Thermodynamic processes

imageOften the system from which we want to extract heat to do work is a gas.  When a gas expands, how much work does it do on its surroundings?

Consider an ideal gas in a cylinder with a movable piston.  The gas occupies a volume V, the temperature is T, and the pressure is P.  Assume that the piston has a cross-sectional area A.  Assume the gas expands by a small amount and the piston moves out by a small amount dy.  The gas exerts a force F = PA on the piston and therefore does work dW = Fdy on the piston.

dW = Fdy = PAdy = PdV

dV is the change in volume of the gas.  dV is positive because the gas expands.  (If the gas is being compresses, then dV is negative and work is done on the gas.)

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If the volume of the gas changes from V1 to V2, then the total work done by the gas is

ΔW = limΔV-->0jPjΔVj = ∫V1V2 PdV.

We sum the work done during a large number of small changes of the volume.  The work depends on how exactly the pressure varies during the expansion process.  If the pressure and volume are known at each step of the expansion process, then the work can be represented as the area under the curve on a PV diagram.


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imageProblem:

Gas in a container is at a pressure of 1.5 atm and a volume of 4 m3.  What is the work done by the gas if it expands at constant pressure to twice its initial volume?

Solution:
W = P(V2 - V1) = P(2V1 - V1) = PV1 = 151500 Pa*4 m3 = 606000 J.
(The work is the area under curve in the PV diagram.  ΔV is positive because the gas expands, and therefore the work W done by the gas is positive.)

imageProblem:

Gas in a container is at a pressure of 1.5 atm and a volume of 4 m3.  What is the work done by the gas if it is compressed at constant pressure to 1/4 of its initial volume?

Solution:
W = P(0.25V1 - V1) = -0.75 PV1 = -454500 J.
(The work is the area under curve in the PV diagram.  ΔV is negative because the gas is compressed, and therefore the work W done by the gas is negative.)

imageProblem:

Gas in a container is at a pressure of 1.5 atm and a volume of 4 m3.  What is the net work done by the gas if it goes through the following cycle?

(i)  The gas expands at constant pressure of 1.5 atm to a volume of 4 m3.
(ii)  At constant volume of 4 m3 the pressure is reduced to 1 atm.
(iii)  The gas is compressed at constant pressure of 1 atm to a volume of 4 m3.
(iv)  At constant volume of 4 m3 the pressure is increased to 1.5 atm.

Solution:
(i)  W = PΔV = 151500 Pa*4 m3 = 606000 J.
(ii)  W = 0 since ΔV = 0. (no displacement <--> no work).
(iii)  W = PΔV = -101000 Pa*4 m3 = -404000 J.
(iv)  W = 0.
Net work done by the gas:  Wnet = 606000 J - 404000 J = 202000 J.

The net work done by the gas in one cycle of a process that returns the gas to its initial condition is the area inside the closed loop in the PV diagram.  If the loop is traversed in a clockwise direction the net work is positive.  If the loop is traversed in a counterclockwise direction the net work is negative.

imageProblem:

Calculate the net work output of a heat engine following path ABCDA in the figure on the right.

Solution:
The net work done is the area inside the closed loop in the PV diagram.
W =  (1.5 * 3)*(106 N/m2)(10-3 m3) = 4.5*103 J,
using area = average height * width.
or
area inside the loop = area of center rectangle + area of top triangle - area of bottom triangle
= (1*3 + 0.6*3/2 - 0.4*3/2)*(106 N/m2)(10-3 m3) =  4.5*103 J.

Problem:

A gas is compressed at a constant pressure of 0.8 atm from 9 liter to 2 liter.  (1 liter = 10-3 m3)  In the process, 400 J of thermal energy leaves the gas.
(a)  What is the work done by the gas?
(b)  What is the change in its internal energy?

Solution:
(a)  W = P(V2 - V1) = -80800 Pa*7*10-3 m3 = -565.6 J . (Work is done on the gas.)
(b)  ΔU = -400 J + 565.6 J = 165.6 J.

Problem:

One mole of an ideal gas does 3000 J of work on its surroundings as it expands isothermally to a final pressure of 1 atm and volume of 25 L.  Determine
(a)  the initial volume and
(b)  the temperature of the gas.

Solution:
(a)  For an isothermal process the temperature is constant. 
Therefore PV = nRT = constant.  Here n = 1.
PV = 101000 Pa*25*10-3 m3 = nRT.
W = nRT ln(V2/V1) for an isothermal process.
W/(nRT) = ln(V2/V1) = 3000 J/(101000 Pa*25*10-3 m3) = 1.19.
V2/V1 = exp(1.19) = 3.28.
V1 = (25*10-3 m3)/3.28 = 7.62*10-3 m3 = 7.62 liter.

(b)  For an ideal gas PV = nRT.  101000Pa*25*10-3 m3 = (8.31 J/K)T.  T = 303.85 K.


A bicycle ergometer is an apparatus that resembles a bicycle and is used to measure the amount of work done by a person while pedaling to rotate a large flywheel, usually the front wheel of a stationary bicycle, against a frictional force.  The work done to overcome the frictional force is converted into thermal energy, which causes the temperature of the flywheel and of other components in contact with the system to increase.  A nylon belt, which is under tension and wound around the wheel, can provide the frictional force.  The tension can be measured with a calibrated spring.  The work W done by the person on the ergometer is the product of the force due to the tension F in the belt times the distance through which this force acts.  The force is tangential to the rim of the wheel.  The distance is therefore 2π times the radius of the flywheel r times the number of revolution N made by the wheel.  We have W = NF2πr.  For the ergometer we have from the first law of thermodynamics ΔU = ΔQ - ΔW.  ΔW is positive if the system does work on its surroundings and is negative if work is done on the system.  Here ΔW is negative, since the person does work on the ergometer.  ΔW = -W.  ΔQ is the heat put into or taken out of the system.  We have ΔU - ΔQ = NF2πr.  If we neglect the heat lost by the system to its surroundings and set ΔQ = 0, then the increase in the thermal energy of the system equals ΔU = NF2πr.  The temperature of the flywheel therefore increases by ΔT = ΔU/mc, where m is the mass of the flywheel and c is its specific heat capacity. 
The ergometer converts useful work into thermal energy.  By analyzing the air exhaled by the person pedaling the ergometer, the rate at which the body uses chemical energy can be measured.  By knowing how much chemical energy the body uses at rest and how much it uses while pedaling, the efficiency of the body in converting chemical energy into useful work can be determined.