### Shortcut Methods

**JEE Exam - Numerical Problems.**

**Determine the number of neutrons in an atom of carbon-14 given that its atomic number is 6 and mass number is 14.**

**Solution**:- The number of neutrons = mass number - atomic number
- = 14 - 6
- = 8 neutrons

**Calculate the binding energy per nucleon of oxygen-16, given its mass defect is 0.138 atomic mass units (amu).**

**Solution**:- Binding energy per nucleon = (Mass defect/Mass number)
- = (0.138 amu / 16 amu)
- = 0.008625 amu/nucleon

**A sample of a radioactive substance has a half-life of 10 days. If the initial activity of the sample is 100 decays per minute, what will be its activity after 20 days?**

**Solution**:- After 1 half-life (10 days), the activity will be 100/2 = 50 decays per minute.
- After another 10 days (20 days total), the activity will be 50/2 = 25 decays per minute.

**CBSE Board Exam - Numerical Problems.**

**An atom has 17 protons and 18 neutrons. What is its atomic number and mass number?**

**Solution**:- Atomic number = number of protons = 17
- Mass number = number of protons + number of neutrons = 17 + 18 = 35

**Calculate the relative atomic mass of an element if it has two isotopes with masses 20 amu and 22 amu, and the respective abundances are 90% and 10%.**

**Solution**:- Relative atomic mass = (mass of isotope 1 x abundance of isotope 1) + (mass of isotope 2 x abundance of isotope 2)
- = (20 amu x 0.90) + (22 amu x 0.10)
- = 20.2 amu

**A radioactive element has a half-life of 5 hours. If 1000 atoms are present initially, how many atoms will remain after 10 hours?**

**Solution:**- After 1 half-life (5 hours), 1000/2 = 500 atoms will remain.
- After another 5 hours (10 hours total), 500/2 = 250 atoms will remain.