Question

established relationships between cp and cv.

Answer 1

We called this relation as Mayer's Formula.

We can establish the relation between specific heat capacity at constant volume (Cv) and specific heat capacity at constant pressure (Cp) of a gas.

For an ideal gas, the relation between Cp and Cv is

                            Cp - Cv = R -------------------> (1)

∴ This relation is known as Mayer's Formula.

Now, To establish the relation, we need to begin from the first law of thermodynamics for 1 mole of gas.

                              ΔQ = ΔU + pΔV

If heat ΔQ  is absorbed at constant volume,

∴ pΔV = 0 and ΔQ = CvΔT for one mole of a gas 

Now, ΔV = 0

Then, Cv = (ΔQ/ΔT)v = (ΔU/ΔT)v = (ΔU/ΔT) -----------------> (2)

where the V is dropped in the last step, since U of an ideal gas depends only on the temperature, not on the volume.

Now, heat ΔQ is absorbed at constant  pressure, then

          ΔQ = CpΔT

          Cp = (ΔQ/ΔT)p = (ΔU/ΔT)p = (ΔU/ΔT)p

Now, p can be the dropped from the first term since U of an ideal gas depends only on T, not on pressure.

Now, by using Eq. (2)

or                   Cp = Cv +p(ΔV/Δp)p --------------------> (3)

Now, for 1 mole of an ideal gas, PV = RT
If the pressure is kept constant

          p(ΔV/ΔT)p = R -----------------------> (4)

From Eq. (2) , (3) and (4)

                      { Cp - Cv = R }

Here, Cp and Cv are molar specific heat capacities of an ideal gas at constant pressure and volume and R is the universal gas constant.

The ratio of Cp and Cv  is notified as γ

                                 γ = Cp/Cv

And it is also known as heat capacity ratio,

      Cv = R/r-1 andCp = γ R/r-1

Answer 2

Answer:

We called this relation as Mayer's Formula.

We can establish the relation between specific heat capacity at constant volume (Cv) and specific heat capacity at constant pressure (Cp) of a gas.

For an ideal gas, the relation between Cp and Cv is

                           Cp - Cv = R -------------------> (1)

∴ This relation is known as Mayer's Formula.

Now, To establish the relation, we need to begin from the first law of thermodynamics for 1 mole of gas.

                             ΔQ = ΔU + pΔV

If heat ΔQ  is absorbed at constant volume,

∴ pΔV = 0 and ΔQ = CvΔT for one mole of a gas 

Now, ΔV = 0

Then, Cv = (ΔQ/ΔT)v = (ΔU/ΔT)v = (ΔU/ΔT) -----------------> (2)

where the V is dropped in the last step, since U of an ideal gas depends only on the temperature, not on the volume.

Now, heat ΔQ is absorbed at constant  pressure, then

         ΔQ = CpΔT

         Cp = (ΔQ/ΔT)p = (ΔU/ΔT)p = (ΔU/ΔT)p

Now, p can be the dropped from the first term since U of an ideal gas depends only on T, not on pressure.

Now, by using Eq. (2)

or                   Cp = Cv +p(ΔV/Δp)p --------------------> (3)

Now, for 1 mole of an ideal gas, PV = RT

If the pressure is kept constant

         p(ΔV/ΔT)p = R -----------------------> (4)

From Eq. (2) , (3) and (4)

                     { Cp - Cv = R }

Here, Cp and Cv are molar specific heat capacities of an ideal gas at constant pressure and volume and R is the universal gas constant.

The ratio of Cp and Cv  is notified as γ

                                γ = Cp/Cv

And it is also known as heat capacity ratio,

     Cv = R/r-1 andCp = γ R/r-1