Calculating Work Done by an Ideal Gas in a Three-Step Transformation

In summary, a monatomic ideal gas with pressure p_1 and temperature T_1 is contained in a cylinder of volume V_1 with a movable piston. The gas undergoes a three-step transformation where it is heated at constant volume until the pressure reaches A p_1, then expanded at constant temperature until the pressure returns to p_1, and finally cooled at constant pressure until the volume returns to V_1. To find the work done during step 2, the ideal gas law can be solved and the work can be expressed in terms of p_1, V_1, and A. The process can also be visualized on a pV plane.
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A monatomic ideal gas has pressure p_1 and temperature T_1. It is contained in a cylinder of volume V_1 with a movable piston, so that it can do work on the outside world.

Consider the following three-step transformation of the gas:

1. The gas is heated at constant volume until the pressure reaches A p_1 (where A >1).
2. The gas is then expanded at constant temperature until the pressure returns to p_1.
3. The gas is then cooled at constant pressure until the volume has returned to V_1.

It may be helpful to sketch this process on the pV plane.

How much work W_2 is done by the gas during step 2?
Express the work done in terms of p_1, V_1, and A.

I know I have to find the integral, but I can't work out how to find pressure as a function of volume.

Please help.

Thank you
 
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  • #2
Solve the ideal gas law :)
 
  • #3
Thanks. Found the answer.
 
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Happy Birthday.
 
  • #6
Thanks guys :smile:
 

Related to Calculating Work Done by an Ideal Gas in a Three-Step Transformation

1. How is work done by an ideal gas calculated in a three-step transformation?

The work done by an ideal gas in a three-step transformation is calculated by finding the area under the corresponding pressure-volume (PV) curve. This can be done by breaking the transformation into three steps and calculating the work done for each step separately, then adding them together to get the total work done.

2. What are the three steps involved in calculating work done by an ideal gas?

The three steps involved in calculating work done by an ideal gas are isothermal, adiabatic, and isobaric. In the isothermal step, the temperature of the gas is kept constant while the volume changes. In the adiabatic step, the gas is insulated so that no heat is exchanged with the surroundings while the volume changes. In the isobaric step, the pressure of the gas is kept constant while the volume changes.

3. How do you calculate work done in the isothermal step?

In the isothermal step, the work done by an ideal gas is given by the formula W = nRT ln(V2/V1), where n is the number of moles of gas, R is the gas constant, T is the temperature, and V1 and V2 are the initial and final volumes, respectively.

4. What is the formula for work done in the adiabatic step?

In the adiabatic step, the work done by an ideal gas is given by the formula W = (P2V2 - P1V1)/(1 - γ), where P1 and P2 are the initial and final pressures, V1 and V2 are the initial and final volumes, respectively, and γ is the adiabatic index.

5. How can the work done in the isobaric step be calculated?

In the isobaric step, the work done by an ideal gas is given by the formula W = P(V2 - V1), where P is the constant pressure and V1 and V2 are the initial and final volumes, respectively.

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