Question

What is the relation between \[{C_P}\] and \[{C_V}\] for methanol?
A.${C_P} > > {C_V}$
B.${C_P} \geqslant {C_V}$
C.${C_V} \geqslant {C_P}$
D.${C_V} > > {C_P} $

Answer 1

Hint: \[{C_P}\] in the system is the amount of heat energy released or absorbed by a unit mass of the substance with the change in temperature at a constant pressure. \[{C_V}\] is the amount of heat released/absorbed per unit mass of a substance.

Complete answer:
According to the first law of thermodynamics:
$\Delta Q = \Delta U + \Delta W$, $\Delta Q$ is the amount of heat that is given to the system, $\Delta U$ is the change in internal energy and $\Delta W$ is the work done.
We can write: $\Delta Q = \Delta U + P\Delta V$ as, $\Delta W = P\Delta V$
Since $\Delta Q = n{C_P}\Delta T$ and $\Delta U = n{C_V}\Delta T$
Therefore, $n{C_P}\Delta T = n{C_V}\Delta T + P\Delta V$
We can put the $P\Delta V$ value from the ideal gas as $nR\Delta T$, the equation will be:
$n{C_P}\Delta T = n{C_V}\Delta T + nR\Delta T$
$n{C_P}\Delta T = n\Delta T({C_V} + R)$
${C_P} - {C_V} = R$ (Where $R$ is universal gas constant)
At ambient pressure and temperature the isobaric specific heat, \[{C_P}\] of liquid methanol is $2.53[KJ/KgK]$, while the isochoric specific heat, \[{C_V}\] is $2.12[KJ/KgK]$. The relation between heat capacities is: ${C_P} \geqslant {C_V}$, as the value is approximately the same so they can be equal.
Hence option B is the correct answer.

Note:
The isobaric is used for the substance, when there is a constant pressure, \[{C_P}\] is an isobaric specific heat because the heat is calculated under the constant pressure, whereas the isochoric is used for the substance, where there is constant volume. \[{C_V}\] is an isochoric specific heat because the heat is calculated under the constant volume.