Question

Suppose 5.0 moles of an ideal monatomic gas expand at a constant temperature of \(22^{\circ} \mathrm{C}\) from an initial volume of \(2.0 \mathrm{~m}^{3}\) to \(8.0 \mathrm{~m}^{3}\) a) How much work is done by the gas? b) What is the final pressure of the gas?

Short answer

Answer: The work done by the gas during the isothermal expansion is approximately 9988.05 Joules, and the final pressure of the gas after the expansion is approximately 1531.18 Pascal.

Step-by-step solution

Identify the values given in the problem

For this problem, you have been given: 1. Initial volume (V1) = 2.0 m³ 2. Final volume (V2) = 8.0 m³ 3. Number of moles (n) = 5.0 moles 4. Temperature (T) = 22°C (we need to convert this to Kelvin)

Convert the temperature to Kelvin

The Celsius temperature should be converted to Kelvin, which can be done using the following formula: T(K) = T(°C) + 273.15 So, T(K) = 22 + 273.15 = 295.15 K

Calculate the work done during the isothermal expansion

Using the formula of work done during an isothermal expansion: W = nRT * ln(V2/V1) We need to find the value of R (Universal gas constant) which is in the appropriate units. Here, the units are moles, Kelvin, and m³. So, we use the value of R = 8.314 J/(mol*K). Now, plug in the values: W = (5.0 moles) * (8.314 J/(mol*K)) * (295.15 K) * ln(8.0 m³ / 2.0 m³) W ≈ 9988.05 J Therefore, the work done by the gas during the isothermal expansion is approximately 9988.05 Joules.

Calculate the initial pressure of the gas

Use the ideal gas law to find the initial pressure of the gas. Rearrange the ideal gas law formula to solve for the initial pressure (P1): P1 = nRT / V1 Plugging in the values: P1 = (5.0 moles) * (8.314 J/(mol*K)) * (295.15 K) / (2.0 m³) P1 ≈ 6124.73 Pa The initial pressure of the gas is approximately 6124.73 Pascal.

Calculate the final pressure of the gas after the expansion

We can use the ideal gas law again to find the final pressure (P2): P2 = nRT / V2 Plugging in the values: P2 = (5.0 moles) * (8.314 J/(mol*K)) * (295.15 K) / (8.0 m³) P2 ≈ 1531.18 Pa Therefore, the final pressure of the gas after the isothermal expansion is approximately 1531.18 Pascal.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry and physics that describes the behavior of an ideal gas. The equation is written as \( PV = nRT \), where:

• \( P \) is the pressure of the gas
• \( V \) is the volume of the gas
• \( n \) is the amount of substance in moles
• \( R \) is the ideal gas constant, approximately 8.314 J/(mol*K)
• \( T \) is the temperature in Kelvin

This law helps predict how a gas will respond to changes in temperature, pressure, and volume, provided the gas behaves ideally. In the exercise above, we used the ideal gas law to calculate the initial and final pressures during the expansion process. Understanding this equation is crucial as it establishes the relationship between four significant properties of gas, effortlessly linking them together.

Work Done

In the context of thermodynamics, work done by a gas during expansion or compression is an important concept. Specifically, for an isothermal expansion, the formula to calculate the work done is:\[ W = nRT \ln\left( \frac{V_2}{V_1} \right) \]Where:

• \( W \) is the work done by the gas
• \( n \) is the number of moles of the gas
• \( R \) is the gas constant
• \( T \) is the temperature in Kelvin
• \( V_2 \) and \( V_1 \) are the final and initial volumes, respectively

This formula emphasizes how the work done depends on the change in volume and the constant temperature. In the solution, this was used to calculate the work done as the gas expanded from a volume of 2.0 m³ to 8.0 m³, resulting in work of approximately 9988.05 Joules. It's essential to understand that during an isothermal process, temperature remains constant even though work is being done.

Moles of Gas

Moles of gas is a measure that quantifies the amount of a substance. In chemistry, it's often used to express how many molecules or atoms are present in a given sample. The concept of 'mole' provides a bridge between the atomic scale and the macroscopic quantities we can measure.In the given exercise, 5.0 moles of ideal monatomic gas were considered. Moles are particularly useful because they allow us to use the ideal gas law effectively. Knowing the number of moles helps in directly calculating other properties of the gas, such as pressure or work done during expansion, as it is directly included in the ideal gas equation \( PV = nRT \). This parameter, along with others, facilitated solving for pressures and work done in the context of the problem.

Pressure Calculation

Pressure calculation in gases involves relating pressure to other variables such as volume, temperature, and number of moles using formulas like the ideal gas law. Pressure ( \( P \)) expresses the force applied per unit area. Understanding pressure is crucial as it is a fundamental variable in thermodynamic equations.In the exercise, we performed two pressure calculations during different gas states. We first used the ideal gas law to determine the initial pressure \( P_1 \), calculated with:\[ P_1 = \frac{nRT}{V_1} \]And post-expansion pressure \( P_2 \):\[ P_2 = \frac{nRT}{V_2} \]Each resulting in values of approximately 6124.73 Pa and 1531.18 Pa, respectively. Calculating pressure ensures the accurate understanding of how the gas behaves when variables change, reflecting its dynamic nature in processes like isothermal expansions.