Question

The density of gas \(A\) is twice that to \(B\) at the same temperature the molecular weight of gas \(B\) is twice that of \(A\). The ratio of pressure of gas \(A\) and \(B\) will be : (a) \(1: 6\) (b) \(1: 1\) (c) \(4: 1\) (d) \(1: 4\)

Short answer

The ratio of pressure of gas A and B is 4:1.

Step-by-step solution

- Understanding the Relationship Between Density and Molecular Weight

The density (d) of a gas is directly proportional to its molecular weight (M) at constant pressure and temperature according to the formula \( d = \frac{MP}{RT} \), where P is the pressure, R is the universal gas constant, and T is the absolute temperature.

- Setting up the Given Information

According to the problem, the density of gas A is twice that of gas B, which we can express as \( d_A = 2d_B \). It is also given that the molecular weight of gas B is twice that of A, which we can express as \( M_B = 2M_A \).

- Deriving the Ratio of Pressures of Gas A and B

Since both gases are at the same temperature, we can assume T is constant for both. By using the proportional relationship from Step 1 for both gases, we can write \( d_A = \frac{M_AP_A}{RT} \) and \( d_B = \frac{M_BP_B}{RT} \). Substituting the known relationships of the densities and molecular weights, we get \( 2d_B = \frac{M_AP_A}{RT} \) and \( d_B = \frac{2M_AP_B}{RT} \). On dividing the first equation by the second, we obtain \( 2 = \frac{P_A}{2P_B} \), which simplifies to \( P_A = 4P_B \). Therefore, the ratio of the pressures of gas A to gas B is 4:1.

Key concepts

Density and Molecular Weight Relationship

Understanding the relationship between the density of a gas and its molecular weight is fundamental to grasping various concepts in chemistry and physics. Gas density is a measure of mass per unit volume, and it’s related to molecular weight through the Ideal Gas Law. Here’s a simple way to break it down: At constant temperature and pressure, if the molecular weight of a gas increases, so does its density. This proportional relationship is essential for predicting how different gases will behave under the same conditions.

In our exercise, we’re told that gas A is twice as dense as gas B (\r\( d_A = 2d_B \)), and that the molecular weight of gas B is twice that of A (\r\( M_B = 2M_A \)). By using these relationships, we can deduce how the properties of one gas compare to the other. It's like having a twin but finding out you both have distinct traits despite your identical appearance. The wonderful thing about this relationship is its predictability, allowing us to solve complex problems with ease.

Ideal Gas Equation

The Ideal Gas Equation, \r\( PV = nRT \), is a cornerstone in understanding gas behavior. It allows us to link together the pressure (P), volume (V), number of moles (n), and absolute temperature (T) of a gas, with R being the universal gas constant. Think of it as a recipe that tells us how gases should behave under ideal conditions. Now, if you modify one of the ingredients (say, increase the pressure), the equation tells us what will happen to the other parts of the recipe (like how the volume will change).

This equation is idealized because it doesn't take into account interactions between gas molecules or the volume occupied by the molecules themselves. Despite these simplifications, it’s incredibly useful for many situations, like our textbook problem. We leverage this equation to relate the molecular weights and densities of the gases to their pressures, helping us solve for the unknown ratio with confidence and a clear understanding of the underlying principles.

Pressure Ratio of Gases

When talking about the pressure ratio of gases, we're comparing the force exerted by one gas to another in a given volume. It's like comparing the strengths of two superheroes by seeing how hard they can each punch through the same block of ice. When we say that the pressure of one gas is twice the pressure of another, we are implying that it exerts twice the force on the walls of its container.

The pressure ratio can tell a lot about how gas will act. If two gases have the same temperature and volume, but different pressures, the one with higher pressure contains more molecules pushing against the walls or is made up of heavier molecules doing the pushing - or a mix of both. In our exercise, we find that the pressure of gas A is four times (\r\( P_A = 4P_B \)) that of gas B. This is a direct result of understanding how molecular weight affects density and, subsequently, how both relate to pressure within the framework of the ideal gas law.