Question

. Under what circumstances is the following statement correct? Equal molar amounts of two different gases at the same temperature, placed in containers of equal volume, have the same pressure.

Short answer

Equal pressures when molar amount, temperature, and volume are the same.

Step-by-step solution

- Understand the Ideal Gas Law

The Ideal Gas Law states: \[ PV = nRT \]Where:- \(P\) is the pressure,- \(V\) is the volume,- \(n\) is the number of moles,- \(R\) is the universal gas constant,- \(T\) is the temperature.

- Analyzing the Given Conditions

Given that the gases have equal molar amounts (i.e., the same number of moles, \(n\)) and are at the same temperature (\(T\)). They are also placed in containers of equal volume (\(V\)).

- Applying the Ideal Gas Law to Each Container

For the first gas, the pressure can be expressed as:\[ P_1V = nRT \]For the second gas, the pressure can be expressed as:\[ P_2V = nRT \]Since \(V\), \(n\), \(R\), and \(T\) are the same for both gases, the pressures \(P_1\) and \(P_2\) must also be equal.

- Conclusion

Under the conditions that all other variables (molar amount, temperature, volume, and the gas constant) are the same, the pressures of the two gases will be equal.

Key concepts

Understanding Pressure

Pressure is a force exerted by gas particles when they collide with the walls of their container. In the context of the Ideal Gas Law, it's denoted by the symbol \(P\). Pressure is crucial because it tells us how much force gas particles are applying. In daily life, we measure pressure in different units like atmospheres (atm), pascals (Pa), or pounds per square inch (psi). Higher pressure means particles collide more frequently and forcefully against the container walls.

Defining Volume

Volume (\(V\)) is the amount of space a gas occupies. In the Ideal Gas Law, volume is crucial as it helps to determine how much space gas particles have to move around. The volume of a gas can be measured in liters (L), cubic meters (m³), or cubic centimeters (cm³). Different volumes under constant temperature and pressure will hold different amounts of gas. In our exercise, the volume is the same for both gases, meaning they have the same amount of space to move around.

Exploring Moles

A mole (\(n\)) is a measurement of the amount of substance and is part of the Ideal Gas Law. One mole contains approximately 6.022 \times 10^{23} particles (Avogadro's number). It's a way to count atoms or molecules. In our scenario, equal molar amounts mean both containers have the same number of gas molecules. This is why the pressure relates directly to the number of moles when temperature and volume remain constant.

The Universal Gas Constant

The universal gas constant (\(R\)) is a constant that appears in the Ideal Gas Law. Its value is approximately 8.314 J/(mol·K). \(R\) provides a relationship factor between the pressure, volume, moles, and temperature of a gas. It helps unify these variables into one equation. Different units of \(R\) are used depending on the units of pressure and volume in a given context. For instance, it's essential when calculating how gases behave under varying conditions.

Importance of Temperature

Temperature (\(T\)) is a measure of the average kinetic energy of gas particles. In the Ideal Gas Law, it's expressed in Kelvin. It directly influences the behavior of gases. Higher temperatures mean gas particles move more energetically, increasing pressure if volume and the number of moles remain constant. In the exercise, both gases are at the same temperature, ensuring they have equivalent kinetic energy. This consistency helps in concluding that pressures remain equal when other factors are constant.