- #1

zenterix

- 651

- 82

- Homework Statement
- Consider an irreversible adiabatic expansion of an ideal gas.

- Relevant Equations
- We start with an adiabatic container containing a gas at ##P_1, V_1##, and ##T_1##. There is a movable piston in the container. The external pressure is ##P_2<P_1##, and thus the piston is held in place by stoppers.

When we remove the stoppers, the gas expands and the piston shoots up and eventually reaches a new final position in which the internal and external pressures are the same.

Apparently we can write

$$\delta q=0\tag{1}$$

$$\delta w=-P_2dV\tag{2}$$

$$dU=C_VdT\tag{3}$$

$$dU=-P_2dV\tag{4}$$

I've studied the origin of all these equations but I just don't feel comfortable with them yet.

Thus, I have a few basic questions still.

##(1)## comes from the fact that the process is adiabatic.

I am confused about when we can or cannot write a differential form.

1) It seems that ##\delta w=-P_{ext}dV## can always be written no matter what (ie no matter if the process is reversible or irreversible). Is this so?

2) In a ##PV## diagram, we only know start and end points. According to a lecture I saw, we can't draw any sort of path connecting them because the process is irreversible.

We can depict the work done in this diagram. I just realized (if it is indeed correct) that when we do this we are not drawing a path but just graphically showing a quantity, the total work done. Is this correct?

3) ##U## is a state function. Suppse we start at a ##P_1,V_1##, and ##T_1## and move to a final ##P_2,V_2##. For our irreversible process we will have a ##T_2##.

For a reversible process we will have some ##T_{2,rev}## which may be different from ##T_2##. Thus, we aren't moving to the same final states necessarily.

(3) comes from writing out the differential form for ##U=U(T,V)## and assuming an ideal gas. Thus, for an ideal gas, (3) is "always" true. But is "always" really always here?

It seems not to matter that the irreversible process we have happens very suddenly and even violently. (3) expresses what happens when we change ##T## infinitesimally.

Reversibility just seems to imply being able to sub in the ideal gas law for ##P_2## in (2) and (4).

I suppose any very rapid process can be considered as a sequence of infinitesimal steps in time, but we are not talking about time in these equations, we are talking about temperature. Temperature does not seem to be necessarily the same throughout the system in this rapid process.

4) With regard to equation (4), I have similar doubts. I guess volume can be seen, like time, to increase infinitesimally if we just break the process into small enough steps. And external pressure ##P_2## is constant, so (4) seems to make sense. But then we're back to my question 1) above.

Apparently we can write

$$\delta q=0\tag{1}$$

$$\delta w=-P_2dV\tag{2}$$

$$dU=C_VdT\tag{3}$$

$$dU=-P_2dV\tag{4}$$

I've studied the origin of all these equations but I just don't feel comfortable with them yet.

Thus, I have a few basic questions still.

##(1)## comes from the fact that the process is adiabatic.

I am confused about when we can or cannot write a differential form.

1) It seems that ##\delta w=-P_{ext}dV## can always be written no matter what (ie no matter if the process is reversible or irreversible). Is this so?

2) In a ##PV## diagram, we only know start and end points. According to a lecture I saw, we can't draw any sort of path connecting them because the process is irreversible.

We can depict the work done in this diagram. I just realized (if it is indeed correct) that when we do this we are not drawing a path but just graphically showing a quantity, the total work done. Is this correct?

3) ##U## is a state function. Suppse we start at a ##P_1,V_1##, and ##T_1## and move to a final ##P_2,V_2##. For our irreversible process we will have a ##T_2##.

For a reversible process we will have some ##T_{2,rev}## which may be different from ##T_2##. Thus, we aren't moving to the same final states necessarily.

(3) comes from writing out the differential form for ##U=U(T,V)## and assuming an ideal gas. Thus, for an ideal gas, (3) is "always" true. But is "always" really always here?

It seems not to matter that the irreversible process we have happens very suddenly and even violently. (3) expresses what happens when we change ##T## infinitesimally.

Reversibility just seems to imply being able to sub in the ideal gas law for ##P_2## in (2) and (4).

I suppose any very rapid process can be considered as a sequence of infinitesimal steps in time, but we are not talking about time in these equations, we are talking about temperature. Temperature does not seem to be necessarily the same throughout the system in this rapid process.

4) With regard to equation (4), I have similar doubts. I guess volume can be seen, like time, to increase infinitesimally if we just break the process into small enough steps. And external pressure ##P_2## is constant, so (4) seems to make sense. But then we're back to my question 1) above.

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