Question

Carbon monoxide is carried around a closed cycle abc in which bc is an isothermal process as shown in the figure. The gas absorbs $7000\;{\rm{J}}$ of heat as its temperature increases from $300\;{\rm{K}}$ to $1000\;{\rm{K}}$ in going from a to b. the quantity of heat rejected by the gas during the process ca is
(a). $4200\;{\rm{J}}$
(b). $5000\;{\rm{J}}$
(c). $9000\;{\rm{J}}$
(d). $9800\;{\rm{J}}$

Answer 1

We should remember that if there is an isobaric process, the pressure should not vary. Similarly, we should also keep in mind that volume of a system will be constant in an isobaric process.

Complete step by step answer:
As the volume remains constant from a to b, the process from a to b is isochoric. For a system the change in heat energy is dependent on the internal energy and the total work done. Thus for isochoric process from a to b, we have,
$\Delta {Q_{ab}} = \Delta {U_{ab}} + \Delta {w_{ab}}$
Here, $U$ is the internal energy and $w$ is the work done. But for an isochoric process, as the volume remains the same, the total change in work done is zero. That is, $\Delta w = 0$. Thus , from a to b,
$\Delta {Q_{ab}} = \Delta {U_{ab}}$
It is given that the gas absorbs $7000\;{\rm{J}}$ of heat, which is $\Delta {Q_{ab}}$, we have,
$\Delta {Q_{ab}} = 7000\;{\rm{J}} = \Delta {U_{ab}}$
Now the formula to calculate the internal energy change is,
$\Delta U = n{C_v}\Delta T$
Here ${C_v}$ is the molar heat capacity at constant volume, $n$ is the number of moles and $T$ is the temperature of the system.
Here as the gas considered is carbon monoxide, which is a diatomic gas, we have,
${C_v} = \dfrac{5}{2}R$
Here $R$ is the gas constant.
Also it is given that the temperature increases from $300\;{\rm{K}}$ to $1000\;{\rm{K}}$. Thus the difference in temperature is $1000\;{\rm{K}} - 300\;{\rm{K = 700}}\;{\rm{K}}$. Thus by substituting for ${C_v}$,and temperature,
$\Delta U = n\dfrac{5}{2}R\Delta T\\\ \Rightarrow 7000\;{\rm{J}} = nR\dfrac{{700 \times 5}}{2}\\\ \Rightarrow nR = 4$
Let us now consider the change from the point c to a. here the pressure is remaining constant. So the process is isobaric. Thus the change in heat energy from the point c to a is calculated as,
$\Delta Q = n{C_p}\Delta T$
Here ${C_p}$ is the molar heat capacity at constant pressure, $n$ is the number of moles and $T$ is the temperature of the system.
As the gas considered is carbon monoxide, which is a diatomic gas, we have,
${C_p} = \dfrac{7}{2}R$
Also it is given that the bc is an isothermal process. So the temperature for b will be same as that of temperature for c. So the change in temperature for the isobaric process ca is $300\;{\rm{K}} - {\rm{1000}}\;{\rm{K}} = {\rm{700}}\;{\rm{K}}$. Thus substituting for ${C_p}$ and the temperature difference in the equation $\Delta Q = n{C_p}\Delta T$, we have,
$\Delta Q = - n\dfrac{7}{2}R700\;{\rm{K}}\\\ \Rightarrow\Delta Q {\rm{ = - nR}}\dfrac{7}{2} \times 700\;{\rm{K}}$
Substituting $4$ for $nR$, we have,
$\Delta Q = - 4 \times \dfrac{7}{2} \times 700\;{\rm{K}}\\\ \therefore\Delta Q = - 9800\;{\rm{J}}$
Therefore the heat rejected is $9800\;{\rm{J}}$ and the correct option is(d).

Note: The molar heat capacity of a gas at constant volume is defined as the heat energy required to warm one mole of a gas through one degree when its volume is kept constant. And at constant volume we get the molar heat capacity ${C_p}$. And the gas constant is defined as the difference between the molar heat capacities at constant pressure and volume.