The Earth loses heat at a rate of 44 TW, about 16 TW of which is radiogenic, the remainder repre

The Earth loses heat at a rate of 44 TW, about 16 TW of...

The Earth loses heat at a rate of $44 \mathrm{TW}$, about $16 \mathrm{TW}$ of which is radiogenic, the remainder representing the loss of primordial heat. Assuming the ${ }^3 \mathrm{He} /{ }^4 \mathrm{He}$ ratio of the atmosphere represents the ratio of which these isotopes are lost from the solid Earth and that all ${ }^4 \mathrm{He}$ is radiogenic and all ${ }^3 \mathrm{He}$ is primordial, how do the rates of loss of radiogenic $\mathrm{He}$ and heat compare?

Key Concepts

Isotopic Ratios as Proxies for Geophysical Processes

The use of isotopic ratios, such as the 3He/4He ratio, serves as a proxy for understanding the dynamics of heat loss and volatile transport within the Earth. By assuming that the atmospheric ratio reflects the loss rates of primordial versus radiogenic components, researchers can gain insights into how energy and mass are partitioned during the thermal evolution of Earth's interior.

Primordial Heat

Primordial heat is the residual thermal energy left over from the formation of the Earth. Unlike radiogenic heat, which is continuously produced via radioactive decay, primordial heat is gradually lost through the Earth’s surface, playing a key role in the planet’s cooling history.

Radiogenic Heat Production

Radiogenic heat production refers to the heat generated inside the Earth by the decay of radioactive isotopes. This process is fundamental for understanding the internal energy budget of the Earth because it continuously produces heat over time, influencing mantle convection and the thermal evolution of the planet.

Helium Isotopic Geochemistry

Helium isotopic geochemistry involves the study of the two main helium isotopes, 3He and 4He, to trace the origin and evolution of volatiles in the Earth. The different origins of these isotopes—3He being largely primordial and 4He being radiogenic—allow scientists to use their relative abundances to infer processes such as mantle degassing and the timing of heat loss.

Transcript

00:01 Now in this question we are looking at how radioactivity is able to heat up the earth and how much it contributes to the internal heating of the earth's core.

00:12 And we are given that uranium 238 is one of the major radioactive source we have in the mantle.

00:20 The first part of this question, we are given that the amount of uranium 238 is about 31 microgram.

00:31 In 1 kilogram of 1 kilogram of mantle right and we want to find out how many isotopes are in this so very simple the number of isotopes or number of uranium 2 3 8 in this 1 kilogram will be taking the mass which are convert 31 times 10 to power minus 8 sorry minus 9 kilograms divided by the mass of each uranium 238 nuclei, which is 238 times 1 .66, just the power minus 27, sorry, kilograms.

01:28 Now you want to put this into your calculator, get about 7 .8, 465 times 10 power of 16, right? yep.

01:53 So this is the number of uranium 238 nuclei in 1 kilogram of mantle.

02:01 Next part of this question, we are given the half -life of uranium -238 to be 4 .47 billion years times 10 power of 9 years.

02:17 And we want to calculate the decay constant.

02:21 Now the decay constant lambda is given as long 2 over the half -life.

02:26 But we want it in terms of per second.

02:29 So what it means that we have to convert our t half into seconds first.

02:34 So to convert into seconds, find the number of seconds in a year, just about 3 .154 times in about 7 seconds per year.

02:52 So, i'll find these two together, should get about 1 .41, about 17 seconds.

03:11 Seconds and using this value we then take a long 2 divided by this so a lambda should be about right 4 .916 right times 10 power of minus 18 all right this is in terms of per second now the question then wants us to use these two values that we've just calculated right the number of you're going to create the decay constant to get the activity in one kilograms of mantle.

04:23 Right, so activity is just lambda times the number of new thigh.

04:31 So we have these two values, 4 .916, and then power of minus 18, multiplied by 7 .845...

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