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,
$$q=nC\mathrm{\Delta }T$$
Where $$q$$ is the heat
$$C$$ is the specific heat and
$$\mathrm{\Delta }T$$ is the temperature change.

Complete step by step solution:
We have the formula $$q=nC\mathrm{\Delta }T$$
So at constant pressure P we can write
$$q_p=n{C}_p\mathrm{\Delta }T$$
The above formula is equal to the change in enthalpy.
Therefore,
$$q_p=n{C}_p\mathrm{\Delta }T=\mathrm{\Delta }H$$
Now at constant volume V we can write
$$q_V=n{C}_V\mathrm{\Delta }T$$
The above formula is equal to the change in internal energy that is,
$$q_V=n{C}_V\mathrm{\Delta }T=\mathrm{\Delta }U$$
For an ideal gas equation we have n = 1 for 1 mole
$$\mathrm{\Delta }H=\mathrm{\Delta }U+\mathrm{\Delta }(PV)$$
$$\Rightarrow \mathrm{\Delta }H=\mathrm{\Delta }U+\mathrm{\Delta }(RT)$$
Hence we can write,
$$\Rightarrow \mathrm{\Delta }H=\mathrm{\Delta }U+R\mathrm{\Delta }(T)$$
Now putting value of $$\mathrm{\Delta }H$$ and $$\mathrm{\Delta }U$$ in the above equation we get,
$$C_P\mathrm{\Delta }T=C_V\mathrm{\Delta }T+R\mathrm{\Delta }T$$
Cancelling $$\mathrm{\Delta }T$$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.
$$C_p=({\displaystyle \frac{\mathrm{\Delta }H}{\mathrm{\Delta }T}}{)}_P$$
Where $$\mathrm{\Delta }H$$ is the change in enthalpy and $$\mathrm{\Delta }T$$ 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.
$$C_V=({\displaystyle \frac{\mathrm{\Delta }U}{\mathrm{\Delta }T}}{)}_V$$
Where $$\mathrm{\Delta }U$$ is the change in internal energy $$\mathrm{\Delta }T$$ 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.