2} \lambda_Y \) into the equation for \( R_X \):
\[ R_X = \left(\frac{1}{2} \lambda_Y\right) N_0 = \frac{1}{2} R_Y \]

### Step 6: Compare the activities
From the above equation, we can conclude that:
\[ R_X = \frac{1}{2} R_Y \]
This means that the activity of sample X is half that of sample Y.

### Conclusion
Thus, the activity of sample X is smaller than that of sample Y.

### Final Answer
**The activity of sample X is smaller than that of sample Y.**

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