By the Sakur-Tetrode equation, the entropy of a monatomic, ideal gas is given by

$$\frac{S}{k_BN}=\ln\left[\frac VN\left(\frac{4\pi m}{3h^2}\frac{U}{N}\right)^{3/2}\right]+\frac52$$

For our purposes, it will make sense to use the ideal gas law to express $V$ in terms of $P$ and $T$, and to express $U=\frac32Nk_BT$, so we get

$$\frac{S}{k_BN}=\ln\left[\frac {k_BT}{P}\left(\frac{2\pi m}{h^2}\cdot k_BT\right)^{3/2}\right]+\frac52$$

So, as we can see, for a constant pressure $P$, the entropy of the ideal gas is a monotonically decreasing function with respect to decreasing $T$; if we decrease $T$, we decrease $S$.

I suspect your confusion comes from thinking that $S$ can never decrease, but this is only the case for isolated systems. If you are forcing an ideal gas to undergo an isobaric compression, then the system is no longer isolated, and so the entropy can decrease (entropy will increase elsewhere though).

As a separate argument, the entropy is a state function, meaning its value only depends on the state, not how you got there. Now, let's consider your isobaric process, and let's say we do an isobaric expansion and then an isobaric compression back to the original state (this is the original state of the system, not the original state of the system as well as the surroundings, which is impossible to achieve). Since entropy is a state function, the entropy ends at where it started. But this means one of two things happened

1. The entropy remained constant the entire time
2. The entropy increased as well as decreased during this process.

If you showed that the change in entropy is non-zero for some part of this, then you have to conclude that it is possible to decrease the entropy of an ideal gas.

By the Sakur-Tetrode equation, the entropy of a monatomic, ideal gas is given by

$$\frac{S}{k_BN}=\ln\left[\frac VN\left(\frac{4\pi m}{3h^2}\frac{U}{N}\right)^{3/2}\right]+\frac52$$

For our purposes, it will make sense to use the ideal gas law to express $V$ in terms of $P$ and $T$, and to express $U=\frac32Nk_BT$, so we get

$$\frac{S}{k_BN}=\ln\left[\frac {k_BT}{P}\left(\frac{2\pi m}{h^2}\cdot k_BT\right)^{3/2}\right]+\frac52$$

So, as we can see, for a constant pressure $P$, the entropy of the ideal gas is a monotonically decreasing function with respect to decreasing $T$; if we decrease $T$, we decrease $S$.

I suspect your confusion comes from thinking that $S$ can never decrease, but this is only the case for isolated systems. If you are forcing an ideal gas to undergo an isobaric compression, then the system is no longer isolated, and so the entropy can decrease (entropy will increase elsewhere though).

As a separate argument, the entropy is a state function, meaning its value only depends on the state, not how you got there. Now, let's consider your isobaric process, and let's say we do an isobaric expansion and then an isobaric compression back to the original state (this is the original state of the system, not the original state of the system as well as the surroundings, which is impossible to achieve). Since entropy is a state function, the entropy ends at where it started. But this means one of two things happened

1. The entropy remained constant the entire time
2. The entropy increased as well as decreased during this process.

If you showed that the change in entropy is non-zero for some part of this, then you have to conclude that it is possible to decrease the entropy of an ideal gas. Also, note that this argument is not dependent on use of an ideal gas specifically.

By the Sakur-Tetrode equation, the entropy of a monatomic, ideal gas is given by

$$\frac{S}{k_BN}=\ln\left[\frac VN\left(\frac{4\pi m}{3h^2}\frac{U}{N}\right)^{3/2}\right]+\frac52$$

For our purposes, it will make sense to use the ideal gas law to express $V$ in terms of $P$ and $T$, and to express $U=\frac32Nk_BT$, so we get

$$\frac{S}{k_BN}=\ln\left[\frac {k_BT}{P}\left(\frac{2\pi m}{h^2}\cdot k_BT\right)^{3/2}\right]+\frac52$$

So, as we can see, for a constant pressure $P$, the entropy of the ideal gas is a monotonically increasing function of $T$. If we decrease $T$, we decrease $S$.

I suspect your confusion comes from thinking that $S$ can never decrease, but this is only the case for isolated systems. If you are forcing an ideal gas to undergo an isobaric compression, then the system is no longer isolated, and so the entropy can decrease (entropy will increase elsewhere though).

By the Sakur-Tetrode equation, the entropy of a monatomic, ideal gas is given by

$$\frac{S}{k_BN}=\ln\left[\frac VN\left(\frac{4\pi m}{3h^2}\frac{U}{N}\right)^{3/2}\right]+\frac52$$

For our purposes, it will make sense to use the ideal gas law to express $V$ in terms of $P$ and $T$, and to express $U=\frac32Nk_BT$, so we get

$$\frac{S}{k_BN}=\ln\left[\frac {k_BT}{P}\left(\frac{2\pi m}{h^2}\cdot k_BT\right)^{3/2}\right]+\frac52$$

So, as we can see, for a constant pressure $P$, the entropy of the ideal gas is a monotonically decreasing function with respect to decreasing $T$; if we decrease $T$, we decrease $S$.

I suspect your confusion comes from thinking that $S$ can never decrease, but this is only the case for isolated systems. If you are forcing an ideal gas to undergo an isobaric compression, then the system is no longer isolated, and so the entropy can decrease (entropy will increase elsewhere though).

As a separate argument, the entropy is a state function, meaning its value only depends on the state, not how you got there. Now, let's consider your isobaric process, and let's say we do an isobaric expansion and then an isobaric compression back to the original state (this is the original state of the system, not the original state of the system as well as the surroundings, which is impossible to achieve). Since entropy is a state function, the entropy ends at where it started. But this means one of two things happened

1. The entropy remained constant the entire time
2. The entropy increased as well as decreased during this process.

If you showed that the change in entropy is non-zero for some part of this, then you have to conclude that it is possible to decrease the entropy of an ideal gas.