Question

If two moles of an ideal gas at a temperature \(546 \mathrm{~K}\), occupy a volume of \(44.8\) litres its pressure must be (a) \(4 \mathrm{~atm}\) (b) \(3 \mathrm{~atm}\) (c) \(2 \mathrm{~atm}\) (d) 1 atm

Short answer

The pressure is (c) 2 atm.

Step-by-step solution

Understand the Ideal Gas Law

The ideal gas law formula is given by \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles of the gas, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

Identify the Known Values

From the problem, we know the following:- Number of moles, \( n = 2 \) moles- Temperature, \( T = 546 \) K- Volume, \( V = 44.8 \) litres- The ideal gas constant, \( R = 0.0821 \) L atm/mol K

Rearrange the Ideal Gas Law

To solve for pressure \( P \), rearrange the ideal gas law equation to \( P = \frac{nRT}{V} \). This will allow us to plug in the known values to find the pressure.

Substitute Known Values and Calculate

Substitute the known values into the rearranged equation:\[ P = \frac{(2 \, \text{mol})(0.0821 \, \text{L atm/mol K})(546 \, \text{K})}{44.8 \, \text{L}} \]Now, calculate the pressure by completing the multiplication and division.

Perform the Calculation

Calculate the numerator: \( 2 \times 0.0821 \times 546 = 89.6764 \)Now divide by the volume:\( P = \frac{89.6764}{44.8} \approx 2.002 \) atm

Interpret the Result

The pressure \( P \) calculated is approximately \( 2 \) atm. This corresponds with option (c) given in the problem.

Key concepts

Moles

In chemistry, the term "moles" is used to express amounts of a chemical substance. Specifically, a mole is an amount of substance that contains as many elementary entities (such as atoms, molecules, ions, or electrons) as there are atoms in 12 grams of pure carbon-12. This number is described by Avogadro's number, which is approximately \( 6.022 \times 10^{23} \).

Knowing the number of moles of a substance is essential when using the Ideal Gas Law. It helps in determining the amount of gas present in a sample to predict how it will behave under certain conditions. In the given exercise, we have 2 moles of an ideal gas.

Here's how the concept of moles is integrated into the Ideal Gas Law:

• The symbol \( n \) represents the moles in the equation \( PV = nRT \).
• The amount of moles is needed for calculations related to gas reactions and pressure calculations.

Moles allow us to relate the mass of a substance to the number of particles present, which is crucial for understanding chemical reactions and gas behaviors.

Temperature

Temperature plays a critical role in gas behavior. It is a measure of how hot or cold something is and is directly related to the kinetic energy of the gas particles. In terms of the Ideal Gas Law, temperature must be measured in Kelvin for consistent calculations because Kelvin starts at absolute zero, the point where all molecular motion stops.

Converting temperature into Kelvin is straightforward; you just add 273.15 to the Celsius temperature. In our exercise, the temperature is provided as \( 546 \) K, indicating that the gas particles are moving at a certain average speed, affecting the volume and pressure.

• At higher temperatures, gas particles move faster, causing an increase in pressure if the volume is constant.
• Temperature affects the energy within the gas, influencing how the gas will expand or be compressed under given conditions.

Remember, using Kelvin helps maintain uniformity in gas law calculations, avoiding negative temperatures that can occur in Celsius.

Ideal Gas Constant

The Ideal Gas Constant, represented by \( R \), is a crucial part of the Ideal Gas Law equation, \( PV = nRT \). The constant \( R \) is used to make the units consistent across pressure, volume, temperature, and moles. Its value varies depending on the units used for pressure and volume.

For calculations using pressure in atmospheres and volume in liters, \( R \) is typically \( 0.0821 \, \text{L atm/mol K} \). This ensures that when the variables from the equation are substituted, the units align properly to calculate pressure accurately.

Key points about the Ideal Gas Constant include:

• It ensures uniformity in the ideal gas equation, allowing for accurate computations.
• Different problems may require different expressions of \( R \) depending upon the units used, such as \( 8.314 \, \text{J/mol K} \) for joules.

Familiarity with the Ideal Gas Constant helps in avoiding errors and achieving precise predictions of gas behavior.

Pressure Calculation

Pressure is the force exerted by the gas per unit area. In the context of the Ideal Gas Law, calculating the pressure involves knowing the moles of gas, the temperature, the volume, and using the ideal gas constant.

The formula to find pressure is derived from the Ideal Gas Law as \( P = \frac{nRT}{V} \). You simply plug in the values you have for number of moles \( n \), temperature \( T \), the Ideal Gas Constant \( R \), and volume \( V \) to find the pressure of the gas.

In the exercise provided, through this calculation:

• The moles \( n = 2 \).
• Temperature \( T = 546 \text{ K} \).
• The Ideal Gas Constant \( R = 0.0821 \text{ L atm/mol K} \).
• Volume \( V = 44.8 \text{ L} \).

By performing multiplication and division as shown in the step-by-step solution, we find that \( P = \frac{89.6764}{44.8} \approx 2.002 \) atm.

This result tells you that under the given conditions, the gas exerts a pressure of approximately 2 atm, fitting the predicted scenario.