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The mass of nitrogen in a given volume of gas can be calculated using the Ideal Gas Law and some basic conversions. The Ideal Gas Law equation is given by \( PV = nRT \), where:

• \( P \) is the pressure,
• \( V \) is the volume,
• \( n \) is the number of moles,
• \( R \) is the gas constant,
• \( T \) is the temperature in Kelvin.

Let's break down how to calculate the mass of nitrogen.First, it's important to convert the provided units to the standard units used in the Ideal Gas Law:

• Convert the volume from \( \mathrm{cm}^3 \) to \( \mathrm{m}^3 \) by multiplying by \(10^{-6}\). So, \( 3000 \, \mathrm{cm}^3 = 3000 \times 10^{-6} \, \mathrm{m}^3 \).
• Convert the temperature from degrees Celsius to Kelvin by adding 273.15. Thus, \( 22.0^\circ \mathrm{C} = 295.15 \, \mathrm{K} \).
• Convert the pressure from atm to pascal by multiplying by \( 1.013 \times 10^5 \). Therefore, \(2.00 \times 10^{-13} \, \mathrm{atm} = 2.00 \times 10^{-13} \times 1.013 \times 10^5 \, \mathrm{Pa} \).

Now, solve for \( n \) using the rearranged Ideal Gas Law: \( n = \frac{PV}{RT} \). Once \( n \) is known, multiply it by the molar mass of nitrogen \( (28.0134 \, \mathrm{g/mol} = 28.0134 \times 10^{-3} \, \mathrm{kg/mol}) \) to find the mass.This approach will yield the mass of nitrogen in the given conditions.

Once we determine the mass of a gas, calculating its density is straightforward. Density (\( \rho \)) is a measure of how much mass is contained within a specific volume and is defined by the equation:\[ \rho = \frac{\text{Mass}}{\text{Volume}} \]For our specific example with nitrogen, now that we have the mass calculated using the Ideal Gas Law, we can plug it into our density formula.Considerations for calculating density:

• Make sure that the mass is in kilograms \((\mathrm{kg})\), which is derived from the number of moles and the molar mass.
• Ensure the volume is in cubic meters \((\mathrm{m}^3)\), which was previously converted from cubic centimeters in our earlier calculation.

Simply substitute the values into the density formula to obtain the density of nitrogen in \( \mathrm{kg/m}^3 \). This measurement gives a clear idea of how concentrated the nitrogen molecules are within the given volume.

Converting units is a crucial step when working with physics equations like the Ideal Gas Law. Using consistent units ensure accurate calculations. Here's how unit conversion applies in our scenario:**Volume Conversion**The volume was initially given in cubic centimeters \((\mathrm{cm}^3)\). To convert to cubic meters \((\mathrm{m}^3)\), multiply by \(10^{-6}\). This is because there are \(100\) centimeters in a meter, so \((\mathrm{cm})^3\) conversion to \((\mathrm{m})^3\) involves \((10^{-2})^3 = 10^{-6}\).**Temperature Conversion**Temperature provided in Celsius \((^\circ \mathrm{C})\) is converted to Kelvin (\(\mathrm{K}\)) by adding 273.15. This is because the Kelvin scale is an absolute temperature scale and reduces temperature-based discrepancies in calculations.**Pressure Conversion**Pressure in atmospheres \((\mathrm{atm})\) is converted to Pascals \((\mathrm{Pa})\) by multiplying by \(1.013 \times 10^5\). The pascal is the SI unit for pressure, ensuring we maintain uniformity with other SI units like meters and kilograms.Unit conversions are not merely procedural steps; they are vital for ensuring the correct application of formulas and achieving reliable results. Always ensure to cross-check your units for consistency across all measurement parameters.