Question

A curve drawn at constant temperature is called an isotherm. This shows relationship between (1) \(P\) and \(\frac{1}{V}\) (2) \(P V\) and \(V\) (3) \(P\) and \(V\) (4) \(V\) and \(\frac{1}{P}\)

Short answer

The correct answer is (3) \( P \) and \( V \).

Step-by-step solution

Understand the Definition

A curve drawn at constant temperature represents an isotherm. To determine the relationship it shows, recall properties of gases under constant temperature, which relate to the ideal gas law.

Review the Ideal Gas Law

The ideal gas law is given by \[ P V = nRT \]where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature. For an isotherm, temperature \( T \) is constant, thus \( nRT \) is a constant.

Isolate Relevant Relationship

Given constant \( nRT \), we get:\[ P V = \text{constant} \]Which indicates an inverse relationship between \( P \) and \( V \).

Identify the Correct Choice

Options (2) and (3) involve the correct variables. Since we have an inverse relationship, the correct answer is (3) \( P \) and \( V \).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in the study of gases. This equation is written as: \[ PV = nRT \] where:

• P
stands for pressure,
• V
signifies volume,
• n
denotes the number of moles of the gas,
• R
is the universal gas constant,
• and T
represents temperature.

This relationship illustrates how pressure, volume, and temperature of an ideal gas are interconnected. Under the ideal gas conditions, meaning no intermolecular forces and complete elastic collisions, this law holds accurately.
For an isotherm, the temperature T stays constant. This simplifies the Ideal Gas Law to demonstrate other relationships, which we will discuss next.

Inverse Relationship

When analyzing gases, an inverse relationship between two properties means that as one increases, the other decreases. With constant temperature, this relationship is best seen by rearranging the Ideal Gas Law: \[ PV = k \] Here, k is a constant since temperature (T), the number of moles (n), and the gas constant (R) do not change.
This rearrangement indicates that pressure (P) is inversely proportional to volume (V).
So, if the volume increases, pressure must decrease to maintain the product PV constant, and vice versa. Many gas behaviors depend on this simple yet critical relationship.
The inverse relationship is a key characteristic of an isotherm.

Constant Temperature Behavior

An Isotherm represents the behavior of a gas under constant temperature. It is a curve that shows how gas properties change while maintaining the same temperature.
In the equation: \[ PV = nRT \]
Since temperature (T) is constant for an isotherm, it acts as a horizontal constraint. In practice, this means that as the gas expands (volume increases), the pressure decreases, and the other way around, aligning with our previous findings about the inverse relationship.
The graph of an isotherm is often plotted as pressure (P) versus volume (V). The curve shows a hyperbolic shape, highlighting that while one attribute increases, the other decreases proportionally.
Observing these constant temperature curves is crucial in understanding gas laws and their real-world applications, like in engines and natural phenomena.