Question

Sketch a graph of pressure versus volume, assuming temperature is constant. Label the vertical axis \(\mathrm{P}\) and the horizontal axis V.

Short answer

The graph of pressure versus volume at constant temperature is a hyperbola, showing an inverse relationship according to Boyle's Law.

Step-by-step solution

Understand the Relationship

At constant temperature, pressure and volume follow the relationship known as Boyle's Law, which states that \( PV = k \), where \( k \) is a constant. Therefore, pressure \( P \) is inversely proportional to volume \( V \). As volume increases, pressure decreases and vice versa, forming a hyperbolic graph.

Define the Axes

Label the vertical axis as \( \mathrm{P} \) (Pressure) and the horizontal axis as \( \mathrm{V} \) (Volume). This defines the graph where pressure is plotted against changes in volume.

Plot the Curve

Start by selecting several values of volume and calculate the corresponding pressure using the equation \( P = \frac{k}{V} \). You do not need to know \( k \) for sketching, just understand it changes the scale, not the shape. For example, if \( V = 1 \), \( 2 \), \( 3 \), the respective pressures \( P \) will be \( k \), \( \frac{k}{2} \), \( \frac{k}{3} \), etc. Plot these points on the graph.

Draw the Graph

Connect the plotted points to form a smooth curve which will be downward-sloping, approaching both axes but never touching them. This curve represents the inverse relationship indicated by Boyle's Law.

Key concepts

Pressure-Volume Relationship

The relationship between pressure and volume is one of the cornerstone concepts in physical chemistry. Boyle's Law succinctly describes this relationship, stating that for a given mass of gas at constant temperature, the product of the pressure and volume is a constant value. This can be expressed mathematically as \( PV = k \), where \( P \) is the pressure, \( V \) is the volume, and \( k \) is a constant.

This means if you were to graph pressure against volume, you would not see a straight line. Instead, you would see a curve known as a hyperbola. This occurs because as volume expands, pressure compresses (and vice versa), due to the fact that their product must always equal the same constant. Such a graph becomes a tangible visualization of how pressure decreases as volume grows, staying true to the inverse nature of their relationship.

In experiments, this principle holds true as long as the temperature remains unchanged. By maintaining this constant, we can clearly observe the elegant balance between pressure and volume.

Constant Temperature

A key condition for observing Boyle's Law is maintaining a constant temperature. Imagine this environment like a perfectly controlled room where no heat is lost or gained. This means thermal energy within the gas remains steady, ensuring its kinetic energy stays constant too.

The reason temperature must be constant is due to its influence on gas particles. When temperatures vary, so does the kinetic energy of the particles. If particles move faster (at higher temperatures), they collide with container walls more frequently and with greater force, changing pressure.

• Constant temperature ensures that the pressure of a gas changes only due to volume alterations, not thermal fluctuations.
• This simplifies the experiment to a pure pressure-volume correlation, adhering strictly to Boyle’s Law.

Understanding the essence of constant temperature helps clarify why the graphs of pressure versus volume appear the way they do. Without temperature control, the graph might display a far more complex relationship.

Inverse Proportionality

Inverse proportionality is a fundamental characteristic of the pressure-volume relationship under Boyle's Law. It signifies that as one variable increases, the other decreases in such a way that their product consistently equals the same constant.

In layman's terms, inverse proportionality is like a seesaw: if one side (volume) goes up, the other side (pressure) must come down. This ensures the system remains balanced and the product of pressure and volume remains unchanged.

The mathematical expression, \( P = \frac{k}{V} \), shows how pressure is calculated as the constant divided by volume. Therefore, doubling the volume will halve the pressure, and tripling the volume will reduce pressure to a third.

• This relationship explains the distinct downward-sloping curve observed in a pressure-volume graph.
• It manifests an intuitive and visual confirmation of how gases behave under constant temperature conditions.

Inverse proportionality thus serves as the backbone of understanding how pressure changes with varying volume, tightly knit by the unyielding strings of Boyle's Law.