Problem 74 - Chemistry: An Atoms-Focused Approach

Question

Liquid Nitrogen-Powered Car Students at the University of North Texas and the University of Washington built a car propelled by compressed nitrogen gas. The gas was obtained by boiling liquid nitrogen stored in a \(182 \mathrm{L}\) tank. What volume of \(\mathrm{N}_{2}\) is released at 0.927 atm of pressure and \(25^{\circ} \mathrm{C}\) from a tank full of liquid \(\mathrm{N}_{2}\) \((d=0.808 \mathrm{g} / \mathrm{mL}) ?\)

Step-by-Step Solution

Answer

Answer: The volume of nitrogen gas released from the tank is 130,467.87 liters.

Step 1: Calculate the mass of liquid nitrogen.

First, let's find the mass of the liquid nitrogen using its volume and density. Mass = Volume × Density. Volume of liquid nitrogen \((V_{LN_{2}}) = 182\) L. Density of liquid nitrogen \((d_{LN_{2}}) = 0.808\) g/mL. Note: We need to convert the volume of liquid nitrogen from liters to milliliters, as the density is given in g/mL. So, \(V_{LN_{2}} = 182 L \times 1000 \mathrm{mL}/\mathrm{L} = 182,000 \mathrm{mL}\). Now let's find the mass of liquid nitrogen: \(\mathrm{Mass} (m_{LN_{2}}) = V_{LN_{2}} \times d_{LN_{2}} = 182,000 \mathrm{mL} \times 0.808 \mathrm{g}/\mathrm{mL} = 146,656 \mathrm{g}\).

Step 2: Convert the mass of liquid nitrogen to moles.

To determine the number of moles, we have to use the molar mass of nitrogen gas \((N_{2})\). The molar mass of nitrogen gas can be calculated as: Molar mass of nitrogen atom \((N) = 14.01 \mathrm{g/mol}\), therefore molar mass of nitrogen gas \((N_{2}) = 2 \times 14.01\mathrm{g/mol} = 28.02\mathrm{g/mol}\). Now, convert the mass of liquid nitrogen to moles: \(\mathrm{Moles} (n_{N_{2}}) = \frac{m_{LN_{2}}}{Molar\; mass\; of\; N_{2}} = \frac{146,656 \mathrm{g}}{28.02 \mathrm{g/mol}} = 5,230.69\mathrm{mol}\).

Step 3: Apply the Ideal Gas Law.

We have to determine the volume of the released nitrogen gas. Let's use the Ideal Gas Law formula: \(PV = nRT\) Where: P = pressure of the gas \((0.927 \mathrm{atm})\) V = volume of the gas (what we need to find) n = number of moles of the gas \((5,230.69\mathrm{mol})\) R = ideal gas constant \((0.0821 \mathrm{L atm/mol K})\) T = temperature of the gas (in Kelvin) Convert the given temperature in Celsius to Kelvin: \(T = 25^{\circ} \mathrm{C} + 273.15 = 298.15\mathrm{K}\). Now, rearrange the Ideal Gas Law formula to solve for the volume: \(V = \frac{nRT}{P}\)

Step 4: Calculate the volume of released nitrogen gas.

Plug in the values and solve for the volume of released nitrogen gas: \(V = \frac{5,230.69\mathrm{mol} \times 0.0821 \mathrm{L\;atm/mol\;K} \times 298.15\mathrm{K}}{0.927\mathrm{atm}} = 130467.87\mathrm{L}\). The volume of nitrogen gas released from the tank is 130,467.87 liters.

Key Concepts

Liquid Nitrogen

Liquid nitrogen is an incredibly cold form of nitrogen that is in a liquid state. It is created and stored at cryogenic temperatures, which are extremely low temperatures necessary for the liquefaction of gases.

• It is predominantly found in the commercial and laboratory sectors.
• The cold temperature is a result of its boiling point at -196°C or 77 Kelvin.

To measure liquid nitrogen, we consider its volume and density, as it helps in calculations involving its conversion to gaseous nitrogen, like in this exercise. In our scenario, liquid nitrogen serves as the starting material, which will boil to form nitrogen gas used in powering the car engine.

Molar Mass

Molar mass is an important concept in chemistry that refers to the mass of one mole of a substance. In simpler terms, it tells us how much one mole of a molecule or atom weighs. The unit is typically g/mol (grams per mole), which easily meshed fits into many chemical calculations.

• For nitrogen gas \( N_2 \), each nitrogen atom has a molar mass of 14.01 g/mol.
• Since nitrogen exists naturally as a diatomic molecule (two atoms), the molar mass of \( N_2 \) becomes 28.02 g/mol.

Understanding molar mass is crucial to converting grams of a substance to moles, as was needed in the second step of our exercise.

Mole Conversion

The conversion between mass and moles is an essential step in many calculations in chemistry. This step involves changing the mass of a chemical in grams to the number of moles, which can be achieved by using the molar mass.For instance, the mass of liquid nitrogen converted to moles is calculated using:\[ \text{Moles of } N_2 = \frac{\text{Mass of } N_2}{\text{Molar Mass of } N_2} \]Where:

• Mass of \( N_2 = 146,656 \, \text{g} \)
• Molar mass of \( N_2 = 28.02 \, \text{g/mol} \)

This conversion allows chemists to easily quantify the relationship between reactants and products in chemical reactions.

Gas Volume Calculation

Calculating the volume of a gas involves using the Ideal Gas Law, a critical equation in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. The formula is:\[ PV = nRT \]Where:

• \( P \) is the pressure of the gas
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the ideal gas constant \( (0.0821 \, \text{L atm/mol K}) \)
• \( T \) is the temperature in Kelvin

To find the volume \( V \), rearrange the formula and substitute the known values as it was executed in the final step of our problem-solving, thus providing us with the volume of nitrogen gas released.