Thermodynamic Systems - Volume, Final Pressure and Final Temperature

[Lewishio]

Homework Statement

I have had a go at the following question but have gone wrong somewhere, any ideas?

A mass of 0.12 kg of air has an initial temperature of 500°C and pressure 0.8 MPa. If the air is
expanded according to the law pV1.2 = c to a final volume of 90 litres, determine

i) its initial volume,
ii) its final pressure,
iii) its final temperature.

For air, take R = 287 Jkgˉ¹ Kˉ¹.

Relevant Equations

pV = mRT
V = mRT/P
T = PV/mR

Mass = 0.12kg
Initial temp = 500°c = 773K
Initial pressure = 0.8 MPa = 800,000 Pa
Final volume = 90L
R = 287 Jkg^-1K^-1

1) Initial Volume

V=mRT/P

0.12 x 287 x 773 / 800,000 = 26,662.12m^3

2) Final Pressure

P2 = P1P2^1.2/V2^1.2

800,000 x 26,662.12^1.2 / 90 = 1,816,095,330 Pa = 1,816 MPa

3) Final Temperature

T=P2V2^1.2/mR

1,816,095,330 x 90 / 0.12 x 287 = 1,291 x 10^11 K

 

[Chestermiller]

Problem Statement: I have had a go at the following question but have gone wrong somewhere, any ideas?

A mass of 0.12 kg of air has an initial temperature of 500°C and pressure 0.8 MPa. If the air is
expanded according to the law pV1.2 = c to a final volume of 90 litres, determine

i) its initial volume,
ii) its final pressure,
iii) its final temperature.

For air, take R = 287 Jkgˉ¹ Kˉ¹.
Relevant Equations: pV = mRT
V = mRT/P
T = PV/mR

Mass = 0.12kg
Initial temp = 500°c = 773K
Initial pressure = 0.8 MPa = 800,000 Pa
Final volume = 90L
R = 287 Jkg^-1K^-1

1) Initial Volume

V=mRT/P

0.12 x 287 x 773 / 800,000 = 26,662.12m^3

You forgot to divide by 800000, and you forgot to convert to liters.

 

[Lewishio]

Hi,

Thank you, how does this look? I think there is a problem with the formula for the final temperature as my answer is not correct.

1) Initial Volume

V=mRT/P
0.12 x 287 x 773 / 800,000 = 0.033m^3

Convert to litres:
0.033 x 1000 = 33L

2) Final Pressure

P2 = P1V1^1.2 / V2^1.2
800,000 x 33^1.2 / 90^1.2 = 240,002.4 Pa

3) Final Temperature

T = PV^1.2 / mR
240,002.4 x 90^1.2 / 0.12 x 287 = 1.27 x 10^11 K

 

[Chestermiller]

Hi,

Thank you, how does this look? I think there is a problem with the formula for the final temperature as my answer is not correct.

1) Initial Volume

V=mRT/P
0.12 x 287 x 773 / 800,000 = 0.033m^3

Convert to litres:
0.033 x 1000 = 33L

2) Final Pressure

P2 = P1V1^1.2 / V2^1.2
800,000 x 33^1.2 / 90^1.2 = 240,002.4 Pa

3) Final Temperature

T = PV^1.2 / mR
240,002.4 x 90^1.2 / 0.12 x 287 = 1.27 x 10^11 K

The final temperature is incorrect. You should be using the ideal gas law. You should be using consistent units with the volume. And you should learn how to do math using parenthesis for the denominator.

 

[Lewishio]

Hi,

I have had another go, how does this look?

PV=mRT rearranged to T= PV/mR

(240,002.4Pa x 0.09m^3) / (0.12kg x 287 Jkg^-1K^-1) = 627.1K = 354.1°c