Five moles of monatomic ideal gas have initial pressure $2.50\times {10}^3$ Pa and initial volume $2.10$ m³. While undergoing an adiabatic expansion, the gas does $1480$ J of work. What is the final pressure of the gas after the expansion?

Verified step by step guidance

1. Identify the type of process: The problem states that the gas undergoes an adiabatic expansion. In an adiabatic process, no heat is exchanged with the surroundings, so the first law of thermodynamics simplifies to \( \Delta U = -W \), where \( \Delta U \) is the change in internal energy and \( W \) is the work done by the gas.
2. Use the adiabatic condition for an ideal gas: For a monatomic ideal gas, the adiabatic condition is given by \( PV^\gamma = \text{constant} \), where \( \gamma = \frac{5}{3} \) for a monatomic gas. This relationship will help us find the final pressure.
3. Calculate the change in internal energy: For a monatomic ideal gas, the change in internal energy \( \Delta U \) is given by \( \Delta U = \frac{3}{2} nR \Delta T \). However, since \( \Delta U = -W \) and \( W = 1480 \text{ J} \), we can use this to find \( \Delta U \).
4. Relate initial and final states using the adiabatic condition: Since \( PV^\gamma = \text{constant} \), we can write \( P_i V_i^\gamma = P_f V_f^\gamma \). We know \( P_i = 2.50 \times 10^3 \text{ Pa} \) and \( V_i = 2.10 \text{ m}^3 \). We need to find \( V_f \) using the work done and then solve for \( P_f \).
5. Solve for the final pressure \( P_f \): Use the relationship \( P_f = P_i \left( \frac{V_i}{V_f} \right)^\gamma \) to find the final pressure after determining \( V_f \) from the work-energy relationship. This will give you the final pressure of the gas after the adiabatic expansion.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics, expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. It describes the relationship between these variables for an ideal gas, allowing us to predict how a gas will behave under different conditions.

Adiabatic Process

An adiabatic process is a thermodynamic process in which no heat is exchanged with the surroundings. For an ideal gas, this means that any change in internal energy is due to work done on or by the system. In an adiabatic expansion, the gas does work on its surroundings, leading to a decrease in internal energy and temperature.

First Law of Thermodynamics

The First Law of Thermodynamics, also known as the law of energy conservation, states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In an adiabatic process, since no heat is exchanged, the change in internal energy is equal to the negative of the work done by the gas.