Question

What is the effect on the pressure of a gas if you simultaneously: (a) Halve its volume and double its Kelvin temperature? (b) Double its volume and halve its Kelvin temperature?

Short answer

(a) Pressure is quadrupled; (b) pressure is reduced to one quarter.

Step-by-step solution

Introduction to the Ideal Gas Law

The behavior of a gas can be described using the Ideal Gas Law, given by the equation \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( T \) is the temperature in Kelvin, \( n \) is the number of moles, and \( R \) is the gas constant. For a fixed amount of gas \((n)\), the equation can be rearranged to \( P \propto \frac{T}{V} \), showing that pressure is proportional to temperature and inversely proportional to volume.

Analyze Part (a): Halving the Volume and Doubling the Temperature

Initially, the pressure of the gas is \( P_1 = \frac{nRT_1}{V_1} \). After the changes, the new volume is \( \frac{V_1}{2} \) and the new temperature is \( 2T_1 \). Substituting these into the proportional relationship \( P_2 = \frac{nR(2T_1)}{\frac{V_1}{2}} \), the expression simplifies to \( P_2 = \frac{4nRT_1}{V_1} = 4P_1 \). Thus, the pressure is quadrupled.

Analyze Part (b): Doubling the Volume and Halving the Temperature

Initially, the pressure of the gas is \( P_1 = \frac{nRT_1}{V_1} \). After the changes, the new volume is \( 2V_1 \) and the new temperature is \( \frac{T_1}{2} \). Substituting these into the proportional relationship \( P_2 = \frac{nR(\frac{T_1}{2})}{2V_1} \), the expression simplifies to \( P_2 = \frac{nRT_1}{4V_1} = \frac{P_1}{4} \). Thus, the pressure is reduced to one quarter.

Key concepts

Gas Pressure

Gas pressure is the force that gas molecules exert when they collide with the walls of their container. It depends on several factors, including the number of gas molecules, their speed, and the volume of the gas container. You can think of it as the result of all these tiny collisions.

When the gas is compressed into a smaller space, the molecules have less room to move. This increases the number of collisions against the container's walls, resulting in higher pressure. Conversely, if the volume of the gas expands, the molecules have more space, causing fewer collisions and lower pressure.

According to the Ideal Gas Law, the pressure \( P \) is also directly proportional to the temperature \( T \), meaning if you raise the temperature, the pressure increases too. This is because hotter molecules move faster, exerting more force on the container walls.

Temperature and Volume Relationship

The relationship between temperature and volume of a gas is foundational in understanding gas behavior. According to Charles's Law, the volume of a gas is directly proportional to its temperature, provided the pressure remains constant.

This means that as the temperature of a gas increases, its volume expands if it can do so freely. Why does this happen? An increase in temperature supplies energy to the gas molecules, making them move faster. That's why heating a balloon makes it larger, but cooling it makes it shrink.

• Temperature increase → Volume increase
• Temperature decrease → Volume decrease

This relationship helps explain why weather balloons get larger as they ascend to higher, colder altitudes even though the temperature drops, because the external pressure drops significantly.

Pressure Changes with Temperature

Pressure changes with temperature through a principle visible in the Ideal Gas Law. When a gas is heated, its particles absorb energy, moving faster and thus exerting more pressure on the container’s walls at constant volume.

To better understand this, consider a sealed rigid container. If its temperature is increased, the faster motion of gas molecules bumps up the internal pressure. This is why running an engine or cooking on a stovetop increases the pressure within the appliance if not properly ventilated.

If instead we look at a flexible container, like a balloon, increased temperature may cause both pressure and volume to increase as the balloon expands to accommodate the faster-moving particles:

• Rise in temperature → Rise in pressure (in a rigid container)
• Rise in temperature → Rise in pressure & volume (in a flexible container)

Thus, temperature significantly influences gas pressure, highlighting the need to carefully consider temperature changes in systems involving gases.