Problem on thermodynamics first law again

[ky92]

Homework Statement

Consider a system consisting of 2.0 mol CO2 (assumed to be a perfect gas) at 298K confined to a cylinder of cross-section 10cm^2 at 10 atm. The gas is allowed to expand adiabatically and reversibly against a constant pressure of 1 atm. Calculate W, Q, ΔU, ΔH, and ΔT when the piston has moved 200cm.

(Ans: W=-1.6KJ, Q=0, ΔU=-1.6KJ, ΔT=-28, ΔH=-2.1KJ)

Homework Equations

The thermodynamics equations

The Attempt at a Solution

as it is adiabatic, Q=0.
ΔV=0.001m(0.2m)=2x10^-4 m3
W=-PexΔV=-101325(2x10^-4)=-20.27 J
ΔU=W+Q=-20.27 J

and i found that my W and ΔU are wrong
what's wrong with my answer?

should Pgas=Pex if the system is in equilibrium?
i just found that it isn't
but in my book :
"To achieve reversible expansion we must match Pex to P at each state: dw=-Pex dV=-P dV"

i am so confused.
please help!

 

[D H]

Calculate the work done by the gas on the piston:

$$W_{\text{on piston}} = \int_0^{l_f} P*A \,dl$$

The work done on the gas is just the additive inverse of the work done by the gas. The piston moves 200 cm, given. The area is a constant, so you essentially need to integrate P*dl. The process is adiabatic. What is the relation between pressure and volume for an adiabatic process? (Hint: You need to use the fact that the gas is CO₂.)

 

[Andrew Mason]

Homework Statement

Consider a system consisting of 2.0 mol CO2 (assumed to be a perfect gas) at 298K confined to a cylinder of cross-section 10cm^2 at 10 atm. The gas is allowed to expand adiabatically and reversibly against a constant pressure of 1 atm. Calculate W, Q, ΔU, ΔH, and ΔT when the piston has moved 200cm.

(Ans: W=-1.6KJ, Q=0, ΔU=-1.6KJ, ΔT=-28, ΔH=-2.1KJ)

You make a good point. The problem is with the question. The gas can expand adiabatically but not reversibly. In order for it to be a reversible expansion, there has to be an infinitessimal pressure difference.

In this case there is a significant pressure difference. So what happens here is that some of the energy in the gas does work on the atmosphere and some does work on the piston. The work done on the piston means the piston acquires kinetic energy (or a combination of kinetic and gravitational potential energy if it is not horizontal). So it is a dynamic rather than a quasistatic or reversible process.

In order to calculate that the work that the gas does on the atmosphere and piston, use the internal pressure of the gas. As DH says, you have to use the relationship between P and V in an adiabatic process (adiabatic condition). Express P as a function of V and then as a function of A and length.

AM

 

[ky92]

for an adiabatic process, Q=0,
ΔU=w+0=w
w=ΔU=CvΔT...

that's all i can think of

i have no idea lol

 

[ky92]

You make a good point. The problem is with the question. The gas can expand adiabatically but not reversibly. In order for it to be a reversible expansion, there has to be an infinitessimal pressure difference.

In this case there is a significant pressure difference. So what happens here is that some of the energy in the gas does work on the atmosphere and some does work on the piston. The work done on the piston means the piston acquires kinetic energy (or a combination of kinetic and gravitational potential energy if it is not horizontal). So it is a dynamic rather than a quasistatic or reversible process.

In order to calculate that the work that the gas does on the atmosphere and piston, use the internal pressure of the gas. As DH says, you have to use the relationship between P and V in an adiabatic process (adiabatic condition). Express P as a function of V and then as a function of A and length.

AM

thank you!
let me think about it...

 

[Andrew Mason]

for an adiabatic process, Q=0,
ΔU=w+0=w
w=ΔU=CvΔT...

that's all i can think of

i have no idea lol

But you also know that

$$PV^{\gamma} = P_0V_0^{\gamma} = K$$

Since PV = nRT, P = nRT/V, so

$$TV^{\gamma-1} = K/nR$$

which is also constant, ie $TV^{\gamma-1} = T_0V_0^{\gamma-1}$

So work out the change in T after an adiabatic expansion of 200 cm (you have to first determine the initial volume and the expanded volume) using this expression. (What is the $\gamma$ for CO₂?)

AM