The ratio of the slopes of isothermal and adiabatic curves is

Fill in the blanks:
$1.$ The heat required to raise the temperature of a body by $1\,K$ is called ...... .
$2.$ $A$ temperature difference of $10\,^{\circ}C$ is equivalent to a temperature difference of ...... on the Fahrenheit temperature scale.
$3.$ For an isothermal process of an ideal gas,$\frac{dP}{P} = .........$

The efficiency of a thermodynamic cycle $1-2-3-1$ (see picture) is $20\%$ and for another thermodynamic cycle $1-3-4-1$ efficiency is equal to $10\%$. Determine the efficiency $\eta$ (in $\%$) of the thermodynamic cycle $1-2-3-4-1$. The gas is assumed to be ideal.

Three samples $X, Y$,and $Z$ of the same gas have equal volumes and temperatures. The volume of each sample is doubled. The process is isothermal for $X$,adiabatic for $Y$,and isobaric for $Z$. If the final pressures are equal for the three samples,find the ratio of the initial pressures. (Take adiabatic exponent $\gamma = 3/2$)

$A$ heating element of resistance $r$ is fitted inside an adiabatic cylinder which carries a frictionless piston of mass $m$ and cross-sectional area $A$. The cylinder contains one mole of a diatomic gas. The temperature of the gas varies with time $t$ as $T = \alpha t + \frac{1}{2} \beta t^2$ (where $\alpha$ and $\beta$ are constants), while the pressure remains constant. The atmospheric pressure above the piston is $P_0$. Then:

Three moles of an ideal monatomic gas perform a cycle $ABCDA$ as shown in the figure. The temperatures of the gas at the states $A, B, C$ and $D$ are $400 \, K, 800 \, K, 2400 \, K$ and $1200 \, K$, respectively. The work done by the gas during this cycle is ($R$ is the universal gas constant). (in $R$)