Question

One mole of an ideal gas, at a temperature of \(0^{\circ} \mathrm{C}\), is confined to a volume of \(1.0 \mathrm{~L}\). The pressure of this gas is a) 1.0 atm. c) \(1 / 22.4\) atm. b) 22.4 atm. d) \(11.2 \mathrm{~atm}\).

Short answer

Answer: The pressure of the gas is approximately 22.4 atm.

Step-by-step solution

Convert the temperature to Kelvin

To find the temperature in Kelvin, we need to add 273.15 to the Celsius temperature: \(0^{\circ} \mathrm{C} + 273.15 = 273.15 \mathrm{K}\).

Identify the given values

We are given the following values: - n (number of moles) = 1 mole - V (volume) = 1.0 L - T (temperature) = 273.15 K

Determine the ideal gas constant in appropriate units

Given that we want our answer in atm and our volume in L, we should use the value of R in the appropriate units: R = 0.0821 \(\dfrac{\mathrm{L~atm}}{\mathrm{mol~K}}\).

Solve for the pressure using the Ideal Gas Law

We can now plug the known values into the Ideal Gas Law equation, \(PV = nRT\), and solve for the pressure, P: \(P = \dfrac{nRT}{V} = \dfrac{(1 \thinspace \mathrm{mole})(0.0821 \thinspace \dfrac{\mathrm{L~atm}}{\mathrm{mol~K}})(273.15 \thinspace \mathrm{K})}{1.0 \thinspace \mathrm{L}}\)

Calculate the pressure

Perform the calculations: \(P = \dfrac{(1)(0.0821)(273.15)}{1} \approx 22.4 \thinspace \mathrm{atm}\) Thus, the pressure of the gas is 22.4 atm, which corresponds to answer (b).

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. In the context of the Ideal Gas Law, thermodynamics explores how temperature, pressure, and volume interact within a gas.

The starting point in our exercise is the temperature of the gas, which must be in Kelvin for the calculations. This is crucial because Kelvin is an absolute temperature scale wherein 0 K is absolute zero, the point where no more thermal energy can be extracted from a system. Converting Celsius to Kelvin is a basic but essential step in thermodynamics, as it ensures the measurements align with the absolute scale used in the Ideal Gas Law.

When solving problems related to gas behavior, understanding that temperature is a measure of the average kinetic energy of the gas particles is vital. An increase in temperature translates to faster-moving particles, which can result in increased pressure if the volume is held constant, as stated by Gay-Lussac's Law, one of the gas laws that merge into the Ideal Gas Law.

Gas Pressure

Gas pressure is a measure of the force exerted by gas particles as they collide with the walls of their container. This force is distributed across the surface area of the container's interior. When discussing gas pressure in relation to the Ideal Gas Law, it's essential to recognize that it is one of the core variables that can change the state of the gas.

In the exercise, we are determining the pressure exerted by one mole of gas in a one-liter container at a given temperature. The pressure is calculated using the Ideal Gas Law by rearranging the formula to solve for it.

Understanding Pressure in the Context of the Ideal Gas Law

It is significant to note that as temperature increases, for a fixed volume, so does pressure, which is described by Amontons's Law, another fundamental component of thermodynamics that feeds into the Ideal Gas Law equation. Our example indicated a surprisingly high-pressure value, which could be attributed to the gas being contained in a fixed and relatively small volume.

Molar Volume

Molar volume is the volume occupied by one mole of a substance (gas, liquid, or solid) at a given temperature and pressure. The concept of molar volume is directly tied to Avogadro's Law, which states that equal volumes of all gases, at the same temperature and pressure, have the same number of molecules.

This concept is particularly handy when dealing with ideal gases, as it allows for the creation of a standard condition, known as standard temperature and pressure (STP), which is typically 0°C and 1 atm for gas measurements. At STP, one mole of an ideal gas occupies 22.4 liters. However, in the exercise context, we are not at STP, as the pressure is significantly higher.

The Implications of Varying Molar Volume

The molar volume can inform us about the density of the gas and its behavior under different conditions. By incorporating the molar volume into the Ideal Gas Law calculations, we acknowledge that gases can expand to fill their containers, and that the space each mole occupies is dependent on the temperature and pressure it is subjected to. The exercise demonstrates a case where a mole of gas occupies a much smaller volume than it would at STP, leading to a proportionate increase in pressure.