Question

Select the incorrect relation. (Where symbols have their usual meanings)
A. \[{{\text{C}}_p} = \dfrac{{\gamma R}}{{\gamma - 1}}\]
B. \[{{\text{C}}_p} - {C_v} = R\]
C. \[\Delta U = \dfrac{{{P_f}{V_f} - {P_i}{V_i}}}{{1 - \gamma }}\]
D. ${C_v} = \dfrac{R}{{\gamma - 1}}$

Answer 1

Hint: We know the ideal gas equation is $PV = nRT$. Here, we have to start from the basic equation and then have to use the ideal gas equation and heat capacity at constant volume to arrive at the final form. Later we have to compare with given options.

Complete step by step answer:
We know the equation \[{{\text{C}}_v} = \dfrac{R}{{\gamma - 1}}\]
is correct.
\[{{\text{C}}_p} = \dfrac{{\gamma R}}{{\gamma - 1}}\] is also correct.Also \[{{\text{C}}_p} - {C_v} = R\]
Now let us check for the third option.
The equation is as follows:
$\Delta U = n{C_v}\Delta T$ ……... (1)
The terms are:
N=no. of moles
${C_v}$= heat capacity at constant volume=$\dfrac{R}{{\gamma - 1}}$$\Delta T = $change in temperature=${T_f} - {T_i}$
Now we substitute these two values in equation (1).
$\Delta U = n\dfrac{R}
{{\gamma - 1}}({T_f} - {T_i})$ ……. (2)
Now according to the ideal gas equation is:
PV=nRT
Where
P is pressure, V is volume, n is no. of moles, R is gas constant and T is temperature.
When the no. of moles are same, then we can write
$nR{T_f} = {P_f}{V_f}$
$nR{T_i} = {P_i}{V_i}$
When we substitute these values in eq 2, we get
\[\Delta U = \dfrac{{{P_f}{V_f} - {P_i}{V_i}}}
{{\gamma - 1}}\]
When we compare all the equations, we conclude that option is D is incorrect
\[\Delta U = \dfrac{{{P_f}{V_f} - {P_i}{V_i}}}
{{1 - \gamma }}\]
Hence, the correct answer is option D, that is, ${C_v} = \dfrac{R}
{{\gamma - 1}}$

So, the correct answer is Option D.

Note: Heat capacity or thermal capacity is defined as the amount of heat which is to be supplied to a given mass of a material and it produces a unit change in its temperature. It is the physical property of matter.
The heat capacity at constant volume is the derivative of the internal energy with respect to the temperature.
The specific heat capacity of a substance is the heat capacity of a sample of the substance divided by the mass of the sample.