The molar specific heat at constant pressure of an ideal gas is $\frac{7}{2}R$. The ratio of specific heat at constant pressure to that at constant volume is: (2006)
Given $C_p = \frac{7}{2}R$. Using $C_p - C_v = R$, we find $C_v = \frac{7}{2}R - R = \frac{5}{2}R$. The ratio of specific heats is $\gamma = \frac{C_p}{C_v} = \frac{7/2 R}{5/2 R} = \frac{7}{5}$.
For hydrogen gas $C_p – C_v = a$ and for oxygen gas $C_p – C_v = b$, so the relation between $a$ and $b$ is given by: (1991)
Here $C_p$ and $C_v$ are specific heats per unit mass. For a gas, $C_p - C_v = \frac{R}{M}$. For hydrogen ($M=2$), $a = \frac{R}{2}$. For oxygen ($M=32$), $b = \frac{R}{32}$. Therefore, $a = 16b$.
For a certain gas the ratio of specific heats is given to be $\gamma = 1.5$. For this gas: (1990)
Given $\gamma = \frac{C_p}{C_v} = 1.5 = \frac{3}{2}$. In work units, $C_p - C_v = \frac{R}{J}$. Substituting $C_v = \frac{2}{3}C_p$, we get $C_p - \frac{2}{3}C_p = \frac{R}{J}$, leading to $\frac{C_p}{3} = \frac{R}{J}$, or $C_p = \frac{3R}{J}$.
The mean free path for a gas, with molecular diameter $d$ and number density $n$ can be expressed as: (2020)
From the kinetic theory of gases, the mean free path $\lambda$ (the average distance a molecule travels between collisions) is derived as $\lambda = \frac{1}{\sqrt{2} \pi n d^2}$.
The number of translational degrees of freedom for a diatomic gas is: (1993)
The center of mass of any molecule can move in three independent directions ($x, y,$ and $z$). Therefore, the translational degrees of freedom for a diatomic gas (or any gas) is always 3.
If for a gas, $\frac{R}{C_V} = 0.67$, this gas is made up of molecules which are: (1992)
We know that $\gamma - 1 = \frac{R}{C_v}$. Given $\frac{R}{C_v} = 0.67 \approx \frac{2}{3}$, we get $\gamma = 1 + \frac{2}{3} = \frac{5}{3}$. A heat capacity ratio of $\frac{5}{3}$ corresponds to a monoatomic gas.
A polyatomic gas with $n$ degrees of freedom has a mean energy per molecule given by: (1989)
According to the law of equipartition of energy, the average energy associated with each degree of freedom per molecule is $\frac{1}{2}kT$. For a molecule with $n$ degrees of freedom, the total mean energy is $\frac{n}{2}kT$.
One mole of an ideal monoatomic gas undergoes a process described by the equation $PV^3 = \text{constant}$. The heat capacity of the gas during this process is: (2016 – II)
For a polytropic process $PV^x = \text{constant}$, the molar heat capacity is $C = C_v + \frac{R}{1-x}$. Here $x=3$ and for a monoatomic gas $C_v = \frac{3}{2}R$. Thus, $C = \frac{3}{2}R + \frac{R}{1-3} = \frac{3}{2}R - \frac{R}{2} = R$.
The ratio of the specific heats $\frac{C_P}{C_V} = \gamma$ in terms of degrees of freedom ($n$) is given by: (2015)
The molar heat capacities are $C_v = \frac{n}{2}R$ and $C_p = C_v + R = \left(\frac{n}{2} + 1\right)R$. Their ratio $\gamma = \frac{C_p}{C_v} = \frac{(\frac{n}{2} + 1)R}{\frac{n}{2}R} = 1 + \frac{2}{n}$.
The mean free path $l$ for a gas molecule depends upon diameter, $d$ of the molecule as (2020-Covid)
The expression for mean free path is $l = \frac{1}{\sqrt{2}\pi n d^2}$. From this formula, it is clear that the mean free path is inversely proportional to the square of the diameter, $l \propto \frac{1}{d^2}$.