Question

HOW DO YOU SEE IT? For a quantity of gas at a constant temperature, the pressure \(P\) is inversely proportional to the volume \(V\) . What is the limit of \(P\) as \(V\) approaches 0 from the right? Explain what this means in the context of the problem. Graph cannot copy

Short answer

The limit of pressure \(P\) as volume \(V\) approaches 0 from the right is infinity (∞). This means if the volume of the gas becomes smaller and smaller, the pressure will become larger and larger, theoretically becoming infinite if the volume reaches zero. However, in reality, the gas would condense into a liquid or solid before the pressure could reach infinity.

Step-by-step solution

Understand the concept of 'Inversely Proportional'

In the context of a gas, when we say that 'Pressure is inversely proportional to Volume', we mean that as the volume gets smaller, the pressure gets larger and vice versa. Formally, this relationship can be stated as \(P = k/V\), where \(k\) is a constant of proportionality. This means that the product of \(P\) and \(V\) will always be the same constant \(k\).

Apply the concept of limit

The question asks for the limit of \(P\) as \(V\) approaches 0 from the right. In mathematical notation, this is written as \(\lim_{V \to 0^{+}} P\). Here, \(0^{+}\) signifies that we are only considering volumes just greater than zero.

Interpret the result

As volume \(V\) approaches 0 from the right, implying that you're making \(V\) smaller and smaller but still greater than zero, the value of \(P\) becomes larger and larger. This is because, with \(P = k/V\), as you're decreasing \(V\), you're increasing \(P\). So, the limit of \(P\) as \(V\) approaches 0 is theoretically infinity (∞).

Key concepts

Limit

Limits help us understand what happens to a function as the input approaches a particular value. In this problem, we're looking at what happens to pressure as volume moves towards zero. When the volume is almost zero, the pressure becomes extremely high. However, we don't reach absolute zero in practice. Limits let us talk about this behavior theoretically.

For example, if we look at the limit of pressure as volume approaches zero from the right, it is written mathematically as \( \lim_{V \to 0^{+}} P \). Here, \( 0^{+} \) means we consider volumes slightly greater than zero. This limit shows that the pressure tends to become infinitely large as we reduce the volume.

Pressure and Volume Relationship

The relationship between pressure \( P \) and volume \( V \) is an example of Boyle's Law, where \( P \) is inversely proportional to \( V \). This relationship is incredibly important in understanding how gases behave under different conditions.

When we say pressure is inversely proportional to volume, it means that if \( V \) decreases, \( P \) increases. Mathematically, this is represented by \( P = \frac{k}{V} \). Here, \( k \) is a constant value that does not change even as pressure and volume do.

This principle is fundamental in many real-life applications such as airbags, syringes, and even breathing, where changes in volume can drastically affect pressure.

Constant of Proportionality

In mathematics and science, the constant of proportionality \( k \) bridges the relationship between two quantities. In our exercise, this constant links pressure and volume in the equation \( P = \frac{k}{V} \).

The key point is that for a fixed amount of gas under constant temperature, \( k \) remains unchanged. Regardless of how \( P \) and \( V \) change individually, their product \( PV \) always equals the constant \( k \).

• Helps maintain the balance between inversely related quantities.
• Allows us to predict one quantity, given the other.
• Is fundamental in many equations and scientific laws.

Understanding how to use this constant helps in solving math problems and understanding physical phenomena.

Limits in Calculus

Calculus provides us with tools like limits to tackle problems involving change. Limits express what happens to a function as we input values approaching a particular point. This is essential in situations where we cannot calculate directly due to divisions by zero or when values are undefined.

For example, in the exercise, \( \lim_{V \to 0^{+}} P = \infty \) demonstrates how calculus helps us deal with scenarios where volume approaches zero but never actually becomes zero. Understanding limits means understanding how functions behave and can be applied in many areas, from physics to economics to engineering. By mastering limits, one gains a clearer picture of how systems behave at extreme values.