Why is the value of the heat capacity ratio $\gamma$ never less than 1? For an monoatomic gas, the value of the heat capacity ratio $\gamma$ is 1.67
For diatomic gases, the value of γ (gamma) is 1.40
For many atomic gases, the value of γ (gamma) is 1.33
It's never less than 1! Why?
 A: During constant volume heating, the gas's volume is fixed, and hence it does no work on its surroundings. During constant pressure heating, the gas is allowed to expand against its surroundings, thereby doing work on the surroundings and hence losing some of the energy that it is gaining from heating. As a consequence, the specific heat at constant pressure has to be larger than the specific heat at constant volume, and so $\gamma>1$.
A: A simple argument would be that the heat capacity ratio for an ideal gas is related to the degrees of freedom, $f$, via,
$$\gamma=1+\frac{2}{f}\equiv\frac{f+2}{f}$$
Since the degree of freedom specifies the number of ways a molecule/gas can move (translational, rotational, vibrational) in a given $d$-dimensional space, then it must be strictly positive ($f>0)$, which means that the adiabatic index will satisfy $\gamma>1$ for all $f$.
A: Constant pressure heat capacity is always greater or equal to the constant volume heat capacity for perfect and real gases and for liquids, i.e., for every simple system described by $N$, $V$, $T$ variables.
The reason is a general purely thermodynamic relation between $C_p$ and $C_V$:
$$
C_P = C_V + \frac{TV\alpha^2}{\chi_T} \tag{1}
$$
where $\alpha$ is the thermal expansion coefficient, and $\chi_T$ is the isothermal compressibility. The identity can be easily derived, and details can be found in every good thermodynamics textbook. Equation $(1)$ implies
$$
C_P \geq C_V  ~~~~~~~ \Rightarrow  ~~~~~~~~~ \gamma=\frac{C_P}{C_V} \geq 1
$$
i.e., $\gamma$ can never be smaller than one.
Notice that there is only one case where the equality to one may hold for finite values of the heat capacities: the rare but not impossible case of zero thermal expansion materials (see, for instance, this paper).
A: $$\gamma=\frac{C_p}{C_v}=\frac{C_v+R}{C_v}=1+\frac{R}{C_v}$$We know that R is a positive constant and the heat capacity at constant volume is greater than zero (a gas internal energy increases with increasing temperature).  So $\gamma>1$.
A: Presumably by $\gamma$ you are referring the to the ratio of the specific heat at constant pressure $c_{p}$ to the specific heat at constant volume $c_V$ as applied to an isentropic process for an ideal gas. For an ideal gas
$$c_{P}=c_{V}+R\tag{1}$$
Thus for an ideal gas, $c_P$ is always greater than $c_V$ by an amount equal to the universal gas constant.
Equation (1) can be derived from the basic definitions of the specific heats and enthalpy, combined with the ideal gas law, as follows:
Specific heat definitions:
For constant pressure:
$$c_p = \frac{dh}{dT}$$
For constant temperature:
$$c_v = \frac{du}{dT}$$
Definition of specific enthalpy (h)
$$h = u + Pv$$
From the ideal gas law
$$Pv=RT$$
Therefore
$$h = u+RT$$
Taking the derivative of the last equation with respect to temperature:
$$\frac {dh}{dT} =\frac {du}{dT}+R$$
Substituting the specific heat definitions into the last equation, we get
$$c_p – c_v = R$$
For data on various gases see:https://www.engineeringtoolbox.com/specific-heat-ratio-d_608.html
Hope this helps.
