For an ideal gas with constant properties undergoing a quasi-static process, which one of the following represents the change of entropy (∆s) from state 1 to 2?

The correct answer/option is represented by the formula shown in the first option:

$$\Delta s=C_p\ln \left(\frac{T_2}{T_1}\right)-R\ln \left(\frac{P_2}{P_1}\right)$$

Image Analysis and Context:
By inspecting the provided images, we identify the expressions as follows:
- Image 0 (the correct option) displays the formula: $\Delta s=C_p\ln (T_2/T_1)-R\ln (P_2/P_1)$.
- Image 1 displays the incorrect relation: $\Delta s=C_v\ln (T_2/T_1)-C_p\ln (V_2/V_1)$.
- Image 2 displays the incorrect relation: $\Delta s=C_p\ln (T_2/T_1)-C_v\ln (P_2/P_1)$.
- Image 3 displays: $\Delta s=C_v\ln (T_2/T_1)+R\ln (V_1/V_2)$.
- Image 4 lists three relations, with the third one confirming the correct entropy change expression.

Step-by-Step Derivation:

We begin with the fundamental Gibbs relation (commonly known as the second $Tds$ relation) for a simple compressible system:

$$Tds=dh-vdP$$

where:
- $T$ is the absolute temperature (K),
- $s$ is the specific entropy,
- $h$ is the specific enthalpy,
- $v$ is the specific volume, and
- $P$ is the pressure.

For an ideal gas, we can express the specific enthalpy change ($dh$) and specific volume ($v$) using standard ideal gas properties:
1. The definition of specific heat capacity at constant pressure ($C_p$) yields:

$$dh=C_{p}dT$$

2. From the ideal gas equation of state ($Pv=RT$), specific volume is:

$$v=\frac{RT}{P}$$

where $R$ is the specific gas constant.

Substituting these expressions back into the $Tds$ equation:

$$Tds=C_{p}dT-\left(\frac{RT}{P}\right)dP$$

Dividing the entire equation by the absolute temperature $T$ isolates the change in specific entropy:

$$ds=C_p\frac{dT}{T}-R\frac{dP}{P}$$

To find the total change in specific entropy ($\Delta s=s_2-s_1$) between state 1 and state 2, we integrate both sides. Since the gas properties (such as $C_p$ and $R$) are constant:

$${\int }_{s_1}^{s_2}ds=C_p{\int }_{T_1}^{T_2}\frac{dT}{T}-R{\int }_{P_1}^{P_2}\frac{dP}{P}$$

Carrying out the integrations yields:

$$s_2-s_1=C_p{\left[\ln T\right]}_{T_1}^{T_2}-R{\left[\ln P\right]}_{P_1}^{P_2}$$

Applying the boundaries and the logarithmic subtraction property ($\ln A-\ln B=\ln (A/B)$):

$$\Delta s=C_p\ln \left(\frac{T_2}{T_1}\right)-R\ln \left(\frac{P_2}{P_1}\right)$$

This successfully derives the exact expression visible in the correct option.