Question

How would you estimate the number of gas molecules in one cubic meter of air in the classroom on an average day?

Answer 1

Here we are using the molar volume concept of an ideal gas at a condition which is in standard laboratory condition at an approximation. We know about the molar mass concept of molecules in which the case in general is about mass as the molecule will be in solid state. An entirely different case from this but does pay similarity with the molar mass concept is the molar volume concept which we will be using here since it is about gas molecules. But to deal with this problem, we should also need a particular volume which is not mentioned in the question. The only clue regarding this which is left behind is about the air in one cubic meter in the classroom on an average day. Here , the important point to keep in mind is that one cubic meter is mathematically the same to thousand litres.

Complete step-by-step answer:
First of all we have to know about the molar volume of an ideal gas here. So, for that we have to keep in mind always that at standard laboratory conditions, $1mole$ of an ideal gas occupies $24.5L$ . It means the exact volume we need here is given by $24.5Lmo{l^{ - 1}}$.

Next we need the particular volume of air here to proceed with the calculation. But it is not given directly but it is given that the gas molecules are filled in one cube meter in the classroom. So, we should know here that $1{m^3} = 1000L$ . Actually we will divide this quantity by molar volume mentioned above.

Next our duty is to find the number of moles of gas molecules. For finding this, we will just divide the volume occupied in the class room which we mentioned above by the molar volume which we found earlier. So, the calculation will be as follows,
Number of moles $= \dfrac{{1000L}}{{24.5Lmo{l^{ - 1}}}}$ $= 41moles$

Actually the complete answer is not found yet as our aim is to find the number of gas molecules. So, we need to consider another important point along with this which is that each mole of a particular gas contains ${N_A}$ gas molecules or Avogadro number of gas molecules whose behavior is assumed to be ideal gas behavior. The same concept will be applied here for finding the number of gas molecules.
Therefore, since there are $41mole$ , number of gas molecules $= 41mole \times {N_A}$
$= 41mole \times 6.022 \times {10^{23}}mo{l^{ - 1}}$

Hence, there are $41mole \times {N_A}$ number of gas molecules.

Note: We have to keep in mind that molar volume of a substance is actually the occupied volume of the substance by amount of substance at given pressure and temperature. As we know, we used the volume occupied by an ideal gas in standard laboratory condition so that it is an exact value which will enable us to answer as we don’t need to find it by another calculation. Next thing we have to note is we cannot move further with the volume given which is in one meter cube so that we should convert it to litres at first.

To proceed with this, we should know that $1{m^3} = 1000L$which will cause no error in further calculation and instead if we move with one meter cube itself in the volume side , then the answer will be wrong for sure. We have to remember that the molar volume concept is related to the molar mass concept. Another important point to keep in mind while dealing with these types of questions is that if we need to find the number of molecules, we should find the number of moles at first and later multiply it with Avogadro number then only the answer will be complete . Same method can be approached in any question regarding finding the number of molecules.