### Figuring for Yourself

Estimate the amount of mass that is converted to energy when a proton combines with a deuterium nucleus to form ^{3}He.

How much energy is released when a proton combines with a deuterium nucleus to produce ^{3}He?

The Sun converts 4 × 10^{9} kg of mass to energy every second. How many years would it take the Sun to convert a mass equal to the mass of Earth to energy?

Assume that the mass of the Sun is 75% hydrogen and that all of this mass could be converted to energy according to Einstein’s equation *E* = *mc*^{2}. How much total energy could the Sun generate? If *m* is in kg and *c* is in m/s, then *E* will be expressed in J. (The mass of the Sun is given in Appendix E.)

In fact, the conversion of mass to energy in the Sun is not 100% efficient. As we have seen in the text, the conversion of four hydrogen atoms to one helium atom results in the conversion of about 0.02862 times the mass of a proton to energy. How much energy in joules does one such reaction produce? (See Appendix E for the mass of the hydrogen atom, which, for all practical purposes, is the mass of a proton.)

Now suppose that all of the hydrogen atoms in the Sun were converted into helium. How much total energy would be produced? (To calculate the answer, you will have to estimate how many hydrogen atoms are in the Sun. This will give you good practice with scientific notation, since the numbers involved are very large! See Appendix C for a review of scientific notation.)

Models of the Sun indicate that only about 10% of the total hydrogen in the Sun will participate in nuclear reactions, since it is only the hydrogen in the central regions that is at a high enough temperature. Use the total energy radiated per second by the Sun, 3.8 × 10^{26} watts, alongside the exercises and information given here to estimate the lifetime of the Sun. (Hint: Make sure you keep track of the units: if the luminosity is the energy radiated per second, your answer will also be in seconds. You should convert the answer to something more meaningful, such as years.)

Show that the statement in the text is correct: namely, that roughly 600 million tons of hydrogen must be converted to helium in the Sun each second to explain its energy output. (Hint: Recall Einstein’s most famous formula, and remember that for each kg of hydrogen, 0.0071 kg of mass is converted into energy.) How long will it be before 10% of the hydrogen is converted into helium? Does this answer agree with the lifetime you calculated in Exercise 16.35?

Every second, the Sun converts 4 million tons of matter to energy. How long will it take the Sun to reduce its mass by 1% (the mass of the Sun is 2 × 10^{30} kg)? Compare your answer with the lifetime of the Sun so far.

Raymond Davis Jr.’s neutrino detector contained approximately 10^{30} chlorine atoms. During his experiment, he found that one neutrino reacted with a chlorine atom to produce one argon atom each day.

- How many days would he have to run the experiment for 1% of his tank to be filled with argon atoms?
- Convert your answer from A. into years.
- Compare this answer to the age of the universe, which is approximately 14 billion years (1.4 × 10
^{10}y). - What does this tell you about how frequently neutrinos interact with matter?