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Q.52

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
Found in: Page 545

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Short Answer

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

See the step by step solution

Step by Step Solution

Step 1: Given Information (Part a)

An ideal-gas process is described by

Volume changes from to

Step 2: Explanation (Part a)

(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:

Step 3: Final Answer (Part a)

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

Step 4: Given Information (Part b)

An ideal-gas process is described by

Amount of gas

Temperature

Using this process from to

Step 5: Explanation (Part b)

(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

Step 6: Final Answer (Part b)

Substituting in our expression found for the work, we have

Step 7: Given Information (Part c)

An ideal-gas process is described by

Step 8: Explanation (Part c)

(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,

Step 9: Final Answer (Part c)

Therefore, the final temperature of the gas is

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