Question

A quantity of \(\mathrm{N}_{2}\) gas originally held at 5.25 atm pressure in a 1.00 -L container at \(26^{\circ} \mathrm{C}\) is transferred to a \(12.5-\mathrm{L}\) container at \(20^{\circ} \mathrm{C}\) . A quantity of \(\mathrm{O}_{2}\) gas originally at 5.25 atm and \(26^{\circ} \mathrm{C}\) in a \(5.00-\mathrm{L}\) container is transferred to this same container. What is the total pressure in the new container?

Short answer

The total pressure in the new container is 2.435 atm.

Step-by-step solution

Convert temperatures to Kelvin

First, we need to convert the given Celsius temperatures to Kelvin, as the ideal gas law uses Kelvin temperatures. For N2 gas: Initial temperature in Celsius = 26°C Initial temperature in Kelvin = 26 + 273.15 = 299.15 K Final temperature in Celsius = 20°C Final temperature in Kelvin = 20 + 273.15 = 293.15 K For O2 gas: Initial temperature in Celsius = 26°C Initial temperature in Kelvin = 26 + 273.15 = 299.15 K

Calculate the number of moles for both gases

Now, we will use the ideal gas law equation to calculate the number of moles (n) for both gases. For N2 gas: Initial pressure (P1) = 5.25 atm Initial volume (V1) = 1.00 L Initial temperature (T1) = 299.15 K Ideal gas constant (R) = 0.08206 L atm / (mol K) Rearrange the ideal gas law equation for number of moles (n): \(numull\)n = \frac{P1 \times V1}{R \times T1} = \frac{5.25 \times 1.00}{0.08206 \times 299.15} = 0.217$ mol For O2 gas: Initial pressure (P1) = 5.25 atm Initial volume (V1) = 5.00 L Initial temperature (T1) = 299.15 K \(numull\)n = \frac{P1 \times V1}{R \times T1} = \frac{5.25 \times 5.00}{0.08206 \times 299.15} = 1.085$ mol

Calculate the final pressures for both gases

Now that we have the number of moles for both gases, we can use the ideal gas law equation again, this time solving for the final pressures (P2). For N2 gas: Final volume (V2) = 12.5 L Final temperature (T2) = 293.15 K \(numull\)P2 = \frac{n \times R \times T2}{V2} = \frac{0.217 \times 0.08206 \times 293.15}{12.5} = 0.406$ atm For O2 gas: Final volume (V2) = 12.5 L \(numull\)P2 = \frac{n \times R \times T1}{V2} = \frac{1.085 \times 0.08206 \times 299.15}{12.5} = 2.029$ atm

Calculate the total pressure in the new container

To find the total pressure in the new container, we simply add the final pressures of both gases. Total pressure = Final pressure of N2 gas + Final pressure of O2 gas \(numull\)Total pressure = 0.406 atm + 2.029 atm = 2.435 atm The total pressure in the new container is 2.435 atm.

Key concepts

Gas Pressure Calculations

Understanding gas pressure calculations is essential in chemistry and physics. In simple terms, the pressure of a gas is the force that the gas exerts on the walls of its container. This force is due to the constant, random motion of gas particles that collide with the container walls.

When we speak of calculating gas pressure, we typically refer to the Ideal Gas Law, given by the equation: \( PV = nRT \), where \(P\) is the pressure of the gas, \(V\) is the volume it occupies, \(n\) is the amount of gas in moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin. To solve for pressure, rearrange the equation to \( P = \frac{nRT}{V} \).

For instance, if we have a given number of moles of a gas at a known temperature and volume, we can calculate the gas pressure. It's pivotal when dealing with combined gases as well, such as finding the total pressure in a container holding a mixture of gases. The solution comes from summing the individual pressures contributed by each gas, described as the partial pressures. This principle reflects Dalton's Law of Partial Pressures, emphasizing that gases act independently of one another when in a mixture.

Kelvin Temperature Conversion

When working with gas laws, temperature must always be in Kelvin. The Kelvin scale is an absolute thermodynamic temperature scale where zero on the scale represents absolute zero. Absolute zero is the point at which atomic and molecular motion theoretically stops. To convert Celsius temperatures to Kelvin, we add 273.15 to the Celsius temperature.

For example, converting 26°C to Kelvin, the calculation would be \( 26°\text{C} + 273.15 = 299.15\text{K}\). It’s important to use Kelvin because it ensures that temperature values are always positive, which is necessary when employing the Ideal Gas Law. Being a proportionality constant, the Ideal Gas Law requires absolute temperature for accurate calculations, and working with other scales like Celsius or Fahrenheit would yield incorrect results.

Molar Volume of a Gas

Molar volume is the volume occupied by one mole of a substance at a given temperature and pressure. For gases, this concept is crucial because the volume changes significantly with changes in temperature and pressure. At standard temperature and pressure (0°C and 1 atm), one mole of any ideal gas occupies 22.4 liters.

In calculations involving the molar volume, Avogadro's hypothesis comes into play stating that equal volumes of gases, at the same temperature and pressure, contain the same number of molecules. Hence, molar volume helps understand the relationship between the number of moles and the volume it occupies under specific conditions. This knowledge aides in converting between the amount of a gas and the volume it occupies using the Ideal Gas Law, as illustrated by the exercises involving the expansion and compression of gases into different container volumes.