In the following P-V diagram two adiabatics cut two isothermals at temperatures T₁ and T₂ (fig.). The value of $\frac{V_a}{V_d}$ will be

(1) $\frac{V_b}{V_c}$

(2) $\frac{V_c}{V_b}$

(3) $\frac{V_d}{V_a}$

(4) V_bV_c

An ideal gas with adiabatic exponent $\left(\mathrm{\gamma }=1.5\right)$ undergoes a process in which work done by the gas is same as increase in internal energy of the gas. The molar heat capacity of gas for the process is –

1. $\mathrm{C}=4\mathrm{R}$

2. $\mathrm{C}=0$

3. $\mathrm{C}=2\mathrm{R}$

4. $\mathrm{C}=\mathrm{R}$

The molar heat capacity for an ideal gas

1.  cannot be negative

2.  must be equal to either ${\mathrm{C}}_{\mathrm{V}}$ or ${\mathrm{C}}_{\mathrm{P}}$

3.  must lie in the range ${\mathrm{C}}_{\mathrm{V}}\le \mathrm{C}\le {\mathrm{C}}_{\mathrm{P}}$

4.  may have any value between $-\infty$ and $+\infty$

An ideal gas expands according to the law ${\mathrm{PV}}^2$ = const. The molar heat capacity C is : 

1. ${\mathrm{C}}_{\mathrm{V}} + \mathrm{R}$ 

2. ${\mathrm{C}}_{\mathrm{V}} – \mathrm{R}$ 

3. ${\mathrm{C}}_{\mathrm{V}} + 2\mathrm{R}$

4. ${\mathrm{C}}_{\mathrm{V}} – 3\mathrm{R}$

The molar heat capacity C for an ideal gas going through a given process is given by C = a/T , where 'a' is a constant. If  $\mathrm{\gamma }= {\mathrm{C}}_{\mathrm{P}}/{\mathrm{C}}_{\mathrm{V}}$ , the work done by one mole of gas during heating from ${\mathrm{T}}_0$ to $\mathrm{\eta } {\mathrm{T}}_0$ through the given process will be:

1.  $\frac{1}{\mathrm{a}} \ln \mathrm{\eta }$

2.  $\mathrm{a} \ln \mathrm{\eta }- \left(\frac{\mathrm{\eta }-1}{\mathrm{\gamma }-1}\right) {\mathrm{RT}}_0$

3.  $\mathrm{a} \ln \mathrm{\eta }-\left(\mathrm{\gamma }-\mathrm{A}\right) {\mathrm{RT}}_0$

4.  none of these

P-V diagram of a diatomic gas is straight line passing through origin. The molar heat capacity of the gas in the process will be

1. 4R

2. 2.5 R

3. 3R

4. $\frac{4\mathrm{R}}{3}$

The pressure of a monoatomic gas increases linearly from $4\times {10}^5$ N/m² to $8\times {10}^5$ N/m² when its volume increases from 0.2 m³ to 0.5 m³. The molar heat capacity of the gas is:
[R = 8.31 J/mol k]  

1. 20.1 J/molK               

2. 17.14 J/molK

3. 18.14 J/molK                                   

4. 20.14 J/molK

At ordinary temperatures, the molecules of a
diatomic gas have only translational and
rotational kinetic energies. At high
temperatures, they may also have vibrational
energy. As a result of this compared to lower
temperatures, a diatomic gas at higher
temperatures will have–

1. lower molar heat capacity.
2. higher molar heat capacity.
3. lower isothermal compressibility.
4. higher isothermal compressibility.

Which of the following shows the correct relationship between the pressure 'P' and density $\rho$ of an ideal gas at constant temperature?

(1) 

(2)

(3) 

(4)