One mole of an ideal monatomic gas, initially at a pressure...

Key Concepts

Reversible Process

A reversible process is an idealized thermodynamic transformation that occurs infinitely slowly, so the system remains in equilibrium at all stages. Although real processes are irreversible, assuming a reversible path allows us to calculate the change in entropy, since entropy is a state function and the exact path taken does not affect its overall change between two equilibrium states.

Entropy Change for an Ideal Gas

For an ideal gas undergoing a process, the change in entropy can be computed using specific formulas that involve the ratios of the final and initial temperatures and volumes (or pressures). This calculation is based on integrating the reversible heat transfer divided by temperature, and it often uses expressions such as ?S = nR ln(V?/V?) + nC_v ln(T?/T?) or variants of it, depending on the known variables.

Entropy

Entropy is a thermodynamic property that quantifies the level of disorder or randomness in a system. It serves as a measure of energy dispersal within a system and indicates the irreversibility of natural processes. In thermodynamics, changes in entropy are key for determining the spontaneity of processes.

Ideal Gas Law

The Ideal Gas Law (PV = nRT) is a fundamental relation that connects the pressure, volume, temperature, and the number of moles of a gas. It provides the basis for relating changes in state variables in ideal gas processes, allowing one to deduce unknown variables when a gas undergoes a process like expansion or compression.

Transcript

00:01 In this problem we have one more of an ideal in monatomic gas.

00:07 And this gas is taken from this initial state with this initial pressure and volume to this final state.

00:18 With this pressure and this volume.

00:20 And we want to compute what is the change in the entropy during the process.

00:26 So let's consider a reversible process and we are going to use the first law of the thermodynamics and also these relations that we know for an ideal gas.

00:42 So from the first law of the thermodynamics, we can divide all this first law by the temperature.

00:54 So doing that we get in the left side of the equation, this expression.

01:04 And then we have the differential change in the internal energy that is given by this expression here.

01:13 So here we have ncbddddd divided by t because we are dividing all the equation by the temperature.

01:22 And then we have the pressure.

01:26 In the pressure, we can compute that pressure using this expression here.

01:32 So we have, instead of the pressure, n .r .t.

01:39 Divided by the volume, right? this is just the pressure.

01:43 Also, we have the differential change in the volume.

01:47 And also we have one temperature in the denominator, because we are dividing all the equation by this temperature.

01:54 So from here we see that we can cancel this temperature.

01:59 And we're going to use this expression because this first term is just a differential change in the entropy from the definition.

02:11 So if we want the change in the entropy, not the differential, but the finite change, we are just going to take the integral of all of this term.

02:25 Right so this is equal to the integral of nc bdd divided by t plus the integral of nr dv divided by v and this is in these integrals are taken from the initial state to the final state of the gas okay so so in the first integral we can take the cb as equal to, remember that for a monatomic gas, this is just 3 over 2r.

03:20 So from here we have in 3 over 2r.

03:26 These both terms are constant, so we can take that terms from out of the integral.

03:33 And then we have the integral of d t divided by t.

03:39 In the second integral also we can take out of the integral these values.

03:45 And we have this dv divided by b.

03:49 Also we know that the number of moles is just one.

03:53 This is one mold.

03:55 So this is one and this is one.

03:57 Okay...