Question

A 50.0 -L tank contains $5.21\mathrm{kg}$ of ${\mathrm{N}}_2$ and $4.49\mathrm{kg}$ of ${\mathrm{O}}_2.$ What is the pressure in the tank at ${24}^{\circ }\mathrm{C}?$

Short answer

The pressure in the tank is approximately $1.61\times {10}^5$ Pa.

Step-by-step solution

Convert temperature to Kelvin

First, we need to convert the temperature from Celsius to Kelvin. The formula for this conversion is: K = °C + 273.15 T = 24°C + 273.15 T = 297.15 K

Calculate the number of moles of N₂ and O₂

Now, we will find the number of moles of each gas using their respective molar masses (N₂ = 28.02 g/mol and O₂ = 32.00 g/mol). First, we need to convert the mass of N₂ and O₂ from kg to g: Mass of N₂ = 5.21 kg * 1000 = 5210 g Mass of O₂ = 4.49 kg * 1000 = 4490 g Now, we calculate the number of moles for each gas: Number of moles of N₂ = mass of N₂ / molar mass of N₂ n_N₂ = 5210 g / 28.02 g/mol = 185.94 mol Number of moles of O₂ = mass of O₂ / molar mass of O₂ n_O₂ = 4490 g / 32.00 g/mol = 140.31 mol

Calculate the total number of moles

Now, we will find the total number of moles present in the tank, which is the sum of the number of moles of N₂ and O₂: n_total = n_N₂ + n_O₂ = 185.94 mol + 140.31 mol = 326.25 mol

Calculate the pressure in the tank

Now, using the ideal gas law (PV = nRT), we can find the pressure in the tank. The volume (V) is given in liters, so we need to convert it to m³. Volume (V) = 50.0 L * (1 m³ / 1000 L) = 0.0500 m³ We have all the necessary information to calculate the pressure (P): P = nRT / V P = (326.25 mol * 8.314 J/mol·K * 297.15 K) / 0.0500 m³ P ≈ 161190 Pa The pressure in the tank is approximately 1.61 x 10^5 Pa.

Key concepts

Gas Pressure Calculation

Understanding the calculation of gas pressure is crucial in various fields of science, particularly chemistry, physics, and engineering. Pressure is a measure of the force exerted by the gas particles against the walls of their container per unit area. It's given in pascals (Pa) in the International System of Units (SI).

The ideal gas law, a cornerstone of gas calculations, provides a clear pathway to finding the pressure of a gas in a container. This fundamental law is expressed as PV = nRT, where P stands for pressure, V is volume, n is the number of moles of gas, R is the universal gas constant (8.314 J/mol·K), and T is the temperature in Kelvin (K).

To calculate the pressure using the ideal gas law, follow these steps. Convert all measurements to the proper SI units—moles for the amount of gas, cubic meters for volume, and Kelvin for temperature. With all the variables in correct units, you can rearrange the equation to solve for pressure: $P=\frac{nRT}{V}$. This scenario typically arises in tasks such as calculating the pressure of different gases in a tank at a given temperature, as seen in the textbook exercise.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It's a fundamental concept in stoichiometry and conversion between the mass of a substance and the number of moles. Each element has a different molar mass, which can be found on the periodic table.

For instance, the molar mass of Nitrogen (N) is 14.01 g/mol, and since Nitrogen gas (\r{N}_{2}\r) consists of two Nitrogen atoms, its molar mass is twice that of a single atom, giving us 28.02 g/mol. Similarly, Oxygen gas (O_2) has a molar mass of 32.00 g/mol because Oxygen's atomic molar mass is 16.00 g/mol.

To calculate the number of moles from mass, you use the formula: $n=\frac{mass}{molarmass}$. This step is pivotal in solving problems like the one in our textbook, where the mass of gas in kilograms must first be converted to grams, and then divided by the molar mass to find the number of moles.

Gas Law Application in Chemistry

The ideal gas law is not just a theoretical equation; it has a broad spectrum of applications in the field of chemistry. From calculating the amount of reactants needed in a chemical reaction to determining the final pressure of a system after a reaction, the law serves as a versatile tool.

One common application is reacting gas stoichiometry, where chemists use the ideal gas law to find the volume or mass of gases at different conditions of temperature and pressure. Also, in environmental studies, it helps estimate pollutant gases' dispersion in the atmosphere. In biochemistry, it's used to calculate respiration rates and gas exchange.

Moreover, the gas law is pivotal in industrial processes, helping to design and control conditions in reactors and storage tanks. Understanding how the gas variables are interconnected through the ideal gas law enables chemists and engineers to predict and manipulate the behavior of gases, optimizing production and ensuring safety in their operations.