The primary source of helium in the U.S. is an underground natural gas field near Amarillo, Texas, which contain a relatively large percentage of helium gas. The^4 _2 He atoms come from alpha decays by underground naturally occurring deposits of radionuclides, such as uranium-238 (^238 _92U). a) Suppose we have a sample that begins with 1.00 kg of pure^238 _92 U. which has neutral atomic mass 238.05 u. What is the initial number N_U, 0 of^238 _92 U atoms in the sample? b) Each^238 _92 U decay produces one alpha particle, which eventually collects two electrons to become a neutral^4 _ 4He atom. Show that after a time t, the number of^4 _ 2 He atoms produced is equal N_U, 0(1 - e^-t ln 2/) where t_1/2 is the half-life to^238 _ 92 U. c) The hall-life of^238 _ 92 U is 4.47 billion years. How many^4 _ 2 He atoms would be produced by these^238 _ 92 U decays in 10.000 years? (Keep all numbers unrounded in your calculator during ibis calculation. Round the result to 3 significant figures.) d) The neutral atomic mass of^4 _ 2 He is 4.0026 u. How many milligrams of helium would this be?