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Physics

Work of gas

1. Explain the work of a gas and the cyclic process.

Solution:

Hot gas or steam can perform work. Work of a gas at constant pressure: W′=mMmRm ΔTW' = \frac{m}{M_m} R_m \, \Delta T

W′=n Rm ΔTW' = n \, R_m \, \Delta T

W′=N k ΔTW' = N \, k \, \Delta T

A process in which the final state of the system is identical to the initial state is called a cyclic process.

Work done by a gas during one cycle:

W′=Q=Q1−Q2W' = Q = Q_1 - Q_2

  • QQ = total heat absorbed by the gas during the cycle from the surroundings

  • Q1Q_1 = heat absorbed from the heater

  • Q2Q_2 = heat released to the cooler

Efficiency of a cyclic process and a heat engine (internal combustion engines, turbines, etc.):

η=W′Q1=Q1−Q2Q1<1\eta = \frac{W'}{Q_1} = \frac{Q_1 - Q_2}{Q_1} < 1

ηmax=T1−T2T1<1\eta_{max} = \frac{T_1 - T_2}{T_1} < 1

Fuel calorific value (heating value):

H=Qm,[H]=J⋅kg−1H = \frac{Q}{m}, \quad [H] = J \cdot kg^{-1}

 

2.Calculate the work of steam during one piston stroke if the steam pressure on the piston is 176.104 Pa and the piston area is 700 cm2. The piston stroke is 55 cm.

Solution:

Analysis:

p = 176.104 Pa, S = 700.10-4 m2, h = 55.10-2 m,

 fyzika-praca-plynu-2.gif

The work of steam during one piston stroke is W = 67.76 kJ

3.Oxygen O2 (Mm = 32.10-3 kg·mol-1) has a mass of 4 kg and a temperature of 0oC. How much will its temperature increase in an isobaric process if the gas performs work of 10.4 kJ?

Solution:

Analysis:

m = 4 kg, Mm = 32.10-3 kg·mol-1, W = 10.4·103 J, Rm = 8.314 J·kg-1·K-1

fyzika-praca-plynu-3.gif 

The gas temperature increases by ΔT = 10 K.

4. What is the mass of fuel consumed by airplane engines on a flight of 500 km if the average speed is 250 km·h-1, the average engine power is 2 MW, engine efficiency is 25%, and the fuel calorific value is 45 MJ·kg-1?

Solution:

Analysis:

s = 500 km = 5·105 m, v = 250 km·h-1 = 69.44 m·s-1,
P = 2·106 W, H = 45·106 J·kg-1, η = 0.25   m = ?

fyzika-praca-plynu-4.gif 

The airplane engines consume 1280 kg of fuel.

5.A combustion engine receives heat of 1·104 J from a heater at 900oC. It transfers 75% of this heat to a cooler at 150oC. What is the engine efficiency? What is the maximum efficiency of an engine operating between these temperatures?

Solution:

Analysis:

T1 = 1173.15 K, T2 = 423.15 K, Q1 = 1·104 J, Q2 = 0.75·104 J

fyzika-praca-plynu-5.gif 

The engine efficiency is η = 25%, maximum efficiency η max = 64%.

6.What is the “second law of thermodynamics”?

Solution:

It is impossible to construct a periodically operating heat engine that would convert all the heat received from a heater into useful work.

It is impossible for heat to spontaneously transfer from a colder body to a hotter body.
              
The “second law of thermodynamics” states that it is impossible to construct a “perpetual motion machine of the 2nd kind,” i.e., a heat engine that operates without losses and with 100% efficiency.

7. Complete the short test:

  • 1.) The molar mass of carbon dioxide CO2 is: (12C, 16O)

          A.) 44·10-3 kg·mol-1    B.) 55·10-3 kg·mol-1    C.) 33·10-3 kg·mol-1

  • 2.) Which quantity does not belong among the basic state variables?

          A.) temperature     B.) volume     C.) time

  • 3.) An ideal gas has volume V at 27oC. At what temperature does it have volume 0.75V?

          A.) –31oC     B.) 48oC    C.) –48oC

  • 4.) An ideal gas has a pressure of 5 MPa at 0oC. Determine its pressure at –50oC.

          A.) 3.04 MPa     B.) 4.08 MPa    C.) 5.03 MPa

  • 5.) What is the calorific value of fuel if burning 200 kg of fuel releases 620·107 J of heat?

          A.) 3.1·107 J·kg-1    B.) 5.4·106 J·kg-1    C.) 6.9·105 J·kg-1

  • 6.) What is the maximum efficiency of a steam turbine if the heater temperature is 530oC and the cooler temperature is 50oC?

          A.) 60 %     B.) 65 %    C.) 70 %

  • 7.) What work does an ideal gas of 1 mol perform if its temperature increases by 1 K at constant pressure?
    A.) 1.38 J       B.) 8.314 J     C.) 6.022 J

Solution:

1A, 2C, 3C, 4B, 5A, 6A, 7B

8.What work is done by the air if, at a constant pressure of 0.15 MPa, its volume increases from 3.2 liters to 5.2 liters?

Solution:

praca-plynu-8

The air does 300 J of work.

9.What work is done by air with a mass of 1.3 grams (Mm = 29·10-3 kg·mol-1) if, in an isobaric process, its temperature increases from 200C to 1000C?

Solution:

praca-plynu-9 

The air does 30 J of work.

10.Determine the amount of substance of an ideal gas which, at constant pressure, when its temperature is increased by 20C, does 8314 J of work.

Solution:

 praca-plynu-10

The amount of substance of the ideal gas is 0.5 kmol.

11.In a heat engine, the gas received 0.5 MJ of heat from the heater and transferred 350 kJ of heat to the cooler.

  • a.) What is the efficiency of this engine?
  • b.) What total work did the engine do in 10,000 cycles?

Solution:

 praca-plynu-11

The efficiency of the engine is 30%. The engine did a total work of 1.5 GJ.

12.During the expansion of a gas with constant mass and at constant pressure 9 MPa, the gas increased its volume from 0.2 m3 to 0.7 m3 and absorbed 6 MJ of heat in the process. Determine the change in the internal energy of the gas.

Solution:

 praca-plynu-12

The internal energy of the gas increased by 1.5 MJ.

13.Air with a given (unknown) mass and molar mass Mm = 29·10-3 kg·mol-1, which is in a cylinder with a movable piston, receives 5 kJ of heat in an isobaric process. What work does the air do in this process? How does the internal energy of the air change? (cp = 103 J·kg-1·K-1)

Solution:

 praca-plynu-13

The air does 1.433 kJ of work. The internal energy of the air increases by 3.567 kJ.

14.The temperature of the cooler of a gas engine is 200C. The engine operates with an efficiency of 40%, which is 1.8 times less than the upper limit of efficiency. Determine the temperature of the heater.

Solution:

 praca-plynu-14

The heater has a temperature of 1047 K.

15.Steam is admitted into the cylinder of a steam engine at constant pressure of 3 MPa. The piston stroke is 0.3 m and the corresponding change in volume is 9·10-3 m3. What work does the steam do in one stroke? With what force due to pressure does the steam act on the piston?

Solution:

 praca-plynu-15

The steam does 27 kJ of work. The steam acts on the piston with a force of 90 kN.

16.An ideal gas of constant mass undergoes the cyclic process ABCDA (diagram). What total work does the gas do in one cycle? Of which individual processes is this cycle composed? Solve for the values:

p2 = 720 kPa, p1 = 450 kPa, V2 = 5.5·10-3 m3, V1 = 2.5·10-3 m3, W/ = ?

praca_plynu-16z

Solution:

p₂ = 720 kPa, p₁ = 450 kPa, V₂ = 5.5·10⁻³ m³, V₁ = 2.5·10⁻³ m³, W′ = ?

a = AB = p₂ – p₁
a = 720 kPa – 450 kPa = 270 kPa = 270·10³ Pa

b = CD = V₂ – V₁
b = 5.5·10⁻³ m³ – 2.5·10⁻³ m³ = 3·10⁻³ m³

W′ = S ABCD = a·b
W′ = a·b
W′ = 270·10³ Pa · 3·10⁻³ m³ = 810 J

W′ = 810 J

The total work done by the gas during one cyclic process is 810 J.

AB, CD – isobaric processes
BC, DA – isochoric processes