You have 2.0 moles of helium gas initially at a temperature...

You have 2.0 moles of helium gas initially at a temperature of $340 \mathrm{K}$ and a pressure of $0.20 \times P_{\text {atm }}$ The gas is compressed isothermally to a pressure of $0.50 \times P_{\text {atm. }}$ a. Find the final volume of the gas. Assume it behaves as an ideal gas. b. Find the work done by the gas. c. How much heat flows into the gas?

You have 2.0 moles of helium gas initially at a temperature of $340 \mathrm{K}$ and a pressure of $0.20 \times P_{\text {atm }}$ The gas is compressed isothermally to a pressure of $0.50 \times P_{\text {atm. }}$
a. Find the final volume of the gas. Assume it behaves as an ideal gas.
b. Find the work done by the gas.
c. How much heat flows into the gas?

Key Concepts

Work in an Isothermal Process

In an isothermal process for an ideal gas, the work done can be computed by integrating the pressure with respect to volume. Because the temperature is maintained constant, the work done by the gas or on the gas is given by W = -nRT ln(V_final/V_initial) or, equivalently, by using pressures W = -nRT ln(P_final/P_initial) under the inverse relationship of volume and pressure. The negative sign indicates work done by the gas is considered negative (i.e., energy is leaving the gas) when the gas is compressed.

First Law of Thermodynamics

The first law of thermodynamics states that energy can neither be created nor destroyed, only converted from one form to another. For a thermodynamic process, it is often written as ?U = Q + W, where ?U is the change in internal energy, Q is the heat exchanged with the surroundings, and W is the work done by the system. In an isothermal process, the change in internal energy ?U is zero for an ideal gas, meaning that the heat absorbed by the gas is exactly equal in magnitude (but opposite in sign) to the work done by the gas.

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed as PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the absolute temperature. This law provides a basic model for understanding and predicting the behavior of gases under various conditions, assuming no interaction between the gas molecules and that the volume occupied by the molecules themselves is negligible.

Isothermal Process

An isothermal process is one in which the temperature of the system remains constant throughout the process. In the context of gases, this means that any change in pressure and volume occurs while the internal energy remains constant, because for an ideal gas the internal energy is solely a function of temperature. This condition simplifies analysis by allowing the use of specific equations, such as integrating the work done by a gas as it expands or contracts at a steady temperature.

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