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Why do atoms get larger and heavier going down the periodic table?

“What you need to be able to do is to describe the trend and explain the trend as you go down the periodic table. Anyone?” Chemistry Teacher Garry asks his class.
“There’s an increase going down the table because the elements get larger and heavier,” one student suggests.

“You’re right, they are getting larger and heavier, but why?” Garry asks again.
“Because there are more protons,” a classmate answers.

“Yes, that’s why they’re getting heavier, but why are they getting bigger?” Garry insists.
“More energy levels?”
“Yes, more energy levels! Better than that: More occupied energy levels, because the energy levels are always there, whether they are occupied or not, but as you go down the grid, the number of occupied energy levels increases, and when the number of occupied energy levels increases, the atomic size is also going to increase.”

“When you go across the grid, it’s a lot trickier to explain. You can see the trend as you go across the periodic table: What happens to the atoms is that they get smaller, there are a few exceptions, but in general, they get smaller. Why?”

The room is silent. One student repeats his question, drawing out the last vowel: “Yes, but why-y-y?”
“Why do you think?”
“I really don’t know,” she answers, “But it's good to learn these things,” she adds, tongue in cheek.

“As you go across the table, the number of protons increases, but the electron shells remain the same, so what happens is that the protons pull the electrons towards them, so there is a greater attraction between the nucleus and the electrons, and the atoms get smaller.”