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  $\mathrm{\gamma }=<mspace\; width="1em"></mspace>{\mathrm{C}}_{\mathrm{P}}/{\mathrm{C}}_{\mathrm{V}}$ , the work done by one mole of gas during heating from ${\mathrm{T}}_0$ to $\mathrm{\eta }<mspace\; width="1em"></mspace>{\mathrm{T}}_0$ through the given process will be:

1.  $\frac{1}{\mathrm{a}}<mspace\; width="1em"></mspace>\ln <mspace\; width="1em"></mspace>\mathrm{\eta }$

2.  $\mathrm{a}<mspace\; width="1em"></mspace>\ln <mspace\; width="1em"></mspace>\mathrm{\eta }-<mspace\; width="1em"></mspace>\left(\frac{\mathrm{\eta }-1}{\mathrm{\gamma }-1}\right)<mspace\; width="1em"></mspace>{\mathrm{RT}}_0$

3.  $\mathrm{a}<mspace\; width="1em"></mspace>\ln <mspace\; width="1em"></mspace>\mathrm{\eta }-\left(\mathrm{\gamma }-\mathrm{A}\right)<mspace\; width="1em"></mspace>{\mathrm{RT}}_0$

4.  none of these

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

1. ${\mathrm{C}}_{\mathrm{V}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>\mathrm{R}$ 

2. ${\mathrm{C}}_{\mathrm{V}}<mspace\; width="1em"></mspace>–<mspace\; width="1em"></mspace>\mathrm{R}$ 

3. ${\mathrm{C}}_{\mathrm{V}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>2\mathrm{R}$

4. ${\mathrm{C}}_{\mathrm{V}}<mspace\; width="1em"></mspace>–<mspace\; width="1em"></mspace>3\mathrm{R}$

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 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. $\mathrm{C}=\mathrm{R}$

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

P-V diagram of a cyclic process ABCA is as shown in figure. Choose the correct statement

(1) $\Delta Q_{A\to B}$ = negative

(2) $\Delta U_{B\to C}$ = positive

(3) $\Delta W_{CAB}$ = negative

(4) All of these

In the following figures, four curves A, B, C and D, are shown. The curves are:

Six moles of an ideal gas perform a cycle shown in figure. If the temperature are T_A = 600 K, T_B = 800 K, T_C = 2200 K and T_D = 1200 K, the work done per cycle is -

(1) 20 kJ

(2) 30 kJ

(3) 40 kJ

(4) 60 kJ

Consider a process shown in the figure. During this process the work done by the system -

(1) Continuously increases

(2) Continuously decreases

(3) First increases, then decreases

(4) First decreases, then increases