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

An ideal gas going through the reversible cycle $\mathrm{a}\to \mathrm{b}\to \mathrm{c}\to \mathrm{d}$, has the V-T diagram as shown below in the figure. Process $\mathrm{d}\to \mathrm{a} \mathrm{and} \mathrm{b}\to \mathrm{c}$ are adiabatic.

The corresponding P-V diagram for the process is (all figures are schematic and not drawn to scale):

1. 

2. 

3. 

4. 

200 g water is heated from $40°\mathrm{C} \mathrm{to} 60°\mathrm{C}$. Ignoring the slight expansion of water, the change in internal energy is close to (Given specific heat of water=4184 J/Kg K) :

1. 8.4 kJ

2. 4.2 kJ

3. 16.7 kJ

4. 167.4 kJ

One mole of diatomic ideal gas undergoes a cyclic process ABC as shown in the figure. The process of BC is adiabatic. The temperature at A, B and C are 400 K, 800 K, and 600 K respectively. Choose the correct statement:

1. The change in internal energy in the process AB is - 350R.

2. The change in internal energy in the process BC is - 500R.

3. The change in internal energy in the whole cyclic process is 250R.

4. The change in internal energy in the process CA is 700R. 

If the temperature of the source and the sink in the heat engine is at 1000 K & 500 K respectively, then the efficiency can be:
1. 20%
2. 30%
3. 50%
4. All of these

The volume and temperature graph is given in
the figure. Pressure for two processes are different, then
which of the following is true?

(1) P₁ = P₂ and P₃ = P₄ and P₃ > P₂
(2) P₁ = P₂ and P₃ = P₄ and P₃ < P₂
(3) P₁ = P₂ = P₃ = P₄
(4) P₁ > P₂ > P₃ > P₄

If \(n\) moles of an ideal gas is heated at a constant pressure from \(50^\circ\text C\) to \(100^\circ\text C,\) the increase in the internal energy of the gas will be:
\(\left(\frac{C_{p}}{C_{v}} = \gamma\   ~\text{and}~\   R = \text{gas constant}\right)\)

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$\ln \left(7\right)$

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. $70\% \mathrm{and}-10°\mathrm{C}$