# Pressure-Volume Diagrams

## Problems

### practice

- One mole of an ideal, monatomic gas runs through a four step cycle. All processes are either isobaric or isochoric. The pressure and volume of the gas at the extreme points in the cycle are given in the table below.
- Sketch the PV graph of this cycle.
- Determine the temperature at state A, B, C, and D.
- Calculate
*W*,*Q*, and Δ*U*on the path A→B, B→C, C→D, D→A and for one complete cycle. (Include the algebraic sign with each value.) - Does this cycle behave more like an engine or a refrigerator?

A four step cycle (isobaric & isochoric) state A B C D *P*(Pa) 100, 000 100, 000 200, 000 200, 000 *V*(m ^{3})0.020 0.060 0.060 0.020 *T*(K) path A→B B→C C→D D→A ABCDA description isobaric isochoric isobaric isochoric closed cycle ∆ *U*(J) *Q*(J) *W*(J) - Write something else.
- Write something different.
- Write something completely different.

### conceptual

- Which requires more work: compressing a gas slowly so that its temperature remains equal to that of the environment (isothermal compression) or compressing a gas quickly so the heat that "wants" to escape does not have time to escape (adiabatic compression)? Justify your answer.

### numerical

- Estimate the power of a relaxed human heart given an average blood pressure of 13 kPa, a volume change of 80 mL per beat, and a pulse rate of 72 beats per minute.
- Estimate the power of a pair of human lungs from the graph below given a respiration rate of 20 breaths per minute.
- One mole of an ideal gas runs through the zilch cycle, which consists of the following four processes…
The zilch cycle in words path description A→B Isothermal expansion at 600 K with 2.30 kJ of work done by the gas. B→C Isochoric pressure drop. C→D Isothermal expansion at 300 K with 1.44 kJ of work done by the gas. D→A Adiabatic compression. ABCDA The gas returns to its original state when the cycle is complete. The zilch cycle has no practical use. It's just an interesting thought problem. Given this information, complete the following table.

The zilch cycle in numbers path description ∆ *U*(kJ)*Q*(kJ)*W*(kJ)A→B isothermal −2.30 B→C isochoric C→D isothermal −1.44 D→A adiabatic ABCDA closed cycle - 0.40 moles of an ideal, monatomic gas runs through a four step cycle. All processes are either adiabatic or isochoric. The pressure and volume of the gas at the extreme points in the cycle are given in the table below.
- Sketch the PV graph of this cycle.
- Determine the temperature at state A, B, C, and D.
- Calculate
*W*,*Q*, and Δ*U*on the path A→B, B→C, C→D, D→A and for one complete cycle. (Include the algebraic sign with each value.) - Does this cycle behave more like an engine or a refrigerator?

A four step cycle (adiabatic & isochoric) state A B C D *P*(Pa) 100, 000 1,462, 000 5,850, 000 400, 000 *V*(m ^{3})0.010 0.002 0.002 0.010 *T*(K) path A→B B→C C→D D→A ABCDA description adiabatic isochoric adiabatic isochoric closed cycle Δ *U*(J) *Q*(J) *W*(J) - 0.040 moles of an ideal, monatomic gas runs through a three step cycle. All processes follow straight line paths on a pressure-volume graph. The pressure and volume of the gas at the extreme points in the cycle are given in the first two rows of the table below.
- Sketch the PV graph of this cycle.
- Determine the temperature at state A, B, and C.
- Calculate
*W*,*Q*, and Δ*U*on the path A→B, B→C, C→A and for one complete cycle. (Include the algebraic sign with each value.) - Does this cycle behave more like an engine or a refrigerator?

A three step cycle (straight line processes) state A B C *P*(kPa) 100 100 200 *V*(m ^{3})0.002 0.001 0.001 *T*(K) path A→B B→C C→A ABCA description isobaric isochoric diagonal closed cycle Δ *U*(J) *Q*(J) *W*(J)