Understanding Boyle's Law

Boyle's Law is a fundamental principle in chemistry and physics that describes the relationship between the pressure and volume of a gas. It is one of the key gas laws, along with Charles's Law and Gay-Lussac's Law, which form the basis of the Ideal Gas Law.

What Boyle's Law States

Boyle's Law specifically states that for a fixed amount (or mass) of gas, held at a constant temperature, the pressure exerted by the gas is inversely proportional to the volume it occupies.

In simpler terms, this means that if you increase the pressure on a gas while keeping its temperature the same, its volume will decrease. Conversely, if you decrease the pressure, the volume will increase. The pressure and volume change in opposite directions.

Mathematical Expression of Boyle's Law

The inverse relationship can be expressed mathematically as:

\( P \propto \frac{1}{V} \)

This proportionality holds true under the condition that the temperature (\(T\)) and the amount of gas (\(n\), typically moles or mass) are kept constant.

To turn this proportionality into an equality, we introduce a constant. Multiplying both sides by \(V\), we get:

\( PV = \text{constant} \)

This equation shows that for a fixed mass of gas at constant temperature, the product of the pressure and volume is always a constant value.

If a gas changes from an initial state (\(P_1, V_1\)) to a final state (\(P_2, V_2\)) while temperature and mass are constant, Boyle's Law implies:

\( P_1V_1 = P_2V_2 \)

Analyzing the Options Based on Boyle's Law

The question asks which expression is a constant according to Boyle's law. Let's look at the given options:

• Option 1: P/T - This expression involves both pressure and temperature. While temperature is related to gas behavior, the ratio P/T is not the constant described by Boyle's Law, which requires temperature to be constant. This ratio is related to other gas laws or combined gas laws.
• Option 2: V/T - This expression involves both volume and temperature. The ratio V/T is the constant described by Charles's Law, which states that for a fixed mass of gas at constant pressure, the volume is directly proportional to the absolute temperature (\(V \propto T\), or \(V/T = \text{constant}\)). This is not Boyle's Law.
• Option 3: PV - This expression is the product of pressure (\(P\)) and volume (\(V\)). As derived from the mathematical statement of Boyle's Law (\(PV = \text{constant}\)) under constant temperature and mass, the product of pressure and volume is indeed a constant.
• Option 4: PT - This expression is the product of pressure and temperature. This combination does not represent a constant according to Boyle's Law or other standard gas laws under the conditions specified by Boyle's Law (constant temperature).

Based on the analysis of Boyle's Law and its mathematical form, the constant is the product of pressure and volume.

Conclusion

According to Boyle's law, with temperature and the amount of gas held constant, the product of pressure (\(P\)) and volume (\(V\)) is a constant.

Revision Table: Key Gas Laws Constants

This table summarizes the constant relationships for the main gas laws, which is helpful for remembering what remains constant under different conditions.

Additional Information: Connecting Gas Laws

Boyle's Law, Charles's Law, and Gay-Lussac's Law can be combined into the Combined Gas Law, which states that for a fixed amount of gas:

\( \frac{PV}{T} = \text{constant} \)

This shows how pressure, volume, and temperature are all related. If any one of these is held constant, the combined gas law simplifies to one of the individual laws:

• If \(T\) is constant, \(\frac{PV}{\text{constant}} = \text{constant}\) which means \(PV = \text{constant}\) (Boyle's Law).
• If \(P\) is constant, \(\frac{\text{constant} \times V}{T} = \text{constant}\) which means \(\frac{V}{T} = \text{constant}\) (Charles's Law).
• If \(V\) is constant, \(\frac{P \times \text{constant}}{T} = \text{constant}\) which means \(\frac{P}{T} = \text{constant}\) (Gay-Lussac's Law).

The most comprehensive relation for ideal gases is the Ideal Gas Law:

\( PV = nRT \)

Where \(n\) is the number of moles of gas and \(R\) is the ideal gas constant. If \(n\) and \(T\) are constant, then \(nRT\) is a constant, leading back to \(PV = \text{constant}\), which is Boyle's Law.