Age Equation Using the Pb²⁰⁷/Pb²⁰⁶ Ratio

The age of a sample can also be determined by using the ratio of Pb²⁰⁷ to Pb²⁰⁶. Starting with the following decay equations for uranium isotopes:

Pb²⁰⁷ from U²³⁵ decay:

\[ ^{207}\text{Pb} = ^{235}\text{U} (e^{\lambda_{235} t} - 1) \]

Pb²⁰⁶ from U²³⁸ decay:

\[ ^{206}\text{Pb} = ^{238}\text{U} (e^{\lambda_{238} t} - 1) \]

By taking the ratio of these two equations, and knowing that the natural abundance ratio of \(^{235}\text{U}/^{238}\text{U}\) is approximately 1/137.8, we get:

\[ \frac{^{207}\text{Pb}}{^{206}\text{Pb}} = \frac{^{235}\text{U}}{^{238}\text{U}} \cdot \frac{(e^{\lambda_{235} t} - 1)}{(e^{\lambda_{238} t} - 1)} \]

Since \(\frac{^{235}\text{U}}{^{238}\text{U}} \approx \frac{1}{137.8}\), we can simplify the equation to:

\[ \frac{^{207}\text{Pb}}{^{206}\text{Pb}} = \frac{1}{137.8} \cdot \frac{(e^{\lambda_{235} t} - 1)}{(e^{\lambda_{238} t} - 1)} \]

Pb²⁰⁷/Pb²⁰⁶ Isotope Ratio and Its Use in Age Dating

• The Pb²⁰⁷/Pb²⁰⁶ isotope ratio provides a direct measurement of geological time. By observing the radiogenic Pb²⁰⁷/Pb²⁰⁶ ratio in a sample, the age of the sample can be determined.

• The relationship between the Pb²⁰⁷/Pb²⁰⁶ ratio and time is implicit, meaning it is based on prior numerical values that have been calculated from the decay constants of U²³⁸ and U²³⁵. Table in the next slide shows the numerical values of the radiogenic Pb²⁰⁷/Pb²⁰⁶ isotope ratio at different geological times (in billions of years, Ga).

Numerical Values of the Radiogenic Pb²⁰⁷/Pb²⁰⁶ Isotope Ratio Over Time

• Source: (Allègre, 2012)

Time (Ga)	\[(e^{\lambda_{238} t} - 1)\]	\[(e^{\lambda_{235} t} - 1)\]	Pb²⁰⁷/Pb²⁰⁶ Radiogenic
0.0	0.0000	0.0000	0.00000
0.2	0.0315	0.2177	0.05012
0.4	0.0640	0.4828	0.05471
0.6	0.0975	0.8056	0.05992
0.8	0.1321	1.1987	0.06581
1.0	0.1678	1.6774	0.07250
1.2	0.2046	2.2603	0.08012
1.4	0.2426	2.9701	0.08879
1.6	0.2817	3.8344	0.09872
1.8	0.3221	4.8869	0.11000
2.0	0.3638	6.1685	0.12298
2.2	0.4067	7.7292	0.13783
2.4	0.4511	9.6296	0.15482
2.6	0.4968	11.9437	0.17436
2.8	0.5440	14.7617	0.19680
3.0	0.5926	18.1931	0.22266
3.2	0.6428	22.3716	0.25241
3.4	0.6946	27.4597	0.28672
3.6	0.7480	33.6556	0.32634
3.8	0.8030	41.2004	0.37212
4.0	0.8599	50.3878	0.42498
4.2	0.9185	61.5752	0.48623
4.4	0.9789	75.1984	0.55714
4.6	1.0413	91.7873	0.63930

Key Points:

• Direct Time Measurement: The Pb²⁰⁷/Pb²⁰⁶ isotope ratio provides an age estimate directly related to time, without requiring knowledge of uranium concentrations.
• Implicit Function: Calculating the age requires the use of pre-determined values based on decay rates.
• The table lists radiogenic Pb²⁰⁷/Pb²⁰⁶ isotope ratios for various times, ranging from 0 Ga to 4.6 Ga. The increase in the Pb²⁰⁷/Pb²⁰⁶ ratio with time reflects the accumulation of radiogenic lead due to the decay of uranium isotopes.

For example:
- At 0.2 Ga, the radiogenic Pb²⁰⁷/Pb²⁰⁶ ratio is 0.05012.
- At 2.0 Ga, the ratio is 0.12298.
- At 4.6 Ga (the approximate age of the Earth), the ratio is 0.63930.

Example

The Pb²⁰⁶/Pb²⁰⁷ ratio of a uranium ore deposit is found to be 13.50. What is the age of the ore, assuming it has remained closed since crystallization and common lead can be ignored?

Solution

1. Given:

   • Measured ratio: \(\frac{^{206}\text{Pb}}{^{207}\text{Pb}} = 13.50\)

2. Convert to Pb²⁰⁷/Pb²⁰⁶:

   • The inverse of the given ratio will give \(\frac{^{207}\text{Pb}}{^{206}\text{Pb}}\):

   \[ \frac{^{207}\text{Pb}}{^{206}\text{Pb}} = \frac{1}{13.50} = 0.07407 \]

3. Compare with the Table:

   • From the table, find the value closest to \(\frac{^{207}\text{Pb}}{^{206}\text{Pb}} = 0.07407\).
   • At 1.0 Ga, \(\frac{^{207}\text{Pb}}{^{206}\text{Pb}} \approx 0.07250\).
   • At 1.2 Ga, \(\frac{^{207}\text{Pb}}{^{206}\text{Pb}} \approx 0.08012\).

4. Interpolate:

   • To estimate the age between 1.0 Ga and 1.2 Ga, interpolate the values:

   \[ \text{Age} = 1.0 + \left( \frac{0.07407 - 0.07250}{0.08012 - 0.07250} \right) \times 0.2 \]

   \[ \text{Age} = 1.0 + \left( \frac{0.00157}{0.00762} \right) \times 0.2 \approx 1.041 \text{ Ga} \]

• The age of the uranium ore deposit is approximately 1.04 Ga (billion years).