Question

At what temperature \(28 \mathrm{~g}\) of \(\mathrm{N}_{2}\) will occupy a volume of 20 litres at 2 atm? (a) \(300.0 \mathrm{~K}\) (b) \(487.2 \mathrm{~K}\) (c) \(289.6 \mathrm{~K}\) (d) \(283.8 \mathrm{~K}\)

Short answer

The temperature at which 28 g of N2 will occupy a volume of 20 litres at 2 atm pressure is 487.2 K.

Step-by-step solution

Use the Ideal Gas Law

To find the temperature, we will use the Ideal Gas Law, which is given by the equation PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature in Kelvin.

Calculate the Number of Moles

First, calculate the number of moles of N2 using the molar mass. The molar mass of N2 is about 28 g/mol, so n = mass / molar mass = 28 g / 28 g/mol = 1 mol.

Rearrange the Ideal Gas Law to solve for T

Rearrange the Ideal Gas Law equation to solve for T: T = PV / (nR).

Insert the Values and Calculate the Temperature

Now, insert the given values into the rearranged equation: T = (2 atm) * (20 L) / (1 mol * 0.0821 L*atm/mol*K). Calculate the temperature in Kelvins.

Check the Answer

After solving the equation, check which of the provided options matches the calculated temperature.

Key concepts

Solving Ideal Gas Law problems

When solving Ideal Gas Law problems, it's important to understand the relationship between the pressure (P), volume (V), number of moles (n), and temperature (T). The formula encapsulating this relationship is the Ideal Gas Law: \( PV = nRT \). Here, 'R' represents the universal gas constant with a value of \( 0.0821 \frac{L \cdot atm}{mol \cdot K} \).

To solve an Ideal Gas Law problem, follow these general steps:

• Identify and note down the given values for P, V, n, and T.
• If any value is missing, rearrange the equation to solve for that variable.
• Ensure all units are consistent, especially temperature in Kelvin and pressure in atmospheres (atm).
• Insert the values into the equation and solve for the unknown.
• Double-check your result against the given options or known physical constraints.

It's important to use the correct units and constants to ensure the calculations are accurate. In the textbook problem, by following these steps, we solve for temperature given the pressure, volume, and number of moles.

Molar mass of gases

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It's a crucial factor when working with gases, as it links the mass of a gas to its amount in moles. To solve problems involving the molar mass of gases:

• Firstly, find the molar mass by summing the atomic masses of the elements in the gas molecule. For \(\mathrm{N}_2\) , the molar mass is approximately 28 g/mol.
• Use the molar mass to convert the mass of the gas into moles using the formula \( n = \frac{\text{mass}}{\text{molar mass}} \).

This conversion is pivotal when using the Ideal Gas Law, where the n term requires moles. The given problem provides the mass of nitrogen gas, which we use along with the molar mass to find the number of moles.

Calculating moles from mass

Calculating moles from mass is a foundational skill in chemistry for quantifying substances. The mole (mol) is the base unit in the International System of Units (SI) for measuring the amount of substance. One mole of any substance contains Avogadro's number of particles (\(6.022 \times 10^{23}\) particles/mol).

To calculate moles from mass:

• Determine the molar mass of the substance.
• Use the formula \( n = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \) to calculate moles.

In our exercise, we have a 28 g sample of nitrogen gas (\(\mathrm{N}_2\)). Using Nitrogen's molar mass of 28 g/mol, we find that the sample contains 1 mole of gas. Accurate calculation of moles is essential when using the Ideal Gas Law equation.

Converting temperature to Kelvin

Temperature measurements in chemistry problems are often required in Kelvin, as it is the SI unit for temperature and used in gas law equations. Kelvin is an absolute temperature scale starting at absolute zero, where all thermal motion ceases.

To convert Celsius to Kelvin, use the formula: \( K = ^\circ C + 273.15 \). However, remember that Fahrenheit conversions require an additional step.

• To convert Fahrenheit to Celsius: \( ^\circ C = (\frac{5}{9}) \cdot (^\circ F - 32) \).
• Then convert Celsius to Kelvin as above.

Most problems, like our textbook example, will provide temperature directly in Kelvin. If not, ensure you convert to Kelvin before using the temperature in the Ideal Gas Law formula.