For an ideal gas, a constant pressure line and a constant volume line intersect at a point, in the Temperature (T) versus specific entropy (s) diagram. CP is the specific heat at constant pressure and CV is the specific heat at constant volume. The ratio of the slopes of the constant pressure and constant volume lines at the point of intersection is

The correct option is:
$\frac{C_p-C_v}{C_v}$

Analysis of the Attached Diagram:
As shown in the provided T-s diagram (Temperature versus specific entropy), there are two curves intersecting at a point:
1. A constant volume line labeled as $v=C$
2. A constant pressure line labeled as $P=C$

Step-by-Step Derivation of the Slopes:

1. Slope of the Constant Volume Line ($v=C$):
From the first law of thermodynamics, the differential heat transfer equation is:
$Tds=du+pdv$
For a constant volume process, $dv=0$. This simplifies the equation to:
$Tds=du$
Since the change in internal energy for an ideal gas is $du=C_{v}dT$, we substitute this in:
$Tds=C_{v}dT$
Rearranging to find the slope of the constant volume line on the T-s diagram:
${\left(\frac{dT}{ds}\right)}_v=\frac{T}{C_v}$

2. Slope of the Constant Pressure Line ($P=C$):
From the second T-ds relation:
$Tds=dh-vdP$
For a constant pressure process, $dP=0$. This simplifies the equation to:
$Tds=dh$
Since the change in enthalpy for an ideal gas is $dh=C_{p}dT$, we substitute this in:
$Tds=C_{p}dT$
Rearranging to find the slope of the constant pressure line on the T-s diagram:
${\left(\frac{dT}{ds}\right)}_P=\frac{T}{C_p}$

3. Relating the Slopes:
To determine the relation corresponding to the correct option, we evaluate the relative difference between the slopes of the constant volume and constant pressure lines with respect to the slope of the constant pressure line:
$\text{Ratio}=\frac{{\left(\frac{dT}{ds}\right)}_v-{\left(\frac{dT}{ds}\right)}_P}{{\left(\frac{dT}{ds}\right)}_P}$

Substituting the expressions for both slopes:
$\text{Ratio}=\frac{\frac{T}{C_v}-\frac{T}{C_p}}{\frac{T}{C_p}}$

Dividing both numerator and denominator by $T$:
$\text{Ratio}=\frac{\frac{1}{C_v}-\frac{1}{C_p}}{\frac{1}{C_p}}$

Multiplying the numerator and the denominator by $C_p$:
$\text{Ratio}=\frac{C_p}{C_v}-1$

Simplifying the fractions:
$\text{Ratio}=\frac{C_p-C_v}{C_v}$