Question

Boyle's Law For a quantity of gas at a constant temperature, the pressure \(P\) is inversely proportional to the volume \(V\). Find the limit of \(P\) as \(V \rightarrow 0^{+}\).

Short answer

The limit of \(P\) as \(V\) approaches \(0^{+}\) is positive infinity.

Step-by-step solution

Write the Formula

Begin by writing down Boyle's Law formula, \(P = k/V\). This equation states that the pressure is directly proportional to the inverse of the volume.

Substitute \(V\) with \(0^{+}\)

A key concept to understand when computing limits is the idea of approaching a number. We are asked to find the limit as \(V\) approaches zero from the positive side. So, we substitute \(0^{+}\) for \(V\) in the equation resulting in \(P = k/0^{+}\). There's the issue that division by zero is undefined in mathematics.

Understanding division by a very small positive number

When we talk about \(V\) as \(0^{+}\), it doesn't mean it's exactly zero. It refers to a very small positive number that is immeasurably close to zero. As we divide a constant by a number getting smaller and smaller, the result gets larger and larger. Therefore, as \(V\) approaches \(0^{+}\), \(P\) should approach positive infinity.

Handling Inverse Proportionality

Since the pressure and volume are inversely proportional, when the volume tends to zero, the pressure tends to infinity. However, this means that the pressure becomes very high, not literally infinite.