A cylinder with a movable piston holds 2.95 mol of argon at...

A cylinder with a movable piston holds 2.95 mol of argon at a constant temperature of 235 $\mathrm{K}$ . As the gas is compressed isothermally, its pressure increases from 101 kPa to 121 $\mathrm{kPa}$ (a) the final volume of the gas, (b) the work done by the gas, and (c) the heat added to the gas.

A cylinder with a movable piston holds 2.95 mol of argon at a constant temperature of 235 $\mathrm{K}$ . As the gas is compressed isothermally, its pressure increases from 101 kPa to 121 $\mathrm{kPa}$ (a) the
final volume of the gas, (b) the work done by the gas, and (c) the
heat added to the gas.

Key Concepts

Isothermal Process

An isothermal process is one in which the temperature remains constant throughout the transformation. For an ideal gas undergoing an isothermal change, the internal energy remains unchanged, which has important consequences in the energy balance of the process.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure, volume, temperature, and number of moles of a gas. This equation is essential for connecting the thermodynamic state variables and solving for unknown quantities when certain parameters are held constant.

Work in an Isothermal Process

For an ideal gas undergoing an isothermal process, the work done by or on the gas can be calculated through the integral of pressure with respect to volume, leading to the formula W = nRT ln(V_final/V_initial). This logarithmic relationship arises due to the inverse dependence of pressure on volume at constant temperature.

First Law of Thermodynamics

The first law of thermodynamics, which states that ?U = Q - W, relates the heat added to a system to its work and change in internal energy. In an isothermal process for an ideal gas, the internal energy remains constant (?U = 0), meaning that the heat added to the system is equal in magnitude to the work done by the gas.

Transcript

00:01 Hello everyone.

00:02 This is problem 84 from chapter 18.

00:04 It tells us the cylinder with a movable piston, holds a number of moles of argon at a constant temperature, and it says the gas is compressed isothermally and the pressure is increasing.

00:16 It wants us to find, in part a, the final volume of the gas.

00:21 Okay, so we know here that we can approximate this gas as an ideal gas.

00:31 So, vf should be equal to nrtf over pf using the ideal gas equation.

00:42 Okay, and we're given pf, we're given the temperature t, which is constant, so it's tf and ti, and we're given the number of moles, so we can find this, and it's just 0 .0476 cubic meters approximately.

01:02 So that's part a.

01:03 Very, very simple, just knowing to you, ideal gas equation.

01:08 Okay, part b.

01:10 That's part a.

01:11 Okay, part b.

01:13 Part b then asks us to find the work done by the gas.

01:18 So whenever we're doing that, we either think about the area under the curve in the pv plane or, you know, just actually do the calculation and involve pdv.

01:30 So because it's not that nice of a shape in the pv plane, it's a curve because it's moving along an isostheron, this is probably going to be our best bet and then now p isn't constant so it's difficult to find what this is but we do know that t is constant so we can put this sorry we can put this actually in terms of t and the variable we're integrating over v here okay so since nrt is constant this is just integral of dv over v which is going to be an rt law vf over vi.

02:22 Now, we don't actually know what vf is or vi.

02:31 We're not given those, or sorry, we did find vf in part a...

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