Question

What is the relationship between ${C_p}$ and ${C_v}$ for an Ideal Gas?

Answer 1

${C_p}$ is the specific heat at constant pressure and ${C_v}$ is the specific heat at constant volume.
At constant pressure we get${C_P} > {C_V}$.
Formula used: We will use the following formula in the solution,
$q = nC\Delta T$
Where $q$ is the heat
$C$ is the specific heat and
$\Delta T$ is the temperature change.

Complete step by step solution:
We have the formula $q = nC\Delta T$
So at constant pressure P we can write
${q_p} = n{C_p}\Delta T$
The above formula is equal to the change in enthalpy.
Therefore,
${q_p} = n{C_p}\Delta T = \Delta H$
Now at constant volume V we can write
${q_V} = n{C_V}\Delta T$
The above formula is equal to the change in internal energy that is,
${q_V} = n{C_V}\Delta T = \Delta U$
For an ideal gas equation we have n = 1 for 1 mole
$\Delta H = \Delta U + \Delta (PV)$
$\Rightarrow \Delta H = \Delta U + \Delta (RT)$
Hence we can write,
$\Rightarrow \Delta H = \Delta U + R\Delta (T)$
Now putting value of $\Delta H$ and $\Delta U$ in the above equation we get,
${C_P}\Delta T = {C_V}\Delta T + R\Delta T$
Cancelling $\Delta T$from the above equation we get
${C_P} = {C_V} + R$
Which gives,
${C_P} - {C_V} = R$
Therefore the relationship between ${C_p}$ and ${C_v}$for an Ideal Gas equation is:
${C_P} - {C_V} = R$
Additional information: ${C_p}$, the specific heat at constant pressure, is the amount of heat energy released or absorbed by a unit mass of the substance with the change in temperature at constant pressure.
${C_p} = {(\dfrac{{\Delta H}}{{\Delta T}})_P}$
Where $\Delta H$ is the change in enthalpy and $\Delta T$ is the change in temperature at constant pressure.
${C_v}$ is the heat energy transfer between a system and its surrounding without any change in the volume of the system.
${C_V} = {(\dfrac{{\Delta U}}{{\Delta T}})_V}$
Where $\Delta U$ is the change in internal energy $\Delta T$ is the change in temperature at constant volume.
At constant pressure ${C_p}$>${C_v}$

Note: We should remember that ${C_p}$ is linked with change in enthalpy and ${C_v}$ is linked with change in internal energy, because if these are swapped then, we will not get the correct relation between both.