Question

What are the conditions under which the relation between volume \((\mathrm{V})\) and number of moles \((\mathrm{n})\) 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 \(\mathrm{V}\)

Short answer

The relation between \( V \) and \( n \) is plotted under constant pressure \( P \) and temperature \( T \) (option a).

Step-by-step solution

Understand the Problem

We need to determine which conditions allow us to plot the relationship between volume \(V\) and number of moles \(n\) of a gas.

Recall the Ideal Gas Law

The ideal gas law is given by \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature.

Isolate the Variables of Interest

To plot \( V \) against \( n \), we need to have \( V = f(n) \). By rearranging the ideal gas law, \( V = \frac{nRT}{P} \). Thus, \( V \) is directly proportional to \( n \) as long as \( R \), \( T \), and \( P \) are constant.

Determine Constant Conditions

For \( V \) to be directly proportional to \( n \), \( P \) and \( T \) must remain constant. This is because the equation \( V = \frac{nRT}{P} \) only holds if \( R \), \( T \), and \( P \) do not change.

Key concepts

Volume and Moles Relationship

The relationship between volume and moles in a gas is one of the fundamental concepts that can be derived from the Ideal Gas Law. According to the Ideal Gas Law, the equation \( PV = nRT \) brings into light that, at a given pressure \( P \) and temperature \( T \), the volume \( V \) of a gas is directly proportional to the number of moles \( n \). This means, larger the amount of the gas, greater will be its volume under the same conditions of pressure and temperature.

To explore this further:

• If the number of moles of the gas doubles, the volume also doubles, provided pressure and temperature remain unchanged.
• This direct proportionality can be expressed mathematically as \( V \propto n \) or \( V = kn \), where \( k \) is a constant that depends on \( R \), \( T \), and \( P \).

Understanding this concept of proportionality is pivotal for comprehending how gases behave under various conditions and is applicable in various scientific and industrial processes.

Gas Laws

The Ideal Gas Law encapsulates the fundamental principles expressed in several individual gas laws: Boyle's Law, Charles's Law, and Avogadro's Law. It is essential for understanding how gases will react to changes in physical conditions.

• Boyle's Law states that the volume of a gas is inversely proportional to its pressure when the number of moles and temperature are constant. This is crucial in scenarios involving compression of gases.
• Charles's Law holds that the volume of a gas is directly proportional to its temperature, given that the pressure and number of moles remain constant. This explains why balloons expand as they heat up.
• Avogadro's Law tells us that the volume of a gas is directly proportional to the number of moles when pressure and temperature are constant. This law forms the basis for understanding the concept of molar volume.

The Ideal Gas Law, \( PV = nRT \), consolidates these laws into one powerful equation and plays a vital role in chemical calculations and predictions regarding gas behavior.

Constant Pressure and Temperature

To effectively study the volume and moles relationship in a gas, maintaining constant pressure and temperature is key. This condition allows scientists to isolate and observe the behavior of gases without external influences from varying pressure or temperature.

By keeping pressure and temperature stable:

• Any changes in the gas's volume can be directly attributed to changes in the number of moles.
• This facilitates a clear understanding of how gases expand or contract with the addition or removal of moles.

In practical applications, achieving constant pressure and temperature might involve using closed, controlled environments such as reaction vessels or syringes. Researchers and engineers utilize this principle to predict and manage reactions in chemical industries, laboratories, and even in environmental studies. Understanding these controlled conditions is underlying many pivotal discoveries in chemistry and physics.