Problem 19 - Chemistry in 30 days

Question

Two glass bulbs ' \(X^{\prime}\) and \(^{\prime} Y^{\prime}\) are connected by a very small tube having a stop cock. Bulb \(X\) has a volume of \(100 \mathrm{~cm}^{3}\) and contains the gas, while bulb \(Y\) is empty. On opening the stop cock, the pressure falls down by \(60 \%\). The volume of bulb \(Y\) must be (a) \(150 \mathrm{~cm}^{3}\) (b) \(250 \mathrm{~cm}^{3}\) (c) \(100 \mathrm{~cm}^{3}\) (d) \(125 \mathrm{~cm}^{3}\)

Step-by-Step Solution

Answer

The volume of bulb Y must be 150 cm³.

Step 1: Understanding the Initial Condition

Initially, bulb X contains gas of volume 100 cm³. The pressure in bulb X is considered as the initial pressure because there is no gas in bulb Y. Let this initial pressure be denoted as \( P_i \).

Step 2: Understanding the Impact of Opening the Stop Cock

When the stop cock is opened, gas moves from bulb X to bulb Y, causing the pressure to drop by 60%. This means the final pressure in both bulbs is only 40% of the initial pressure \( P_i \). Thus, the final pressure can be expressed as \( P_f = 0.4 P_i \).

Step 3: Applying the Ideal Gas Law

Since the system is closed, the initial and final states must obey the ideal gas law. Therefore, \( P_i \times V_x = P_f \times (V_x + V_y) \), where \( V_x = 100 \) cm³ and \( V_y \) is the volume of bulb Y.

Step 4: Substituting the Known Values

Substitute the final pressure in terms of initial pressure into the equation: \( P_i \times 100 = 0.4 P_i \times (100 + V_y) \).

Step 5: Solving for \( V_y \)

Cancel \( P_i \) from both sides (assuming \( P_i eq 0 \)) to simplify: \( 100 = 0.4 \times (100 + V_y) \). Solve this equation for \( V_y \).

Step 6: Calculating \( V_y \)

Multiply and rearrange the terms: \( 100 = 40 + 0.4 V_y \). Subtract 40 from both sides: \( 60 = 0.4 V_y \). Divide by 0.4 to find \( V_y \): \( V_y = 150 \).

Key Concepts

Pressure

Pressure is the force exerted by a gas due to its molecules hitting the walls of its container. In a gas system, pressure can significantly change when the gas is allowed to occupy a larger volume or when restricted to a smaller one.

• A direct relationship exists between pressure and the number of gas molecules within a given volume (more molecules mean higher pressure).
• According to Boyle's Law, at constant temperature, the pressure of a gas is inversely proportional to its volume.
• In the context of our exercise, initially, the bulb labeled X has all the gas, leading to a specific initial pressure, denoted as \( P_i \).
• Once the bulbs are connected, the gas spreads out, reducing the pressure by 60% due to the increased volume it can now occupy.

Working with the ideal gas law, it’s important to recognize how pressure directly affects the overall calculations when volumes change.

Volume

Volume is the space that a gas occupies. It plays a crucial role in determining its pressure when constrained in a system like a gas bulb.

• Initially, the gas is confined to a volume of 100 cm³ in bulb X, defining the initial state of pressure and volume.
• When bulb Y is introduced, and gas is allowed to flow, the total volume available for the gas changes to \(100 + V_y\), where \(V_y\) is the volume of bulb Y.
• This expansion of volume is the reason behind the drop in pressure observed.
• The exercise specifically demonstrates how volume changes impact pressure under the application of Boyle’s Law within a closed system.

Understanding the relationship between pressure and volume is key to mastering gas behavior.

Gas System

A gas system is a setup in which gases are contained and manipulated under various conditions. Understanding a gas system involves recognizing how gases respond to changes in their environment like pressure and volume adjustments.

• In the ideal gas law formula \( PV = nRT \), where \(n\) is the number of moles, and \(R\) is the gas constant, both pressure \(P\), and volume \(V\) directly impact the state of the gas at temperature \(T\).
• In our exercise, the gas system consists of bulbs X and Y, initially separated, with bulb X containing the gas.
• When the stop cock is opened, the gas expands into bulb Y, demonstrating the behavior of gases when given more space, impacting pressure as per Boyle’s Law.

Remember that gas behavior in a system is predictable using these fundamental principles of gas laws.

Closed System

A closed system is one in which no gas enters or leaves; only changes in pressure, volume, and temperature occur while maintaining the same number of gas molecules.

• It's important to note that in a closed system, although gases can move from one part of the system to another, no external substances are added.
• The exercise scenario specifies a closed system; the gas initially in bulb X just redistributes between the two bulbs without any gas loss to the outside.
• This is key to understanding the conservation of the amount of gas as well as to successfully applying the ideal gas law to determine the unknowns (such as the volume of bulb Y).
• In real-world applications, closed systems allow scientists and engineers to predict how gases behave under various manipulations.

Recognizing and understanding the constraints of a closed system is essential for problem-solving in thermodynamics.