Problem 99 - A Complete Resource Book in Chemistry for JEE Main

Question

At a certain temperature for which \(\mathrm{RT}=25 \mathrm{~L}\) atm. \(\mathrm{mol}^{-1}\), the density of a gas, in \(\mathrm{g} \mathrm{L}^{-1}\), is \(\mathrm{d}=2.00 \mathrm{P}+\) \(0.020 \mathrm{P}^{2}\), where \(\mathrm{P}\) is the pressure in atmosphere. The molecular weight of the gas in \(\mathrm{g} \mathrm{mol}-1\) is (a) 60 (b) 75 (c) 50 (d) 35

Step-by-Step Solution

Answer

The molecular weight of the gas is 50 g/mol.

Step 1: Understand the Problem

We are given the density equation for a gas depending on the pressure \( P \), and the relation \( RT = 25 \, \mathrm{L} \cdot \mathrm{atm} \cdot \mathrm{mol}^{-1} \). We need to find the molecular weight of the gas.

Step 2: Recall the Ideal Gas Law

The ideal gas law is given by \( PV = nRT \). We can express the number of moles \( n \) in terms of mass \( m \) and molecular weight \( M \): \( n = \frac{m}{M} \), so \( PV = \frac{mRT}{M} \) which implies \( M = \frac{mRT}{PV} \).

Step 3: Relate Density to Ideal Gas Law

Density \( d \) is defined as \( \frac{m}{V} \). Substituting into the ideal gas law gives \( PV = \frac{dVRT}{M} \), which simplifies to \( d = \frac{PM}{RT} \). Rearrange to find \( M \): \( M = \frac{dRT}{P} \).

Step 4: Substitute Given Density Formula

We know \( d = 2.00P + 0.020P^2 \) and \( RT = 25 \, \mathrm{L} \cdot \mathrm{atm} \cdot \mathrm{mol}^{-1} \). Substitute these into the molecular weight formula: \( M = \frac{(2.00P + 0.020P^2)(25)}{P} \).

Step 5: Simplify the Expression

Divide each term in the numerator by \( P \): \( M = \frac{2.00P \cdot 25 + 0.020P^2 \cdot 25}{P} = 50 + 0.5P \).

Step 6: Find the Molecular Weight

Given options are independent of \( P \), assume \( P = 1 \) for simplicity, then \( M = 50 + 0.5 \times 1 = 50.5 \), which approximates to 50.

Key Concepts

Density Equation

The density equation relates how much mass of a gas occupies a certain volume at a given pressure. In this context, density is described by the equation \( d = 2.00P + 0.020P^2 \), where \( d \) is the density in grams per liter, and \( P \) is the pressure in atmospheres.
This equation shows us that density isn't static; it varies with pressure. As pressure increases, the term \( 0.020P^2 \) becomes more significant. So, density increases with pressure, but not in a linear fashion. The second term involves pressure squared, which makes density grow faster as pressure rises.
Understanding how density changes with pressure helps when using the Ideal Gas Law to calculate other properties of a gas, like its molecular weight. Let's see how that works!

Molecular Weight Calculation

Calculating the molecular weight of a gas can be achieved by relating it to the density equation and the Ideal Gas Law. The molecular weight \( M \) tells you how much one mole of a substance weighs in grams. Using both the density equation and Ideal Gas Law, we have:

• The Ideal Gas Law suggests that: \( PV = \frac{mRT}{M} \) where \( m \) is the mass and \( V \) the volume.
• By linking the density \( d = \frac{m}{V} \) to the Ideal Gas Law, it's expressed as \( d = \frac{PM}{RT} \).
• Rearranging gives us the formula, \( M = \frac{dRT}{P} \).

You substitute the given density equation into the molecular weight formula \( M = \frac{(2.00P + 0.020P^2)(25)}{P} \) to get \( M = 50 + 0.5P \).
For selecting molecular weight based on the problem, choose a convenient pressure, often \( P = 1 \). Here, it yields \( M = 50.5 \), approximating to an option provided.
This method exemplifies how dynamic calculations can be when multiple factors, such as density and pressure, come into play.

Gas Laws

Gas laws describe how gases behave under different conditions of pressure, volume, and temperature. The Ideal Gas Law is a cornerstone of these principles, expressed as \( PV = nRT \), connecting pressure \( P \), volume \( V \), amount of moles \( n \), and temperature \( T \).
It's particularly useful because it gives a simple relationship between macroscopic properties of gases, assuming ideal behavior. The equation allows us to predict how a gas will respond under changes in any of these parameters if the others remain constant.

• For constant temperature and volume, Gay-Lussac's law applies: pressure varies directly with temperature.
• At constant temperature and pressure, Avogadro's law tells us that volume varies with the number of moles.

The Ideal Gas Law can adapt these laws to estimate other properties, like molecular weight, by associating density and applying it over a range of pressures.
These relationships guide chemists and scientists to predict behaviors in natural and controlled environments, essential for fields from meteorology to pharmaceuticals.