Problem 23 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Reducing the pressure from \(1.0\) atm to \(0.5\) atm would change the number of molecules in one mole of ammonia to (a) \(75 \%\) of initial value (b) \(50 \%\) of initial value (c) \(25 \%\) of initial value (d) None of these

Step-by-Step Solution

Answer

The number of molecules in one mole of ammonia would be 50% of the initial value.

Step 1: Understanding the Relationship between Pressure and Number of Molecules

According to the Ideal Gas Law, which is given by the equation 'PV = nRT', where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature, if the temperature and the volume are kept constant, the pressure of a gas is directly proportional to the number of moles of the gas. This means that if the pressure is reduced, the number of moles (and thus the number of molecules) present will also be reduced proportionally.

Step 2: Calculating the Percentage Change in Number of Molecules

Given that the initial pressure is 1.0 atm and it is reduced to 0.5 atm, the pressure is halved. Assuming the temperature and volume remain constant and applying the direct proportionality from the Ideal Gas Law, the number of molecules would also be halved. Therefore, the number of molecules in one mole of ammonia would become 50% of its initial value.

Step 3: Selecting the Correct Answer

Since the number of molecules in a mole of ammonia is halved when the pressure is reduced from 1.0 atm to 0.5 atm, the correct answer is that the number of molecules would be 50% of the initial value, which corresponds to option (b).

Key Concepts

Pressure and Number of Molecules

When studying the impact of pressure on a gas, it's essential to understand its microscopic behavior, specifically how pressure and the number of molecules are related. In an enclosed system, gas molecules constantly move and collide with the walls of their container. These collisions create force over an area, which is what we experience as pressure.

If we have a fixed volume of gas and we decrease the pressure, we infer that fewer collisions are occurring. This implies there's a reduction in the number of gas molecules within that volume, assuming all other conditions remain constant. Conversely, increasing the pressure would suggest more molecules are present to exert force on the container walls. Ideal Gas Law problems utilize this concept to relate how changing one variable, like pressure, affects others, notably the number of molecules.

Gas Laws

Gas laws are fundamental principles that describe the behavior of gases under various conditions. One of the most well-known equations embodying these principles is the Ideal Gas Law, represented as \( PV = nRT \). This law connects the physical properties of pressure (\(P\)), volume (\(V\)), the number of moles (\(n\)), temperature (\(T\)), and the ideal gas constant (\(R\)).

Within the Ideal Gas Law, the relationship between these variables allows us to predict how a change in one will affect the others. For example, if the temperature and volume of a gas remain constant, a decrease in pressure will linearly decrease the number of moles of the gas, according to the equation. This helps explain why, in the textbook problem, halving the pressure also halves the number of molecules. Therefore, mastering the gas laws is crucial for scientists and engineers who work with gaseous systems, ensuring accurate predictions and manipulations of gas behavior.

Mole Concept

The mole concept is a cornerstone in chemistry that assists scientists in counting particles, such as atoms, molecules, or ions. A mole is defined as the amount of substance that contains exactly \(6.022 \times 10^{23}\) entities (Avogadro's number).

When dealing with gases, the mole concept becomes particularly handy because it allows chemists to relate a gas's macroscopic properties, like pressure, to the microscopic number of particles, bypassing the impractical task of counting each molecule. As we saw in the exercise, when the pressure dropped from 1.0 atm to 0.5 atm, this would theoretically halve the number of moles, and correspondingly, the number of molecules would also be halved, not in absolute number but in proportion to what a 'mole' represents. Hence, understanding the mole concept enables us to make sense of changes in gas properties on a particle level and solve Ideal Gas Law problems easily.