Q.52

Expert-verifiedFound in: Page 545

Book edition
4th

Author(s)
Randall D. Knight

Pages
1240 pages

ISBN
9780133942651

52. An ideal-gas process is described by , where is a CALC constant.

a. Find an expression for the work done on the gas in this process as the volume changes from to .

b. of gas at an initial temperature of is compressed, using this process, from to . How much work is done on the gas?

c. What is the final temperature of the gas in ?

a) The expression for the work done on the gas in this process as the volume changes from to is

b) The amount of work done on the gas is

c) The final temperature of the gas in is

An ideal-gas process is described by

Volume changes from to

(a) The work is equal to minus the area under the diagram. As an integral, we can express it to be

Knowing the formula giving the pressure as a function of the volume, we are left to only integrate:

Therefore, the expression for the work done on the gas in this process as the volume changes is

An ideal-gas process is described by

Amount of gas

Temperature

Using this process from to

(b) In this case the only thing we need to find is the constant and then substitute.

First, let's take the function providing the pressure as a function of the volume:

First, let's take the function providing the pressure as a function of the volume:

Now we can express the initial pressure in terms of the initial volume and initial temperature , from the ideal gas law:

Substituting in the case of the initial parameters, we find the constant to be

Numerically, disregarding the units, this constant will be

Substituting in our expression found for the work, we have

An ideal-gas process is described by

(c) Knowing the relation between the pressure and volume, we can write:

We can use this to substitute in the ideal gas law:

Since we have the ratio of pressures given by the ratio of volumes, we can now express the unknown temperature by the ratio of volumes and the known initial one, finding:

In our numerical case, we have which means that the final temperature will be,

Therefore, the final temperature of the gas is

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