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 volume doubles, the final pressure P2 can be found using P1V1 = P2V2.
4. Step 4: For isobaric expansion (ii), the pressure remains constant (P2 = P1). Use the ideal gas law to find the final temperature T2 by setting up the equation P1V1/T1 = P1V2/T2 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 P2 and temperature 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 kelvins. This law is fundamental for understanding how changes in one property affect the others in gas behavior.

Types of Thermodynamic Processes

Thermodynamic processes can be classified into isothermal (constant temperature), isobaric (constant pressure), and adiabatic (no heat exchange). Each process has distinct characteristics that affect how a gas behaves when it expands or contracts. Understanding these processes is crucial for calculating changes in temperature and pressure during gas expansion.

Monatomic Ideal Gas Properties

Monatomic ideal gases, such as helium or neon, consist of single atoms and exhibit specific heat capacities that differ from diatomic or polyatomic gases. For monatomic gases, 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 determining temperature changes during various thermodynamic processes.