Question

Under constant temperature, graph between \(\mathrm{p}\) and \((1 / \mathrm{V})\) is (A) Hyperbola (B) Circle (C) Parabola (D) Straight line

Short answer

The relationship between pressure (\(p\)) and the inverse of volume (\(\frac{1}{V}\)) under constant temperature can be represented by the equation \(p = \frac{k}{V}\), which shows a direct relationship. Thus, the graph between \(p\) and \(\frac{1}{V}\) is a straight line. The correct answer is (D) Straight line.

Step-by-step solution

Boyle's law can be represented by the following equation: \[pV = k\] Where: - \(p\) is the pressure of the gas - \(V\) is the volume of the gas - \(k\) is a constant that depends on the amount of gas and the temperature As the temperature is kept constant, the value of \(k\) is also constant in this case. #Step 2: Rearrange the equation to show the relation between p and 1/V#

Since we need to find the graph between p and (1/V), we'll need to rewrite the equation in terms of these two variables. To do so, we take the inverse of both sides of the equation: \[\frac{1}{p}\frac{1}{V} = \frac{1}{k}\] Multiplying both sides of the equation by \(V\), we get: \[\frac{1}{p} = \frac{V}{k}\] Now, multiplying both sides of the equation by \(p\), we get the final equation: \[p = \frac{k}{V}\] #Step 3: Identify the type of graph between p and 1/V#

The equation that we obtained in the previous step is: \[p = \frac{k}{V}\] From this equation, we can see that the relationship between pressure and the inverse of volume is a direct relationship, as one goes up, the other also goes up. This is a characteristic of a straight line. Hence, the graph between pressure (\(p\)) and the inverse of volume (\(1/V\)) under constant temperature is a straight line. #Step 4: Choose the correct answer#

Based on our analysis, the correct answer is: (D) Straight line

Key concepts

Pressure-Volume Relationship

Boyle's Law plays an important part in understanding the pressure-volume relationship in gases. The law states that for a given mass of an ideal gas, the pressure (\(p\)) and volume (\(V\)) are inversely proportional under constant temperature. This means when the volume of a gas decreases, its pressure increases, and vice versa, when the temperature is held steady.

The mathematical expression of Boyle's Law is given by:

• \[pV = k\]

Where \(k\) is a constant unique to the gas, dependent on its specific conditions like temperature and mass.

This relationship implies that if you were to double the volume (\(V\)), the pressure (\(p\)) would halve, provided the temperature stays the same. Knowing this helps us predict how gas behaves in closed spaces, like balloons or syringes, when the volume is manipulated.

Ideal Gas Law

While Boyle's Law focuses on the pressure-volume relationship, the Ideal Gas Law provides a more comprehensive equation for the state of an ideal gas. It unites pressure, volume, temperature, and the number of particles within a single equation:

• \[pV = nRT\]

Here, \(p\) is the pressure, \(V\) is the volume, \(n\) denotes the amount of gas (in moles), \(R\) is the ideal gas constant, and \(T\) represents the temperature in Kelvins. The beauty of this equation lies in its ability to describe the behavior of an ideal gas under various conditions by holding some variables constant while varying others.

In scenarios where the number of gas particles remains constant and pressure changes with volume at a fixed temperature, the Ideal Gas Law simplifies down to Boyle’s Law. This highlights how individual gas laws are sometimes specific cases of the more general Ideal Gas Law.

Graphical Representation in Physics

Graphing the pressure-volume relationship under constant temperature is a powerful visual tool for physics students. In this context, plotting pressure (\(p\)) against the inverse of volume (\(1/V\)) results in a straight line. This occurs because the mathematical rearrangement of Boyle's equation, \(p = \frac{k}{V}\), reflects a linear equation when considering pressure as the dependent variable and \(1/V\) as the independent variable.

The straight-line graph indicates a direct proportionality between pressure and the inverse of volume. Here are some key points to consider when interpreting such a graph:

• The slope of the line is directly related to the constant \(k\).
• A steeper slope indicates a higher constant, reflecting a stronger pressure response for changes in volume.
• Understanding these graphs helps in visualizing and predicting how gases will behave as the conditions they’re under change.

Graphs provide an intuitive hook for visual learners, making abstract concepts more concrete through visual data.