If \(15 \mathrm{mg}\) of \(\mathrm{N}_{2} \mathrm{O}_{3}\) is added to \(4.82 \times 10^{20}\) molecules of \(\mathrm{N}_{2} \mathrm{O}_{3}\) the total volume occupied by the gas at STP is (a) \(0.044 \ell\) (b) \(0.022 \ell\) (c) \(0.22 \ell\) (d) \(0.44 \ell\)

Understanding the concept of molar mass is crucial for solving chemistry problems involving the quantity of a substance. Molar mass is defined as the mass (in grams) of one mole of any chemical element or compound.
To figure out the molar mass of a compound like dinitrogen trioxide (\(N_{2}O_{3}\)), one must sum the molar masses of all the atoms present in the formula. Nitrogen (N) has a molar mass of approximately 14 g/mol, while oxygen (O) has a molar mass of about 16 g/mol. Therefore, the molar mass of one mole of \(N_{2}O_{3}\) is calculated as the sum of twice the molar mass of nitrogen plus three times the molar mass of oxygen: \[Molar mass of N_{2}O_{3} = (2 \times 14 g/mol) + (3 \times 16 g/mol) = 76 g/mol.\]
When you are given the mass of a substance, like 15 mg, you must convert it to grams since molar mass units are in grams per mole. Then, you can determine how many moles are in that given mass by using the molar mass, which acts as a conversion factor between grams and moles.

Standard Temperature and Pressure (STP) are commonly used conditions in chemistry to report and compare the behavior of gases. At STP, the standard temperature is 0°C (273.15 K), while the standard pressure is 1 atmosphere (atm).

Significance of STP

Under STP conditions, gases exhibit universal behavior which allows for easier calculation and comparison. One of the most important aspects of STP in relation to gases is that one mole of any ideal gas occupies 22.4 liters (L) of volume. This knowledge simplifies the process of calculating the volume of a gas at STP if the number of moles is known, as the volume of a gas is directly proportional to the number of moles present at these conditions.

The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and amount of moles of an ideal gas. The law is represented by the equation: \[PV = nRT\], where P is the pressure, V is the volume, n is the number of moles of the gas, R is the ideal gas constant, and T is the temperature in Kelvin.
When working with gas problems at STP, the ideal gas law allows us to find out the volume occupied by the gas with ease. Since the temperature and pressure are set at standard conditions, the only variables are the number of moles and the volume. Using the law, we can affirm that at STP the constant \(R\) multiplied by standard temperature and divided by standard pressure results in 22.4 L/mol, consistent with the volume occupied by one mole of any ideal gas at these conditions.

The mole is a basic unit in chemistry used to express amounts of a chemical substance. Calculating moles can be done in two common ways: from mass and from the number of molecules.

From Mass

To calculate moles from mass, divide the mass of the substance by its molar mass. This process requires a conversion of weight (often from milligrams to grams) and then the use of the compound's molar mass as a conversion factor, as seen in the exercise with \(N_{2}O_{3}\).

From number of Molecules

To calculate moles from the number of molecules, you must use Avogadro's number (6.022 × 10²³), which is the number of molecules in one mole of a substance. By dividing the number of molecules by Avogadro's number, you get the number of moles.
Summing the calculated moles from mass and the calculated moles from the number of molecules gives the total moles of the substance, which can then be used with the ideal gas law at STP to find volume.