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:
\(\left(\frac{C_{p}}{C_{v}} = \gamma\   ~\text{and}~\   R = \text{gas constant}\right)\)

1.	\(\dfrac{50nR}{\gamma - 1}\)	2.	\(\dfrac{100nR}{\gamma - 1}\)
3.	\(\dfrac{50n\gamma R}{\gamma - 1}\)	4.	\(\dfrac{25n\gamma R}{\gamma - 1}\)

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 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

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. 

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

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. 

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

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. $70\% \mathrm{and}-10°\mathrm{C}$