Work Done During Isothermal Expansion

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Graphical Method
Analytical Method
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Consider that one gram molecule of a perfect gas is taken in a cylinder having perfectly conducting walls and bottom, provided with a piston. Let the cylinder be placed on a source of heat at temperature T°A. If the piston is now moved slowly outwards, the gas expands, does some work and tends to cool but it absorbs required amount of heat from the source to keep it at the same temperature. The expansion is thus isothermal.

Let the piston of area of cross-section A move through a small distance dx so that the gas expands by a small amount dV. As the expansion is small, the pressure of the gas remains , practically, constant, say P.

Then the force acting on the piston, F = P×A

and small work done by the gas,

dW = F dx

= PA dx

dW = P dV [Since, A dx = dV]

P-V Diagram of Isothermal Expansion of an Ideal Gas

(a) Graphical Method

The isotherm PQ for the process is shown in below figure.

The net work done to expand the gas from a volume V₁ to volume V₂ will be,

W = Area ABDC

Thus, area occupied below the curve in between AB and CD gives the required value of work done.

(b) Analytical Method

Analytically, we can be calculated by integrating dW between the limits V₁ to V₂.

$W=\int_{0}^{W}dW=\int_{v_1}^{v_2}PdV$

For a perfect gas, PV = RT or P = RT/V

Substituting P = RT/V in the above equation, we get,

$W = \int_{V^{_{1}}}^{V_{2}}\frac{RT}{V} dV$

As temperature is constant during isothermal expansion and R is a gas constant,

$W =RT \int_{V^{_{1}}}^{V_{2}}\frac{dV}{V} = RT [log_{e}V]_{V_{1}}^{V_{2}}$

= RT (logeV₂ -logeV₁)

= RT loge (V₂/V₁)

W = 2.3026 RT log10 (V₂/V₁)

According to first law of thermodynamics, heat used up for the expansion of gas = W/J

So, H = 2.3026 (RT/J) log10 (V₂/V₁)

Since P₁V₁ = P2V₂, thus, V₂/V₁= P1/P2

Substituting forV₂/V₁, H = 2.3026 (RT/J) log10 (P₁/P₂)

On the other hand, if the gas is compressed, the work is done on the gas and is therefore negative. The work done for isothermal compression is given by,

W = -2.3026 RT log10 (V₂/V₁)

So, W =2.3026 RT log10 (V₁/V₂)

Therefore, H = 2.3026 (RT/J) log10 (V₁/V₂)

and H = 2.3026 (RT/J) log10 (P₂/P₁)

Problem 1:-

Calculate the work done of n moles of a vander Waals gas in an isothermal expansion from V_i to V_f.

Concept:-

The work done W of gas in an isothermal expansion from volume V_i to V_f is defined as,

W = -∫_V_i^V^f p dV

The gas equation for van der Wall gas is,

(p+n²a/V²) (V-nb) = nRT

Here p is the pressure, V is the volume, R is the gas constant, T is the temperature, n is the number of moles, a and b are the van der Walls gas constant.

Solution:-

First we have to find out the pressure p of the gas.

From van der Walls gas equation (p+n²a/V²) (V-nb) = nRT, the pressure p will be,

p = (nRT/(V-nb)) – (n²a/V²)

To obtain the work done of n moles of a van der Walls gas in an isothermal expansion from volume V_i to V_f will be,

W = -∫_V_i^V^f p dV

= -∫_V_i^V^f [((nRT/(V-nb)) – (n²a/V²)] dV

= [- nRT ln(V-nb) – an²/V] _V_i^V^f (Since, ∫ (1/ V-nb) dV = ln(V-nb) and ∫ 1/V² dV = - 1/V)

= (- nRT ln V_f –nb/ V_i –nb) –an²(1/V_f- 1/V_i)

From the above observation we conclude that, the work done of n moles of a van der Walls gas in an isothermal expansion from volume V_i to V_f would be (- nRT ln V_f –nb/ V_i –nb) –an²(1/V_f- 1/V_i).

Problem 2:-

A quantity of ideal gas occupies an initial volume V₀at a pressure p₀ and a temperature T₀. It expands to volume V₁ (a) at constant pressure, (b) at constant temperature, and (c) adiabatically. Graph each case on a pV diagram. In which case is Q greatest? In which case is W greatest? least? In which case is ?E_int greatest? Least?

Concept:-

Isobaric Process:-

The process in which, the change in volume and temperature of a gas takes place at a constant pressure is called an isobaric process.

Isothermal Process:-

The process in which, the change in pressure and volume takes place at constant temperature is called an isothermal change.

Adiabatic Process:-

The process in which, the change in pressure and volume and temperature takes place without any heat entering or leaving the system is called adiabatic change.

Solution:-

(a) Isobaric Process:-

The process in which, the change in volume and temperature of a gas takes place at a constant pressure is called an isobaric process.

A quantity of ideal gas occupies an initial volume V₀ at a pressure p₀ and a temperature T₀. It expands to volume V₁ at constant pressure. The pV diagram for this process as shown in below figure1.

Figure 1:

(b) Isothermal Process:-

The process in which, the change in pressure and volume takes place at constant temperature is called an isothermal change.

A quantity of ideal gas occupies an initial volume V₀ at a pressure p₀ and a temperature T₀. It expands to volume V₁ at constant temperature. The pV diagram for this process as shown in below figure 2.

Figure 2:

P-V diagram of isothermal process

(c) Adiabatic Process:-

The process in which, the change in pressure and volume and temperature takes place without any heat entering or leaving the system is called adiabatic change.

A quantity of ideal gas occupies an initial volume V₀ at a pressure p₀ and a temperature T₀. It expands to volume V₁ adiabatically. The pV diagram for this process as shown in below figure 3.

Figure 3:

P-V diagram of adiabatic process

Q is greatest for constant pressure process and least for adiabatic. W is greatest (in magnitude, it is negative for increasing volume process) for constant pressure process and least for adiabatic. ΔEint is greatest for constant pressure (for which it is positive), and least for adiabatic (for which is negative).

Problem 3:-

A sample of gas expands from 1.0 to 5.0 m³ while its pressure decreases from 15 to 5.0 Pa. How much work is done on the gas if its pressure changes with volume according to each of the three processes shown in the pV diagram in below figure.

A sample of gas expands from 1.0 to 5.0 m3 while its pressure decreases from 15 to 5.0 Pa as shown in the pV diagram.

Concept:-

Work is a path function. So work done on the gas depends upon the path. The area under pv diagram gives the work done on the gas between initial state i and final state f.

Work done W is defined as,

W = -pdV

Here p is the pressure and dV is the change in volume.

Solution:-

(1) In the process 1, the work done W₁ will be zero for vertical path.

So W₁ = 0

The work done is present only in horizontal path.

Thus the work done W₂ for horizontal path will be,

W₂= -pdV (negative sign is due to work is done on the gas

= - (15 Pa) (4m³)

= -60 Pa.m³

= (-60 Pa.m³) (1 J/1 Pa.m³)

= -60 J

Therefore the net work done W on the gas will be,

W = W₁ + W₂

= 0 J+(-60 J)

= -60 J

From the above observation we conclude that, the net work done on the gas would be -60 J.

(2) For the process 2, work done is negative of the area under the curve.

So the area A under the curve for process 2 will be,

A = ½ (15 Pa+5 Pa) (4m³)

= 40 Pa.m³

= (40 Pa.m³) (1 J/1 Pa.m³)

= 40 J

As for the process 2, work done is negative of the area under the curve; therefore the net work done on the gas will be -40 J.

(3) In the process 3, the work done W3 will be zero for vertical path.

So W₃ = 0 J

The work done is present only in horizontal path.

Thus the work done W₄for horizontal path will be,

W₄= -pdV (negative sign is due to work is done on the gas)

= - (5 Pa) (4m³)

= -20 Pa.m³

= (-20 Pa.m³) (1 J/1 Pa.m³)

= -20 J

Therefore the net work done W on the gas will be,

W = W₃ + W₄

= 0 J+(-20 J)

= -20 J

From the above observation we conclude that, the net work done on the gas would be -20 J.

Conceptual Question

Real gases always cool when making a free expansion, whereas an ideal gas does not. Explain.

Solution:-

In a free expansion of an ideal gas, the gas is initially in one side of the container, and when the stopcock is opened, the gas expands into the previously evacuated half. No weights can be raised in this process results no work is done. The process is adiabatic since the container is insulated. Thus W = 0 and Q = 0. So, according to first law of thermodynamics the internal energy will be zero.

The internal energy of an ideal gas undergoing a free expansion remains constant. This signifies that the temperature of the ideal gas does not change in a free expansion. But the internal energy of a real gas undergoing a free expansion does not remain constant results a change in temperature will occur in the expansion of real gas. That is why real gases always cool when making a free expansion, whereas an ideal gas does not.

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