2 . 2 .00 moles of an ideal gas is at a (true) pressure of 2 .00 x 10 5 Pa and i

Question

2. 2.00 moles of an ideal gas is at a (true) pressure

of 2.00x 105 Pa and is enclosed in a chamber of

volume of 20.0 liter (1 liter =10 -3 m3).

  The gas is (1) first expanded at constant pressure

to twice its original volume, then (2) compressed isothermally back to its original volume, and

finally, (3) cooled at constant volume to its original pressure.

a) On a properly labeled P-V

diagram show the various

steps of the process described

use the sketch alongside. (10)

b) What is the temperature during the isothermal

process (step 2)? (10)

c) Calculate the maximum pressure attained in

this entire process and show where it happens on

the P-V diagram above.(10)

d) Calculate the work done and the heat flow during

the steps 1and 3 (indicating, for each, whether the

work is done on the gas or by the gas and

whether the heat is given to or taken from the gas). (10

Explanation / Answer

Hi,

For this exercise we should remember that and ideal gas can be described by: PV = nRT where P is the pressure, V is the volume, n are the mols of the substance, T is the temperature and R is a constant which value is:

8.314 Pa*m3 / (K*mol)

a) Sadly I cannot show you the diagram, but I can describe it. The first step is said to be at constan pressure, so it would be a line parallel to the V axis (an horizontal line).

The second step is a compression, but at constant temperature. If you see the equation for an ideal gas an put P in terms of V, you would see that (for this case) everything else is constant so you will have a function like:

P(V) = A/V; where A is constant an is equal to nRT

So in the graphic you will see something very similar to 1/X (if you were using a Cartesian plane).

For the final part, you will have a vertical line, because the pressure is going down but at constan volume.

(b) To answer this we must know the temperature at end of the first step, because it will be the same during the whole duration of step 2. To do that we do the following:

P1V1 = nRT1 ::::: T1 = P1V1 / (nR) = (2*105 Pa)*(2*20*10-3 m3)/[(8.314 Pa*m3 / (K*mol))*(2 mol)] = 481.1 K

(c) The maximum pressure happens at the end of the second step and its value is:

P2V2 = nRT2 :::::: P2 = nRT1/V2 = (2 mol)*(8.314 Pa*m3 / (K*mol))*(481.1 K) / (20*10-3 m3 ) = 4*105 Pa

(d) In the first step the gas is expanded and doesn't change its temperature, while the pressure is also constant so there's no heat flow. Besides, in this case there's work and it is done by the gas. This work can be calculated as:

W = P*(V1 - V0) = 2*105 Pa * (2*20 - 20) *10-3 m3 = 4000 J

In the second step there's no work because the volumen doesn't change. On the other hand the temperature is changing along wiht the volume so there is a heat flow. This heat in this case is given from the gas. This heat can be calculated as:

Q = n*Cp*(T2 - T0) ; but we don't know the value of Cp (Heat capacity at constan pressure)

However, if we assume that the gas a is a diatomic gas we know that Cp = 7/2 R and so we have the following:

Q = (7/2)*(2 mol)*(8.314 Pa*m3 / (K*mol))*(481.1 - 240.6) K =14000 J

I hope it helps.

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