Coefficient of isothermal elasticity Eθ and coefficient of adiabatic elasticity Eϕ are related by (γ = Cₚ/Cᵥ)

The correct option is E_ϕ = γ E_θ.

To understand the relationship between the coefficient of isothermal elasticity ($E_{\theta }$) and the coefficient of adiabatic elasticity ($E_{\varphi }$), let us derive their expressions from thermodynamic principles.

The bulk modulus or coefficient of elasticity of a gas is generally defined as the ratio of volume stress (change in pressure $dP$) to volume strain (fractional change in volume $-\frac{dV}{V}$):
$E=-V\frac{dP}{dV}$

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 $V$:
$P+V\frac{dP}{dV}=0$
$-V\frac{dP}{dV}=P$
Thus, the isothermal elasticity is equal to the pressure of the gas:
$E_{\theta }=P$ ---- (Equation 1)

2. Adiabatic Elasticity ($E_{\varphi }$):
For an adiabatic process, there is no heat exchange with the surroundings. The equation of state for an ideal gas undergoing an adiabatic process is:
$P{V}^{\gamma }=\text{constant}$
where $\gamma =\frac{C_p}{C_v}$ is the adiabatic index (ratio of specific heats).
Differentiating both sides with respect to $V$:
$V^{\gamma }\frac{dP}{dV}+P\left(\gamma V^{\gamma -1}\right)=0$
Dividing the entire equation by $V^{\gamma -1}$:
$V\frac{dP}{dV}+\gamma P=0$
$-V\frac{dP}{dV}=\gamma P$
Thus, the adiabatic elasticity is:
$E_{\varphi }=\gamma P$ ---- (Equation 2)

Conclusion:
Substituting Equation 1 ($P=E_{\theta }$) into Equation 2, we get:
$E_{\varphi }=\gamma E_{\theta }$
This shows that the adiabatic elasticity of a gas is $\gamma$ times its isothermal elasticity.