Question

An ideal gas undergoes two processes in a pistoncylinder device as follows: 1-2 Polytropic compression from \(T_{1}\) and \(P_{1}\) with a polytropic exponent \(n\) and a compression ratio of \(r=V_{1} / V_{2}\) 2-3 Constant pressure expansion at \(P_{3}=P_{2}\) until \(V_{3}=V_{1}\) (a) Sketch the processes on a single \(P\) -V diagram. (b) Obtain an expression for the ratio of the compressionto-expansion work as a function of \(n\) and \(r\) (c) Find the value of this ratio for values of \(n=1.4\) and \(r=6\).

Short answer

(a) Sketch the processes on a single P-V diagram. 1. Draw the axes: Draw a horizontal axis labeled "V" for volume and a vertical axis labeled "P" for pressure. 2. Plot points 1 and 3: Plot the initial point (1) with coordinates (\(V_1\), \(P_1\)), and the final point (3) with coordinates (\(V_1\), \(P_3\)). 3. Draw process 1-2: Draw a curve between points 1 and 2, representing the polytropic compression process with given exponent n and compression ratio r. 4. Draw process 2-3: From point 2, draw a horizontal line to point 3, representing the constant pressure expansion process. (b) Obtain an expression for the ratio of the compression to expansion work as a function of n and r. 1. Find work done in process 1-2: The work done during the polytropic compression process can be calculated using the equation: \(W_{12} = \frac{P_1V_1 - P_2V_2}{(n-1)}\). 2. Find work done in process 2-3: The work done during the constant pressure expansion process can be calculated using the equation: \(W_{23} = P_2(V_3 - V_2)\). 3. Find the ratio of \(W_{12}\) to \(W_{23}\): Divide \(W_{12}\) by \(W_{23}\) to find the ratio: \(\frac{W_{12}}{W_{23}} = \frac{\frac{P_1V_1 - P_2V_2}{(n-1)}}{P_2(V_3 - V_2)}\). (c) Find the value of this ratio for values of n = 1.4 and r = 6. 1. Calculate the ratio for n = 1.4, r = 6: Plug in the values of n and r into the equation for the ratio of the compression to expansion work to obtain: \(\frac{W_{12}}{W_{23}} = \frac{\frac{P_1V_1 - P_2V_2}{0.4}}{P_2(V_3 - V_2)}\). Furthermore, use the given compression ratio (\(r = \frac{V_1}{V_2}\)) and the fact that \(V_3 = V_1\) to substitute and obtain the final value of the ratio.

Step-by-step solution

Draw the axes.

First, draw the horizontal and vertical axes. Label the horizontal axis as "V" for volume and the vertical axis as "P" for pressure. Step 2: Plot the initial point (1) and final point (3)

Plot points 1 and 3

Plot the initial point (1) with coordinates (\(V_1\), \(P_1\)), and the final point (3) with coordinates (\(V_1\), \(P_3\)). Step 3: Draw the polytropic compression process (1-2)

Draw process 1-2

Draw a curve starting from point 1 with coordinates (\(V_1\), \(P_1\)) and ending at an intermediate point 2 with coordinates (\(V_2\), \(P_2\)), representing the polytropic compression process with given exponent n and compression ratio r. Step 4: Draw the constant pressure expansion process (2-3)

Draw process 2-3

From point 2, draw a horizontal line to point 3 with coordinates (\(V_1\), \(P_3=P_2\)), representing the constant pressure expansion process. (b) Obtain an expression for the ratio of the compression to expansion work as a function of n and r. Step 1: Find the work done during the polytropic compression process

Find work done in process 1-2

The work done during the polytropic compression process can be calculated using the equation: \(W_{12} = \frac{P_1V_1 - P_2V_2}{(n-1)}\). Step 2: Find the work done during the constant pressure expansion process

Find work done in process 2-3

The work done during the constant pressure expansion process can be calculated using the equation: \(W_{23} = P_2(V_3 - V_2)\). Step 3: Find the ratio of the compression to expansion work

Find the ratio of \(W_{12}\) to \(W_{23}\)

To find the ratio of the work done during the compression process compared to the expansion process, divide \(W_{12}\) by \(W_{23}\): \(\frac{W_{12}}{W_{23}} = \frac{\frac{P_1V_1 - P_2V_2}{(n-1)}}{P_2(V_3 - V_2)}\). (c) Find the value of this ratio for values of n=1.4 and r=6. Step 1: Calculate the value of the ratio for given values of n and r

Calculate the ratio for n=1.4, r=6

Plug in the values of n and r into the equation for the ratio of the compression to expansion work to obtain: \(\frac{W_{12}}{W_{23}} = \frac{\frac{P_1V_1 - P_2V_2}{0.4}}{P_2(V_3 - V_2)}\). Furthermore, use the given compression ratio (\(r = \frac{V_1}{V_2}\)) and the fact that \(V_3 = V_1\) to substitute and obtain the final value of the ratio.