Problem 19 - A Complete Resource Book in Chemistry for JEE Main

Question

What are the conditions under which the relation between volume (V) and number of moles (n) of gas is plotted? ( \(\mathrm{P}=\) pressure; \(\mathrm{T}\) - temperature) (a) constant \(\mathrm{P}\) and \(\mathrm{T}\) (b) constant \(\mathrm{T}\) and \(\mathrm{V}\) (c) constant \(\mathrm{P}\) and \(\mathrm{V}\) (d) constant \(\mathrm{n}\) and \(\mathrm{V}\)

Step-by-Step Solution

Answer

The relation between volume and moles is plotted under constant pressure and temperature (option a).

Step 1: Understanding the Context

In this problem, we need to identify the condition under which the relationship between volume ( V ) and number of moles ( V ) in a gas is plotted. The options consider different variables that can be kept constant, such as pressure ( P ), temperature ( T ), and volume ( V ).

Step 2: Ideal Gas Law

The ideal gas law equation is \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature. This equation helps us relate these variables.

Step 3: Analyzing the Options

The relationship between volume and number of moles according to the ideal gas law is directly proportional when pressure and temperature are constant. This means that when \( P \) and \( T \) are held constant, changes in \( n \) cause proportional changes in \( V \).

Step 4: Identifying the Correct Condition

For the relation \( V \propto n \) (volume is directly proportional to the number of moles), conditions must keep pressure and temperature constant. This condition is expressed in option (a).

Step 5: Confirming the Selection

Under constant pressure and temperature, the number of moles and volume relationship is linear, following \( V = \frac{nRT}{P} \). This verifies that option (a), constant \( P \) and \( T \), is correct.

Key Concepts

Volume and Moles Relationship

The relationship between volume and moles in a gas is a fascinating concept in chemistry. It stems from the principle that under certain conditions, the volume of a gas is proportional to the number of moles present in the system. But what are these conditions exactly?
To explore this, let us imagine a scenario where both the pressure and temperature are held constant. In this setting, when we increase the number of moles of gas, we observe that the volume also increases proportionally. This direct relationship is a fundamental principle in understanding the behavior of gases.

• If we double the number of moles of gas, the volume doubles as well, assuming constant pressure and temperature.
• This is important in practical applications like blowing up balloons or filling tires, where adding more gas equates to more volume.

In essence, this relationship underscores how gases behave predictably under controlled conditions, highlighting the intrinsic connection between volume and the number of moles.

Ideal Gas Equation

The ideal gas equation, often represented as \( PV = nRT \), is a cornerstone of chemistry that links several key variables: pressure \( P \), volume \( V \), number of moles \( n \), the ideal gas constant \( R \), and temperature \( T \).
This equation acts as a bridge to understanding how gases react to changes in these variables. For instance, the equation reveals that by manipulating one variable while keeping others constant, we can predict changes in the gas's behavior.

• \( P \) and \( V \) are directly related to \( n \) and \( T \). When one changes, the others must adjust to maintain the balance expressed by the equation.
• The ideal gas constant \( R \) provides a standardized way to understand these changes regardless of the specific gas involved.

In practical terms, the ideal gas law not only helps in solving theoretical problems but also has real-world applications, such as in calculating the amount of fuel needed for a rocket launch or determining the right conditions for chemical reactions in industries.

Direct Proportionality of Volume and Moles

Direct proportionality is a key concept when examining the relationship between gas volume and moles. This idea means that if we were to graph the volume against the number of moles, we would see a straight line passing through the origin, indicating a proportional relationship.
This is vividly demonstrated through the concept where keeping pressure and temperature constant ensures that volume changes as a direct consequence of changes in moles.

• An increase in moles results in an increase in volume; similarly, a decrease in moles leads to a decrease in volume.
• Mathematically, this is represented by \( V = k \times n \), where \( k = \frac{RT}{P} \) denotes a constant when \( P \) and \( T \) remain unchanged.

This simple relationship allows scientists and engineers to predict and manipulate the conditions needed for various processes, from simple laboratory experiments to large-scale industrial operations. Understanding this direct proportionality is crucial for anyone studying or working with gases, as it forms the basis for more complex concepts and applications.