Question

(Molar specific heat capacity at constant pressure) - (Molar specific heat capacity at constant volume) is equal to:

• A. R
• B. P
• C. V
• D. T

Answer 1

The correct option is A

R

Explanation for correct option:

(a) R

• ${\mathrm{C}}_{\mathrm{p}}$ is the abbreviation used to indicate a substance's molar heat capacity at constant pressure.
• ${\mathrm{C}}_{\mathrm{v}}$ is the term used to express molar heat capacity at constant volume.
• As a result, these two factors describe the molar heat capacity as pressure and temperature fluctuate.

Derivation of the relation between ${\mathrm{C}}_{\mathrm{p}}$ and ${\mathrm{C}}_{\mathrm{v}}$:

Step 1:

• We may deduce the following from the equation $\mathrm{q}=\mathrm{n}\mathrm{C}\mathrm{T}$
• The heat $\mathrm{q}$ necessary to cause a $\mathrm{T}$ variation in temperature of one mole of any substance is represented by the formula $\mathrm{q}=\mathrm{n}\mathrm{C}\mathrm{T}$.
• The constant $\mathrm{C}$ here is referred to as the body's molar heat capacity.

Step 2: Internal heat equation for ${\mathrm{C}}_{\mathrm{p}}$

• We have at constant pressure P:

$\mathrm{qP}=\mathrm{n}{\mathrm{C}}_{\mathrm{p}}∆\mathrm{T}$

• This is equal to the change in enthalpy, i.e.

${\mathrm{C}}_{\mathrm{p}}∆\mathrm{T}=\mathrm{H}=\mathrm{n}$ ………………..(1)

Step 3: Internal heat equation for ${\mathrm{C}}_{\mathrm{v}}$

• Similarly, we have at constant volume $\mathrm{V}$:

$\mathrm{qV}=\mathrm{n}{\mathrm{C}}_{\mathrm{v}}∆\mathrm{T}$

• This number equals the change in internal energy, i.e.

${\mathrm{C}}_{\mathrm{V}}∆\mathrm{T}=\mathrm{U}=\mathrm{n}$ …………………(2)

Step 4: Finding enthalpy for $\mathrm{n}=1$

• For one mole ($\mathrm{n}=1$) of an ideal gas, we know that

$$\begin{gathered} \mathrm{H}=\mathrm{U}+\left(\mathrm{pV}\right)=\mathrm{U}+\left(\mathrm{RT}\right) \\ \mathrm{H}=\mathrm{U}+\mathrm{R}\mathrm{T} \end{gathered}$$

• As a result, $\mathrm{H}=\mathrm{U}+\mathrm{R}\mathrm{T}$

Step 5: Finding the relation between ${\mathrm{C}}_{\mathrm{p}}$ and ${\mathrm{C}}_{\mathrm{v}}$:

• Substituting the values of H and U from the previous equations (1) and (2),

$$\begin{gathered} {\mathrm{C}}_{\mathrm{p}}∆\mathrm{T}={\mathrm{C}}_{\mathrm{v}}∆\mathrm{T}+\mathrm{R}∆\mathrm{T} \\ {\mathrm{C}}_{\mathrm{p}}={\mathrm{C}}_{\mathrm{v}}+\mathrm{R} \\ {\mathrm{C}}_{\mathrm{p}}–{\mathrm{C}}_{\mathrm{v}}=\mathrm{R} \end{gathered}$$

So, the correct option is (a) R.