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)

During an adiabatic compression, 830 J of work is done on 2 moles of a diatomic ideal gas to reduce its volume by 50%. The change in its temperature is nearly:

$(\mathrm{R}=8.3 {\mathrm{JK}}^{-1} {\mathrm{mol}}^{-1})$

1. 40 K

2. 33 K

3. 20 K

4. 14 K

A Carnot engine absorbs 1000 J of heat energy from a reservoir at $127°\mathrm{C}$ and rejects 600 J of heat energy during each cycle. The efficiency of the engine and temperature of the sink will be:

1. $20\% \mathrm{and}-43°\mathrm{C}$

2. $40\% \mathrm{and}-33°\mathrm{C}$

3. $50\% \mathrm{and}-20°\mathrm{C}$

4. $70\% \mathrm{and}-10°\mathrm{C}$

One mole of an ideal monoatomic gas is compressed isothermally in a rigid vessel to double its pressure at room temperature, $27°C$.The work done on the gas will be:

(1) 300R

(2) 300R$\ln \left(6\right)$ 

(3) 300R$\ln \left(2\right)$

(4) 300R$\ln \left(7\right)$

Consider a spherical shell of radius R and temperature T. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume$\mathrm{u}=\frac{\mathrm{U}}{\mathrm{V}}\propto {\mathrm{T}}^4 \mathrm{and} \mathrm{pressure} \mathrm{P}=\frac{1}{3}\left(\frac{\mathrm{U}}{\mathrm{V}}\right)$. If the shell now undergoes an adiabatic expansion, the relation between T and R is:

1. $\mathrm{T}\propto {\mathrm{e}}^{-3\mathrm{R}}$

2. $\mathrm{T}\propto \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{R}$}\right.$

3. $\mathrm{T}\propto \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\mathrm{R}}^3$}\right.$

4. $\mathrm{T}\propto {\mathrm{e}}^{-\mathrm{R}}$