The specific heat at constant pressure and at constant volume for an ideal gas are Cp and Cv and its adiabatic and isothermal elasticities are Eϕ and Eθ respectively. The ratio of Eϕ to Eθ is

The correct option is Cp/Cv.

To find the ratio of the adiabatic elasticity ($E_{\varphi }$) to the isothermal elasticity ($E_{\theta }$) for an ideal gas, we can derive the expressions for both elasticities using thermodynamic principles.

The bulk elasticity (or bulk modulus) of a gas is generally defined as:
$E=-V\frac{dP}{dV}$
where $P$ is the pressure and $V$ is the volume of the gas.

1. Isothermal Elasticity ($E_{\theta }$):
For an isothermal process, the temperature remains constant. The equation of state for an ideal gas undergoing an isothermal process is given by Boyle's Law:
$PV=\text{constant}$
Differentiating both sides with respect to $V$, we get:
$P+V\frac{dP}{dV}=0$
This simplifies to:
$-V\frac{dP}{dV}=P$
Therefore, the isothermal elasticity is:
$E_{\theta }=P$

2. Adiabatic Elasticity ($E_{\varphi }$):
For an adiabatic process, there is no heat exchange. The equation of state is:
$P{V}^{\gamma }=\text{constant}$
where $\gamma =\frac{C_p}{C_v}$ is the ratio of specific heat capacity at constant pressure ($C_p$) to that at constant volume ($C_v$).
Differentiating both sides with respect to $V$, we get:
$V^{\gamma }\frac{dP}{dV}+P\left(\gamma V^{\gamma -1}\right)=0$
Dividing the entire equation by $V^{\gamma -1}$:
$V\frac{dP}{dV}+\gamma P=0$
This simplifies to:
$-V\frac{dP}{dV}=\gamma P$
Therefore, the adiabatic elasticity is:
$E_{\varphi }=\gamma P$

3. Ratio of Elasticities:
Now, we find the ratio of the adiabatic elasticity to the isothermal elasticity:
$\frac{E_{\varphi }}{E_{\theta }}=\frac{\gamma P}{P}=\gamma$
Since $\gamma =\frac{C_p}{C_v}$, we have:
$\frac{E_{\varphi }}{E_{\theta }}=\frac{C_p}{C_v}$