Question

What is the relationship between \[{C_p}\] and \[{C_v}\] for an Ideal Gas?

Answer 1

Hint: \[{C_p}\] 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.