Question

6.00 liters of a monatomic ideal gas, originally at \(400 . \mathrm{K}\) and a pressure of \(3.00 \mathrm{~atm}\) (called state 1 ), undergo the following processes: \(1 \rightarrow 2\) isothermal expansion to \(V_{2}=4 V_{1}\) \(2 \rightarrow 3\) isobaric compression \(3 \rightarrow 1\) adiabatic compression to its original state Find the pressure, volume, and temperature of the gas in states 2 and \(3 .\) How many moles of the gas are there?

Short answer

Question: Calculate the properties of the gas in state 2 and state 3 after undergoing isothermal expansion, isobaric compression, and adiabatic compression. Also, determine the number of moles of the gas. Answer: The number of moles of the gas is approximately 0.548 moles. In state 2, the properties of the gas are: \(P_2 = 0.750 ~\text{atm}\), \(V_2 = 24.00 ~\text{L}\), and \(T_2 = 400 ~K\). In state 3, the properties of the gas are: \(P_3 = 0.750 ~\text{atm}\), \(V_3 = 12.74 ~\text{L}\), and \(T_3 = 183.11 ~K\).

Step-by-step solution

Given Information

The given information is: - Initial state (state 1): \(P_1 = 3.00 ~\text{atm}\), \(V_1 = 6.00 ~\text{L}\), \(T_1 = 400 ~\text{K}\) - Process 1-2: Isothermal expansion (constant temperature) with \(V_2 = 4V_1\) - Process 2-3: Isobaric compression (constant pressure) - Process 3-1: Adiabatic compression to the original state

Apply general ideal gas law to state 1

To find the number of moles, we will apply the ideal gas law first to the initial state: PV = nRT, and R is the ideal gas constant, 0.0821 \((\text{L} \cdot \text{atm}) / (\text{mol} \cdot \text{K})\) \(3.00 ~\text{atm} \cdot 6.00 ~\text{L} = n \cdot 0.0821 (\text{L} \cdot \text{atm}) / (\text{mol} \cdot \text{K}) \cdot 400. ~\text{K}\) Solving for n, we get: \(n = \frac{3.00 \cdot 6.00}{0.0821 \cdot 400} = \frac{18}{32.84} \approx 0.548 ~\text{moles}\)

Apply ideal gas law to state 2 (isothermal expansion)

Since the process from state 1 to state 2 is isothermal (constant T), the final temperature in state 2 is the same as the initial temperature, i.e., \(T_2 = T_1 = 400 ~K\). Now, we simply apply the ideal gas law in state 2: \(P_2V_2 = nRT_2\) Given \(V_2 = 4V_1 = 4 \times 6.00 ~\text{L} = 24.00 ~\text{L}\), we can solve for \(P_2\). \(P_2 = \frac{nRT_2}{V_2} = \frac{0.548 \times 0.0821 \times 400}{24.00} \approx 0.750 ~\text{atm}\) So, in state 2, \(P_2 = 0.750 ~\text{atm}\), \(V_2 = 24.00 ~\text{L}\), and \(T_2 = 400 ~K\).

Apply ideal gas law to state 3 (isobaric process)

Since the process from state 2 to state 3 is isobaric (constant P), \(P_3 = P_2 = 0.750 ~\text{atm}\). We can apply the ideal gas law in state 3: \(P_3V_3 = nRT_3\) We know that process 3-1 is adiabatic, so we can use the relation for adiabatic processes: \(P_1V_1^{\gamma} = P_3V_3^{\gamma}\) For a monatomic ideal gas, \(\gamma = \frac{5}{3}\). Now, we have \(P_1V_1^{\frac{5}{3}} = P_3V_3^{\frac{5}{3}}\). Given \(P_1 = 3.00 ~\text{atm}\), \(V_1 = 6.00 ~\text{L}\), and \(P_3 = 0.750 ~\text{atm}\), we can solve for \(V_3\): \(V_3 = \left(\frac{P_1}{P_3}\right)^{\frac{3}{5}}V_1 = \left(\frac{3.00}{0.750}\right)^{\frac{3}{5}} \times 6.00 ~\text{L} \approx 12.74 ~\text{L}\) Now, we can find \(T_3\) using the ideal gas law: \(T_3 = \frac{P_3V_3}{nR} = \frac{0.750 \times 12.74}{0.548 \times 0.0821} \approx 183.11 ~\text{K}\) So, in state 3, \(P_3 = 0.750 ~\text{atm}\), \(V_3 = 12.74 ~\text{L}\), and \(T_3 = 183.11 ~K\). The final results are: - Number of moles of gas: 0.548 moles - In state 2: \(P_2 = 0.750 ~\text{atm}\), \(V_2 = 24.00 ~\text{L}\), and \(T_2 = 400 ~K\) - In state 3: \(P_3 = 0.750 ~\text{atm}\), \(V_3 = 12.74 ~\text{L}\), and \(T_3 = 183.11 ~K\)

Key concepts

Isothermal Expansion

The isothermal expansion process occurs when a gas expands while maintaining a constant temperature. This means the internal energy of the gas does not change because it stays at the same temperature. In this process, the gas absorbs heat energy from the surroundings, which allows it to expand. For the Ideal Gas Law, the formula is given by: \[ P_1V_1 = P_2V_2 \] since \( T_1 = T_2 \).

- **Isothermal means constant temperature:** The temperature, \( T \), remains constant during this process.

- **Pressure times volume is constant:** Because the temperature is constant, the product of pressure and volume does not change.

- **Heat is exchanged with surroundings:** The gas exchanges heat with its surroundings to keep the temperature constant.

This type of expansion is reversible if done slowly so that the system maintains equilibrium with the environment. It is important to ensure the gas maintains the same temperature throughout the process for the ideal gas to behave predictably.

Isobaric Process

An isobaric process occurs when a gas undergoes a change at a constant pressure. For example, in a piston-cylinder apparatus, this can happen when the piston is free to move, allowing the volume to change while the pressure remains constant. The Ideal Gas Law formula for this process becomes: \[\frac{V_2}{T_2} = \frac{V_3}{T_3}\] which indicates the volume and temperature are directly proportional.

- **Constant pressure:** The pressure, \( P \), remains unchanged, making the process isobaric.

- **Volume and temperature relationship:** The change in volume is directly related to the change in temperature, given the pressure is constant.

- **Work done by gas:** Work is done by the gas when it expands or compresses at this constant pressure.

Since the process maintains constant pressure, it is simple to calculate the amount of work done by using the formula: \[ W = P(V_3 - V_2) \].

This helps in understanding how gases behave under different processes and is crucial in thermodynamics to analyze engines and other mechanical systems.

Adiabatic Process

In an adiabatic process, a gas changes state without exchanging heat with its environment. This means the total heat content remains constant and any change in internal energy is due to work done on or by the gas. Here, the formula that defines the relationship between pressure, volume, and temperature is: \[ P_1V_1^\gamma = P_3V_3^\gamma \] where \( \gamma \) is the adiabatic index (ratio of specific heats, \( C_p/C_v \)). For a monatomic ideal gas, \( \gamma = 5/3 \).

- **No heat exchange:** An adiabatic process occurs when the system is perfectly insulated, keeping heat exchange with surroundings to zero.

- **Volume and pressure change impact:** As there is no heat exchange, a change in volume leads to a change in temperature and pressure.

- **Work is done on or by the gas:** The process usually results in a temperature change due to work done. Adiabatic processes can either be fast or involve all heat exchange through work.

This type of expansion or compression is common in thermodynamics when studying cycles like the Carnot or Otto cycle. A deep understanding of adiabatic processes is crucial in designing systems like refrigerators and heat engines, as it determines their efficiency and performance.