### Video Transcript

Which of the following equations is correct for atomic transmutation via beta decay? (A) Where 𝑑 equals 𝑏 minus one. (B) Where 𝑎 minus 𝑐 equals 𝑑. (C) Where 𝑏 equals 𝑑. (D) Where 𝑐 equals 𝑎 minus one. Or (E) where 𝑎 equals 𝑐.

The nuclear equation given involves atomic transmutation, which is the transformation of atoms of one element into atoms of another element. The question deals specifically with beta decay, which is a type of radioactive decay where a neutron in an unstable nucleus is transformed into a proton and an electron is emitted, also called a beta particle.

From the nuclear equation given, 𝑋 represents the chemical symbol of the unstable nucleus, where the letter 𝑎 represents the mass number of particle 𝑋 and the letter 𝑏 represents the number of protons. 𝑋 undergoes decay to form an atom of a new element, represented by 𝑌, whose mass number is represented by the letter 𝑐 and number of protons represented by the letter 𝑑. Parent particle 𝑋 forms daughter particle 𝑌, and a beta particle is emitted, which we could alternatively represent as an electron with a mass number of zero and a charge of negative one.

To determine which of the answers is correct, we can use the information that the reactants and products in a nuclear equation must be equal in total mass number, represented here by the letter 𝐴, and must be equal in total charge, represented here by the letter 𝑍. This means that from the given equation, the total mass number of the reactants must be equal to the sum of the mass numbers of the products, which is 𝑐 plus zero. 𝑐 plus zero equals 𝑐. And therefore, for the total mass number of the reactants to equal the total mass number of the products, 𝑎 must equal 𝑐.

The same must be true for the total charge. The total charge of the reactants, or 𝑏, must be equal to the sum of the charges of the products, which is 𝑑 plus negative one. If we rewrite this equation, we get that 𝑏 equals 𝑑 minus one. Looking at the answer choices, we can see that answer choice (E) matches one of our derived equations. Therefore, answer choice (E), where 𝑎 equals 𝑐, is correct.