The efficiency of Carnot engine is $50\%$ and temperature of sink is $500\, K$. If the temperature of source is kept constant and its efficiency is to be raised to $60\%$, then the required temperature of the sink will be........ $K$

$Assertion :$ The isothermal curves intersect each other at a certain point.
$Reason :$ The isothermal changes takes place rapidly, so the isothermal curves have very little slope.

If $R =$ universal gas constant, the amount of heat needed to raise the temperature of $2\, mol$ of an ideal monoatomic gas from $273\, K$ to $373\, K$ when no work is done is

Which of the following graphs correctly represents the variation of $\beta  =  - \left( {\frac{{dV}}{{dP}}} \right)/V$ with $P$ for an ideal gas at constant temperature ?

One mole of an ideal diatomic gas undergoes a transition from $A$ to $B$ along a path $AB$ as shown in the figure

The change in internal energy of the gas during the transition is

A Carnot engine operating between temperatures $T_1$ and $T_2$ has efficiency $\frac {1}{6}$ . When $T_2$ is lowered by $60\,K$ ; its efficiency increases to $\frac {1}{3}$. Then $T_1$ and $T_2$ are respectively