First, here's a link to a good periodic table: http://www.ptable.com. That will come in handy.
Now, here's a step-by-step solution to the logic puzzle on sporcle.com.
1. Tantalum (Ta) automatically appears in A1.
2. From clue A1, we know that the group 11 metals, traditionally known as the "coinage metals," appear in the other three corners: copper (Cu), silver (Ag), and gold (Au). Of those three, the two that are in the upper-right and lower-right corners must have chemical symbols beginning with ‘A’ because they’re in column E. So copper, with symbol Cu, must be in the lower-left corner, A5. (Although the newly-discovered element roentgenium is also a group 11 element, it falls outside the range of atomic numbers used in this quiz, as stated in the instructions.)
3. Clue A5 tells us that the non-radioactive noble gases fill row 2. There are five non-radioactive noble gases: helium (He), neon (Ne), argon (Ar), krypton (Kr), and xenon (Xe). The element in E2 must be a noble gas whose chemical symbol starts with ‘A’ (because of clue A1), so E2 must be argon.
4. Clue E2 tells us that reversing the letters in the symbol for the element E4 gives the symbol for the element in B5. The element in E4 has a symbol beginning with ‘A’ (because of clue A1), so the five possible pairs for E4 and B5 are astatine (At) and tantalum (Ta), argon (Ar) and radium (Ra), silver (Ag) and gallium (Ga), actinium (Ac) and calcium (Ca), and aluminum (Al) and lanthanum (La). The first two pairs are ruled out because tantalum and argon already appear in A1 and E2, respectively. The third pair is ruled out because silver must appear in either E1 or E5 (because of the coinage metals requirement in clue A1). The fourth pair is ruled out because calcium can't appear in B5; calcium (with atomic number 20) can't appear to the right of copper (with atomic number 29) in row 5 because of clue A5. Only one pair remains, so aluminum must be in E4, and lanthanum must be in B5.
5. Row 5, from left to right, is ordered by increasing atomic number, as per clue A5. Now that lanthanum (with atomic number 57) appears in B5, silver (with atomic number 47) can’t appear in E5. Therefore silver must be in E1, and gold must be in E5.
6. From clue B5, all elements in row 3 have names beginning with ‘M’, while from clue E1, all elements in column C have symbols beginning with ‘H’. The only element fitting both requirements, mercury (Hg), must therefore be in C3.
7. From clue A5, all elements in row 2 are noble gases, while from clue E1, all elements in column C have symbols beginning with ‘H’. The only element fitting both requirements, helium (He), must therefore be in C2.
8. From clue E5, the element in E3 is an alkaline earth metal, while from clue B5, the element in E3 must have a name starting with ‘M’. The only alkaline earth metal whose name begins with ‘M’ is magnesium, so that must go in E3.
9. From clue C3, the elements in B1, C1, and C5 are named for European capitals. The only three elements with that property are hafnium (Hf, named for Copenhagen), holmium (Ho, named for Stockholm), and lutetium (Lu, named for Paris). C1 and C5 must have symbols starting with ‘H’ as per clue E1, so lutetium must go in B1.
10. From clue E4, the element that’s the primary component of air, namely nitrogen (N), is in row 1. The only two vacant spaces left in row 1 are C1 and D1, but the element in C1 must have a symbol starting with ‘H’ as per clue E1. Therefore nitrogen must go in D1.
11. From clue C2, the quiz contains three elements with one-letter symbols, and they all appear in the same column. Nitrogen (N) already appears in D1, so the other two must also be in column D. From clue E3, the most electronegative element (fluorine, F) and the element also known as wolfram (tungsten, W) both appear in the quiz, so they must be in column D because they have one-letter symbols. They can’t be in row 1 (occupied by nitrogen), row 2 (reserved for a noble gas), or row 3 (reserved for an element whose name starts with ‘M’), so they must be in D4 and D5, in some order. From clue A5, the element in D5 must have a greater atomic number than the element in B5, namely lanthanum with atomic number 57. That rules out fluorine because it has atomic number 9. So fluorine must go in D4, and tungsten (with atomic number 74) must go in D5.
12. Clue D1 says the two elements that are liquids at 20 degrees Celsius at standard atmospheric pressure, namely mercury and bromine, are adjacent to each other. Mercury is already in C3. The vacant cells adjacent to mercury are B2 and D2, which are reserved for noble gases (because of clue A5); B3 and D3, which are reserved for elements whose names begin with ‘M’ (because of clue B5); C4, which is reserved for an element whose symbol begins with ‘H’ (because of clue E1); and B4. Bromine (Br) doesn’t fit any of the reserved categories, so it must go in B4.
13. Clue D5 says that the element in C1 is named for Niels Bohr’s hometown. Bohr was born, raised, and educated in Copenhagen, and he lived his entire life there except for a brief exile during World War II. Hafnium is named for Copenhagen, so hafnium must go in C1. Using the list from step 9, the only remaining element named for a European capital is holmium, so holmium must go in C5 as per clue C3.
14. Clue E1 says that all elements in column C have symbols starting with ‘H’. Hafnium, helium, mercury, and holmium have already been placed, but C4 is still vacant. The only two remaining elements whose symbols begin with ‘H’ are hydrogen (H) and hassium (Hs). Hydrogen is ruled out because clue C2 says that the quiz uses three elements with one-letter symbols, and all three have already been used: nitrogen, fluorine, and tungsten. That leaves hassium to go in C4.
15. Clue C1 says that hafnium, in C1, is not adjacent to the noble gas whose name is from the Greek for ‘hidden’, namely krypton. According to clue A5, krypton must be in row 2, where the only vacant spaces are A2, B2, and D2. B2 and D2 are both adjacent to hafnium in C1, so krypton must be in A2.
16. According to clue A5, neon must be in row 2. According to clue C4, the quiz has only two elements with four-letter names; gold is already included at E5, so neon must be the other. Moreover, clue C4 means that gold and neon cannot be in the same diagonal. Of the two vacant spaces in row 2, B2 is in the same diagonal as gold (located in E5), so that space is ruled out. Neon must therefore be in D2.
17. In accordance with clue A5, the only remaining non-radioactive noble gas, xenon, must therefore be in B2, the only remaining space in row 2.
18. The element in A4 is radioactive (from clue B2) and an alkaline earth metal (from clue E5). Because the quiz is limited to elements with atomic numbers 1 to 110, the only element with both properties is radium, so radium must go in A4.
19. Because of clue B5, the three remaining spaces in row 3 (A3, B3, and D3) must be filled with elements whose names begin with ‘M’. Magnesium and mercury have already been used, so the remaining four elements that could occupy those three spaces are manganese, meitnerium, mendelevium, and molybdenum. From clue C5, mendelevium must be the unused element because it is named for a man, Dmitri Mendeleyev. From clue B2, meitnerium is not in a cell adjacent to bromine, which is located in B4; so meitnerium cannot be in A3 or B3. Meitnerium must therefore be in D3.
20. From clue A2, the element in B3 has a greater atomic number than the element in A3. As discussed in step 19, those two spaces must be occupied by the two remaining elements beginning with ‘M’, namely manganese and molybdenum. The atomic numbers of manganese and molybdenum are 25 and 42, respectively, so manganese must be in A3, and molybdenum must be in B3. Alternatively, clue D2 says that the element in B3 derives its name from the ancient Greek word for ‘lead,’ so B3 must be molybdenum, leaving A3 as manganese.
The clues in A3, B3, D3, and D4 are superfluous.
I hope you enjoyed the puzzle.