A monoatomic gas is supplied with the heat \(Q\) very slowly, keeping the pressure constant. The work done by the gas will be: 
1. \({2 \over 3}Q\)
2. \({3 \over 5}Q\)
3. \({2 \over 5}Q\)
4. \({1 \over 5}Q\)

An ideal gas expands isothermally from a volume \(V_1\) to \(V_2\) and then is compressed to the original volume \(V_1\) adiabatically. The initial pressure is \(P_1\) and the final pressure is \(P_3.\) The total work done is \(W.\) Then:
1. \(𝑃 _3 > 𝑃 _1 ,    𝑊 > 0\)
2. \(𝑃 _3 < 𝑃 _1 ,    𝑊 < 0\)
3. \(𝑃 _3 > 𝑃 _1 ,    𝑊 < 0\)
4. \(𝑃 _3 = 𝑃 _1 ,    𝑊 = 0\)

Work done by a system under isothermal change from a volume V₁ to V₂ for a gas which obeys Vander Waal's equation $(V-\beta n)\left(P+\frac{{\displaystyle \alpha n^2}}{{\displaystyle V}}\right)=nRT$

(1) $nRT{\log }_e\left(\frac{{\displaystyle V_2-n\beta }}{{\displaystyle V_1-n\beta }}\right)+\alpha n^2\left(\frac{{\displaystyle V_1-V_2}}{{\displaystyle V_1{V}_2}}\right)$

(2) $nRT{\log }_{10}\left(\frac{{\displaystyle V_2-\alpha \beta }}{{\displaystyle V_1-\alpha \beta }}\right)+\alpha n^2\left(\frac{{\displaystyle V_1-V_2}}{{\displaystyle V_1{V}_2}}\right)$

(3) $nRT{\log }_e\left(\frac{{\displaystyle V_2-n\alpha }}{{\displaystyle V_1-n\alpha }}\right)+\beta n^2\left(\frac{{\displaystyle V_1-V_2}}{{\displaystyle V_1{V}_2}}\right)$

(4) $nRT{\log }_e\left(\frac{{\displaystyle V_1-n\beta }}{{\displaystyle V_2-n\beta }}\right)+\alpha n^2\left(\frac{{\displaystyle V_1{V}_2}}{{\displaystyle V_1-V_2}}\right)$

A cylindrical tube of uniform cross-sectional area A is fitted with two air tight frictionless pistons. The pistons are connected to each other by a metallic wire. Initially the pressure of the gas is P₀ and temperature is T₀, atmospheric pressure is also P₀. Now the temperature of the gas is increased to 2T₀, the tension in the wire will be

(1) 2 P₀A

(2) P₀A

(3) $\frac{P_{0}A}{2}$

(4) 4 P₀A

An insulator container contains 4 moles of an ideal diatomic gas at temperature T. Heat Q is supplied to this gas, due to which 2 moles of the gas are dissociated into atoms but temperature of the gas remains constant. Then

1. Q = 2RT

2. Q = RT

3. Q = 3RT

4. Q = 4RT

The volume of air increases by 5% in its adiabatic expansion. The percentage decrease in its pressure will be -

(1) 5%

(2) 6%

(3) 7%

(4) 8%

The temperature of a hypothetical gas increases to $\sqrt{2}$ times when compressed adiabatically to half the volume. Its equation can be written as

(1) PV^3/2 = constant

(2) PV^5/2 = constant

(3) PV^7/3 = constant

(4) PV^4/3 = constant

Two Carnot engines A and B are operated in succession. The first one, A receives heat from a source at T₁ = 800 K and rejects to sink at T₂ K. The second engine B receives heat rejected by the first engine and rejects to another sink at T₃ = 300 K. If the work outputs of two engines are equal, then the value of T₂ is -

(1) 100K

(2) 300K

(3) 550K

(4) 700K

When an ideal monoatomic gas is heated at constant pressure, fraction of heat energy supplied which increases the internal energy of gas, is 

(1) $\frac{2}{5}$

(2) $\frac{3}{5}$

(3) $\frac{3}{7}$

(4) $\frac{3}{4}$