Problem 85 - Introductory Chemistry Atoms First

Question

Consider two identical \(1-\mathrm{L}\) containers, both at room temperature \((300 \mathrm{~K})\). One of them contains 1 mole of helium gas, and the other contains 1 mole 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

The pressures in both the helium and hydrogen containers are the same, as per the Ideal Gas Law \(PV = nRT\), because they have equal volume (1 L), equal number of moles (1 mole), and equal temperature (300 K). The formula results in equal pressure values for both containers: \(P_{He} = P_{H2} = \frac{1mol \times R \times 300K}{1L}\).

Step 1: Ideal Gas Law Formula

First, we need to understand the Ideal Gas Law formula: \(PV = nRT\) where: - P is the gas pressure, - V is the gas volume, - n is the number of moles of the gas, - R is the ideal gas constant, and - T is the gas temperature in Kelvin.

Step 2: Compare the Gases

In our case, we have the following information for both helium and hydrogen gases: - V (volume) = 1 L - n (number of moles) = 1 mole - T (temperature) = 300 K And the ideal gas constant, R, is the same for all ideal gases. So, our comparison should be based only on the pressure (P), as all other values are equal.

Step 3: Plug Values into the Formula

For helium gas (He): \(P_{He} \times 1L = 1mol \times R \times 300K\) \(P_{He} = \frac{1mol \times R \times 300K}{1L}\) For hydrogen gas (H₂): \(P_{H2} \times 1L = 1mol \times R \times 300K\) \(P_{H2} = \frac{1mol \times R \times 300K}{1L}\)

Step 4: Compare Pressures

Now let's compare the pressures: \(P_{He} = P_{H2} = \frac{1mol \times R \times 300K}{1L}\) Since both pressures are equal, we can conclude that: Pressures in both helium and hydrogen containers are the same.

Key Concepts

Pressure Comparison

In the exploration of the ideal gas law, comparing pressures involves understanding how different gases behave under identical conditions, such as temperature and volume. Here, we are dealing with two gases, helium and hydrogen, in identical containers at the same temperature of 300 Kelvin and a fixed volume of 1 liter. The number of moles for both gases is also equivalent, set at 1 mole each. Using the ideal gas law, \(PV = nRT\), the only variable left that could differ between these two scenarios is usually pressure. However, if all other parameters are the same, then the pressure must also be the same.

• Both containers have the same volume: 1 L
• Temperature outside and inside matches: 300 K
• Each gas contains exactly 1 mole

Thus, according to the ideal gas law, the pressures in the helium and hydrogen containers are equal, as demonstrated by substituting the known values into the formula. This results in equal pressure readings for both gases under these conditions.

Helium Gas

Helium gas is a noble gas, part of a group of gases that are known for being very stable and unreactive. Its stability makes it a prime candidate for usage in various scientific applications.

• Helium molecules are monoatomic, meaning each molecule consists of a single atom
• Due to being a light gas, it often has a lower density compared to more complex molecules

Despite its lightness, when discussing pressure with the ideal gas law, Helium behaves just like other gases. Its physical properties like the atomic mass don't affect the pressure calculated through the formula \(PV = nRT\) as long as the conditions mentioned earlier are identical between the gases considered.

Hydrogen Gas

Hydrogen gas is the simplest and lightest element, consisting of molecules that are diatomic, meaning each molecule is composed of two hydrogen atoms (H₂). This diatomic nature doesn't change the application of the ideal gas law; all ideal gases behave similarly under the law if conditions such as temperature, volume, and moles remain constant.

• Hydrogen is highly flammable and reactive, hence it’s used in fuel cells and industry
• Diatomic hydrogen has more internal energy states compared to monoatomic gases

Like helium, hydrogen will demonstrate the same pressure under similar conditions due to the neutrality of its internal structure in the context of the ideal gas equation. It’s important for students to see that although the molecular nature differs, in ideal conditions, this doesn't affect pressure.

Gas Constant

The gas constant, denoted as \(R\), is a crucial constant in the ideal gas equation. It serves as the linchpin ensuring that comparisons between different gases remain valid when using the ideal gas law. The gas constant has the same value for all ideal gases, meaning that it unifies the equation.

• It bridges the relationship between energy (as seen in temperature), volume, and pressure
• Its consistent value enables broad applications of the ideal gas law across various gases

In the ideal gas law equation, \(PV = nRT\), \(R\) maintains its role by staying constant regardless of the gas involved — whether it's helium or hydrogen. It's important to recognize that \(R\) does not vary with the type of gas, ensuring the validity of calculations and comparisons like those discussed in the exercise.