Problem 86 - Introductory Chemistry Atoms First

Question

Consider two identical \(1-\mathrm{L}\) containers, both at room temperature \((300 \mathrm{~K})\). One of them contains 1.0 \(\mathrm{g}\) of helium gas, and the other contains \(1.0 \mathrm{~g}\) of hydrogen gas. Is the pressure higher in the helium container, higher in the hydrogen container, or the same in the two containers?

Step-by-Step Solution

Answer

In conclusion, by using the ideal gas law equation and comparing the number of moles for each gas, we find that the pressure is higher in the hydrogen container due to a greater number of moles compared to the helium container.

Step 1: Find the number of moles of helium and hydrogen

To find the number of moles, we will use the formula: \(n = \frac{mass}{molar \thinspace mass}\) The molar mass of helium is 4.00 g/mol, while the molar mass of hydrogen is 2.00 g/mol. Given that the container contains 1.0 g of each gas, we can find the number of moles: For helium: \(n_{He} = \frac{1.0 \thinspace g}{4.00 \thinspace g/mol} = 0.25 \thinspace mol\) For hydrogen: \(n_{H_2} = \frac{1.0 \thinspace g}{2.00 \thinspace g/mol} = 0.50 \thinspace mol\)

Step 2: Use the ideal gas law equation to find the pressure

Now, we will substitute the values into the ideal gas law equation: \(PV = nRT\) Since both containers have the same volume, the volume term (V) will be the same for both the helium and hydrogen gases. We can compare the pressure based on the number of moles because the temperature and the gas constant (R) are the same for both containers. For this reason, we can write the following equation and solve it for pressure (P): \(P = \frac{nRT}{V}\) For helium: \(P_{He} = \frac{0.25 \thinspace mol \cdot 8.314 \thinspace J/mol \cdot K \cdot 300 \thinspace K}{1 \thinspace L}\) For hydrogen: \(P_{H_2} = \frac{0.50 \thinspace mol \cdot 8.314 \thinspace J/mol \cdot K \cdot 300 \thinspace K}{1 \thinspace L}\)

Step 3: Compare the pressures

We can now compare the pressures in each container. Since the number of moles of hydrogen gas is greater than the number of moles of helium gas (0.50 mol vs 0.25 mol), and all the remaining factors are the same, it follows that the pressure in the hydrogen container will be greater than the pressure in the helium container. In conclusion, the pressure is higher in the hydrogen container.

Key Concepts

Molar Mass

Molar mass is defined as the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It can be thought of as the 'atomic or molecular weight' of a substance when measured in moles. The molar mass is a bridge between the mass of a substance and the number of moles since it tells us how much one mole of a substance weighs. For instance, the molar mass of helium is 4.00 g/mol, meaning one mole of helium atoms weighs 4 grams.

Understanding molar mass is crucial when comparing different substances under the same conditions. In the given exercise, helium and hydrogen have different molar masses, which directly influence the number of moles present in a fixed mass of each gas. This concept is fundamental when using the ideal gas law to compare pressures, as seen in the solution provided.

Avogadro's Law

Avogadro's law states that equal volumes of all gases, under the same conditions of temperature and pressure, contain an equal number of molecules. In other words, the volume of a gas is directly proportional to the number of moles of gas present, as long as the temperature and pressure are constant.

This key principle allows us to understand how a gas's volume can change in response to the quantity of matter present. When applying Avogadro's law to the exercise, we see how, theoretically, if both gases were at the same pressure and temperature and in identical containers, they would occupy the same volume — but the number of molecules within that volume would be different for helium and hydrogen due to their distinct molar masses.

Gas Pressure

Gas pressure is the force that the gas exerts on the walls of its container, and it results from molecules colliding with the container's boundary. Measured in units such as pascals (Pa) or atmospheres (atm), gas pressure depends on several factors, such as the number of gas molecules, the volume of the container, and the temperature.

In the provided exercise, even though the volume and temperature of the helium and hydrogen gases are the same, their pressures differ due to the number of moles of gas in each container. This disparity directly correlates to the molar mass of each gas and the unique number of collisions that the gas particles make with the walls of their respective container.

Universal Gas Constant

The universal gas constant is a physical constant denoted by the letter 'R' and is a key component of the ideal gas law equation. Its value is 8.314 J/mol·K (joules per mole-kelvin). The universal gas constant relates the energy scale to the molecular scale and provides the link between the pressure, volume, temperature, and number of moles of a gas.

When dealing with the ideal gas law, as seen in the exercise solution, this constant remains the same across all calculations for different gases if the temperature and pressure are maintained constant. The consistency of the universal gas constant value ensures that the relationships described by the ideal gas law hold true universally for ideal gases.