Understanding Gas Pressure and Volume at Constant Temperature

The question asks what happens to the pressure of a gas when its volume is reduced while keeping the "heat constant". In the context of ideal gases and simple thermodynamic processes, "constant heat" often implies a constant temperature process, also known as an isothermal process. We can analyze this situation using the fundamental gas laws.

Boyle's Law: The Pressure-Volume Relationship

For a fixed amount of gas kept at a constant temperature, there is a specific relationship between its pressure and volume. This relationship is described by Boyle's Law.

• Boyle's Law states that the pressure exerted by a fixed mass of an ideal gas is inversely proportional to its volume, provided the temperature is kept constant.
• Mathematically, this can be written as:

\text{Pressure} \propto \frac{1}{\text{Volume}} \quad \text{at constant temperature and mass}

Or, more formally:

P \propto \frac{1}{V} \quad (\text{when } T \text{ and } n \text{ are constant})

This inverse relationship means that if the volume increases, the pressure decreases, and if the volume decreases, the pressure increases, as long as the temperature and the amount of gas remain unchanged.

Applying Boyle's Law to the Question

The question states that the volume of the gas is reduced, and the process occurs under "constant heat" (implying constant temperature). According to Boyle's Law, if the temperature and the amount of gas are constant, pressure and volume are inversely proportional.

• Initial state: Let the initial pressure be \(P_1\) and the initial volume be \(V_1\).
• Final state: Let the final pressure be \(P_2\) and the final volume be \(V_2\).
• Given: The volume is reduced, which means \(V_2 < V_1\).
• According to Boyle's Law: \(P_1V_1 = P_2V_2\) (constant value).
• We can rearrange this to find the relationship between \(P_2\) and \(P_1\): \(P_2 = P_1 \times \frac{V_1}{V_2}\).
• Since \(V_2 < V_1\), the ratio \(\frac{V_1}{V_2}\) is greater than 1 (\(\frac{V_1}{V_2} > 1\)).
• Therefore, \(P_2 > P_1\). The final pressure is greater than the initial pressure.

This confirms that when the volume of a gas is reduced at constant temperature, its pressure increases.

Reasoning from Kinetic Theory

The increase in pressure when volume is reduced at constant temperature can also be understood from the kinetic theory of gases.

• At constant temperature, the average kinetic energy (and thus the average speed) of the gas molecules remains constant.
• Pressure is caused by the collisions of gas molecules with the walls of the container.
• When the volume of the container is reduced, the molecules have less distance to travel between collisions with the walls.
• This leads to more frequent collisions with the walls per unit time.
• More frequent collisions mean a greater force exerted on the walls over time, resulting in increased pressure.

Based on Boyle's Law and the kinetic theory explanation, reducing the volume of a gas at constant temperature leads to an increase in its pressure.

Revision Table: Key Gas Laws Concepts

Additional Information: Isothermal Processes

The scenario described in the question is an example of an isothermal process, where the temperature of the system remains constant throughout. For an ideal gas undergoing an isothermal process:

• The internal energy of the gas remains constant (\(\Delta U = 0\)) because the internal energy of an ideal gas depends only on its temperature.
• According to the First Law of Thermodynamics (\(\Delta U = Q - W\)), where \(Q\) is the heat added to the system and \(W\) is the work done by the system: Since \(\Delta U = 0\), \(Q = W\).
• If the volume is reduced, work is done on the gas (compression), meaning \(W\) is negative. The work done by the gas is \(-W\), which is positive. So, in compression, \(W\) (by gas) is negative, thus \(Q\) must be negative. This means heat is removed from the gas to keep the temperature constant.
• Conversely, if the volume increases (expansion), work is done by the gas (\(W\) is positive), and heat must be added to the gas (\(Q\) is positive) to maintain constant temperature.

So, while the question mentions "constant heat", in a rigorous thermodynamic sense for volume reduction to maintain constant temperature, heat must actually be *removed* from the system. However, in many introductory contexts, "constant heat" is used loosely to imply "constant temperature" in relation to the ideal gas laws like Boyle's Law.