# Adiabatic expansion of a balloon

## Homework Statement

A flexible balloon contains 0.375 mol of hydrogen sulfide gas H2S. Initially the balloon of H2S has a volume of 6750 cm^3 and a temperature of 29.0 C. The H2S first expands isobarically until the volume doubles. Then it expands adiabatically until the temperature returns to its initial value. Assume that the H2S may be treated as an ideal gas with C_p = 34.60 J/mol*K and gamma = 4/3.

What is the final volume in meters cubed?

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Andrew Mason
Homework Helper

## Homework Statement

A flexible balloon contains 0.375 mol of hydrogen sulfide gas H2S. Initially the balloon of H2S has a volume of 6750 cm^3 and a temperature of 29.0 C. The H2S first expands isobarically until the volume doubles. Then it expands adiabatically until the temperature returns to its initial value. Assume that the H2S may be treated as an ideal gas with C_p = 34.60 J/mol*K and gamma = 4/3.

What is the final volume in meters cubed?
Show us your thoughts on this first.

This a first Law problem.

Can you draw a PV diagram of this two stage process? What is the Work done by the gas in the first stage? What is the heat flow into the gas in each stage? What is the total change in internal energy of the gas?

If you answer those questions, you will be able to answer the question.

AM

as a continuation of this problem, I tried to solve it by using the equation:

v2^(1/3) = [t1*v1^(1/3)]/t2

Found t2 by:
po=pf=t1/v1=t2/2v1
2t1v1/v1=t2
2t1=t2

so... i got v2^(1/3) = {[299*(7050cm3 *10-9 m3)^1/3] / 598}
v2={0.00958}^3
=8.81 * 10-7

The answer machine says wrong. What did I do wrong??? Thanks in advance!

You've been given a gamma, why do you think this is?

Recheck the ideal gas adiabatic relations. As this is

Andrew Mason
Homework Helper
I may have confused you. This does not require application of the first law. Sorry about that.

Determine the temperature after the first expansion (PV=nRT). Then apply the adiabatic condition (PV^{\gamma} = nRTV^{\gamma-1} = K)to determine the volume when T is back to the original T.

AM

I'm confused. What have I done wrong with my equation.

By the way, I'm a different person asking the same question. :)

What is K, the original temperature? And how do you get P, through the p0=t1/v1 equation?

Andrew Mason
Homework Helper
I'm confused. What have I done wrong with my equation.

By the way, I'm a different person asking the same question. :)

What is K, the original temperature? And how do you get P, through the p0=t1/v1 equation?
K is a constant. The adiabatic condition is PV^{gamma} = constant.

As far as your equation is concerned, I am not sure what you are doing. You haven't provided any explanation. You have to break the question into two parts. The first part is simple. You can easily find T after the isobaric expansion. To find the volume after the adiabatic expansion you apply the adiabatic condition.

AM

I found help elsewhere. Your way with the K constant is actually quite unnecessary. It's faster to use just V1 and gamma. Thanks for coming back and trying to help I guess.

Andrew Mason
Homework Helper
I found help elsewhere. Your way with the K constant is actually quite unnecessary. It's faster to use just V1 and gamma. Thanks for coming back and trying to help I guess.
You appear to be applying the adiabatic condition but you seem to be unaware of it.

[/tex]PV^{\gamma}=K[/tex]

If you substitute P = nRT/V this becomes:

[/tex]nRTV^{\gamma-1} = K[/tex]

This means that [/itex]TV^{\gamma-1}[/itex] does not change, so:

[/tex]T_2V_2^{\gamma-1} = T_3V_3^{\gamma}[/tex]

T1 = 302K
T2 = 604K
[/itex]V2 = 6750 cm^3 = 6.75 x 10^3 (x 10^{-6}) m^3 = 6.75 x 10^{-3} m^3[/itex]
V3 = 13.50 x 10^{-3}m^3

So:

[/tex](604)(1.350 x 10^{-2})^{1/3} = (302)V_3^{1/3}[/tex]

[/tex]V_3^{1/3} = .476 m^3[/tex]

[/tex]V_3 = .78 m^3 = 7.8 x 10^5 cm^3[/tex]

You started out ok but your numbers are wrong. Also the temperature of 29C is 273+29 = 302K.

(Latex seems to be down so I have put a / in front of the tags so you can see them)
AM

You appear to be applying the adiabatic condition but you seem to be unaware of it.

[/tex]PV^{\gamma}=K[/tex]

If you substitute P = nRT/V this becomes:

[/tex]nRTV^{\gamma-1} = K[/tex]

This means that [/itex]TV^{\gamma-1}[/itex] does not change, so:

[/tex]T_2V_2^{\gamma-1} = T_3V_3^{\gamma}[/tex]

T1 = 302K
T2 = 604K
[/itex]V2 = 6750 cm^3 = 6.75 x 10^3 (x 10^{-6}) m^3 = 6.75 x 10^{-3} m^3[/itex]
V3 = 13.50 x 10^{-3}m^3

So:

[/tex](604)(1.350 x 10^{-2})^{1/3} = (302)V_3^{1/3}[/tex]

[/tex]V_3^{1/3} = .476 m^3[/tex]

[/tex]V_3 = .78 m^3 = 7.8 x 10^5 cm^3[/tex]

You started out ok but your numbers are wrong. Also the temperature of 29C is 273+29 = 302K.

(Latex seems to be down so I have put a / in front of the tags so you can see them)
AM
How can V_3 = .78 m^3
if
V_3^(1/3) = .476 m^3

3sqrt(V_3) = .476 m^3

V_3 = (.476 m^3)^3 = 0.1078 ?????

I have trouble finding delta_U

delta_U = nCv delta_T

T = 300 K
n = 0.35 mol
Cv = 25.95 J/mol*K