A cylinder contains 0.100 mol of an ideal monatomic gas. Initially the gas is at 1.00 * 10^5 Pa and occupies a volume of 2.50 * 10^-3 m^3. (b) If the gas is allowed to expand to twice the initial volume, find the final temperature (in kelvins) and pressure of the gas if the expansion is (i) isothermal; (ii) isobaric; (iii) adiabatic.

Verified step by step guidance

1. Step 1: Identify the initial conditions of the gas. Here, the initial number of moles (n) is 0.100 mol, initial pressure (P1) is 1.00 * 10^5 Pa, and initial volume (V1) is 2.50 * 10^-3 m^3.
2. Step 2: Calculate the final volume (V2) after expansion. Since the volume is doubled, V2 = 2 * V1.
3. Step 3: For isothermal expansion (i), use the ideal gas law, PV = nRT, where T remains constant. Since the temperature is constant, the final pressure can be found using P1V1 = P2V2, solving for P2.
4. Step 4: For isobaric expansion (ii), the pressure remains constant (P2 = P1). Use the ideal gas law to find the final temperature by setting up the equation nRT1/V1 = nRT2/V2 and solving for T2.
5. Step 5: For adiabatic expansion (iii), use the adiabatic condition PV^gamma = constant, where gamma (γ) for a monatomic gas is 5/3. Use this relation along with the initial conditions to find the final pressure and temperature. You can use P1V1^gamma = P2V2^gamma to find P2, and then use the ideal gas law to find T2.

Key Concepts

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of an ideal gas through the equation PV = nRT. Here, P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. This law is fundamental for understanding how gases behave under different conditions, especially during expansions and compressions.

Types of Thermodynamic Processes

Thermodynamic processes can be classified into isothermal (constant temperature), isobaric (constant pressure), and adiabatic (no heat exchange). Each type of process affects the gas's temperature and pressure differently during expansion. Understanding these processes is crucial for predicting the behavior of the gas when it undergoes changes in volume.

Monatomic Ideal Gas Properties

Monatomic ideal gases consist of single atoms and exhibit specific heat capacities that differ from diatomic or polyatomic gases. For a monatomic gas, the molar specific heat at constant volume (Cv) is 3/2 R, and at constant pressure (Cp) is 5/2 R. These properties are essential for calculating temperature changes and pressures during various thermodynamic processes.