Thermodynamics, isothermal irreversible

[Faiq]

Homework Statement

Can you please show how to solve this question (I am not asking it for homework. I am asking because it would help me understand better). Determine the entropy change in sys, surr, uni, when a sample of helium has of mass M grams at 298K and 1 bar doubles its volume in isothermal irreversible expansion against external pressure =p

The complete question can be found on https://chemistry.stackexchange.com/q/84590/29265

Homework Equations

The Attempt at a Solution

 

[Chestermiller]

Homework Statement

Can you please show how to solve this question (I am not asking it for homework. I am asking because it would help me understand better). Determine the entropy change in sys, surr, uni, when a sample of helium has of mass M grams at 298K and 1 bar doubles its volume in isothermal irreversible expansion against external pressure =p

The complete question can be found on https://chemistry.stackexchange.com/q/84590/29265

Homework Equations

The Attempt at a Solution

Based on your effort in the link you cited, we are permitted by PF rules to continue without further effort.

Let's assume temporarily that the constant external pressure p = 0.5 bars so that, if the gas pressure is suddenly dropped from 1 bar to p at time zero and the gas subsequently allowed to expand irreversibly while it re-equilibrates with the (ideal reservoir) at 298 K, the final volume will be double the initial volume. (Later we can talk about how to solve if the constant external pressure p is made less than 0.5 bars, and the gas expansion is forced to stop after doubling).

Let's begin by using the ideal gas law to determine the initial volume in terms of M. Using the value of $R=0.08314\ \frac{liter.bar}{K.mole}$, what is the initial volume of helium in terms of M? In terms of M, what is the final volume at 0.5 bars and 298 K?

Chet

 

[Faiq]

Initial volume = $M*0.08314*298/4$= 6.19M liters
Final Volume = $M*0.08314*298/2$= 12.39M liters

 

[Chestermiller]

Initial volume = $M*0.08314*298/4$= 6.19M liters
Final Volume = $M*0.08314*298/2$= 12.39M liters

OK. So STATE 1 is:
0.25M moles, 298 K, 6.19M liters

STATE 2 is:
0.25M moles, 298 K, 12.39M liters

Before we calculate $\Delta S$ for the system, let's first apply the first law of thermodynamics to determine the change in entropy of the surroundings in this irreversible process.

For an expansion process, the equation for the work W done by the system on the surroundings is:
$$W=\int{p_{ext}dV}$$ where $p_{ext}$ is the externally applied pressure. In our irreversible process, at time zero, we suddenly drop the external pressure from 1 bar to 0.5 bars, and then hold the external pressure constant while the gas expands, until the system re-equilibrates both mechanically and thermally. So, in our process, $p_{ext}$ is constant at 0.5 bars. Based on this, what do you get for the work W done by the system on the surroundings (in terms of M, expressed in liter-bars)?

What is the change in internal energy for this "isothermal" change between STATES 1 and 2? Based on the first law, what is the amount of heat Q transferred from the surroundings to the system in our irreversible process?