Question

For a fixed amount of perfect gas, which of these statements must be true?
(A) \[E\] and \[H\] each depend only on \[T\].
(B) \[{C_p}\] is constant.
(C) \[PdV = nRdT\] for every infinitesimal process.
(D) Both (A) and (C) are true.

Answer 1

Hint: We need to know the characteristics of a perfect gas. A perfect gas is also known as an ideal gas. For a gas to be behave as an ideal gas:
-The gas particles should have negligible volume.
-The gas particles should be equally sized and do not possess intermolecular forces, either attraction or repulsion with other gas particles.
-The gas particles should move random motion and perfectly obey Newton’s Laws of Motion.
-The gas particles should have perfect elastic collisions with no energy loss.
-In order to conclude whether a gas is ideal or not, four variables are taken into consideration and a law is made out of it known as The Ideal Gas law. The four variables are Pressure ($P$) , Volume (\[V\]), number of moles of gas \[\left( n \right)\] and Temperature (\[t\]) . It gives us a simple equation-the ideal gas equation as \[PV = nRT\]. For an ideal gas, \[\dfrac{{PV}}{{nRT}} = 1\].

Complete step by step answer:
For an ideal gas which obeys the equation \[PV = nRT\], the internal energy \[E\] increases as a system’s temperature increases, the molecules will move faster, thus have more kinetic energy. The internal energy is equal to the heat of the system and hence depends only on temperature.
Using the definition of enthalpy that is Enthalpy, the sum of the internal energy and the product of the pressure and volume of a thermodynamic system and the equation of state of ideal gas to yield,
\[H = E + PV\]
Or, \[H = E + RT\] ( for 1 mole of an ideal gas, i. e, n=1)
Since R is constant and \[E\]depends only on \[T\], therefore, H is also a function of \[T\] and depends only on temperature.
Since \[E\] and \[H\] depend only on the temperature for an ideal gas, the constant volume and constant pressure specific heats \[{C_v}\] and \[{C_p}\] also depend on the temperature only.
\[{C_v} = {C_v}\left( T \right)\] \[{C_p} = {C_p}\left( T \right)\]
For an ideal gas, the definitions of \[{C_v}\] and \[{C_p}\] are given as follows:
\[{C_v} = {\left( {\dfrac{{\partial E}}{{\partial T}}} \right)_V}\] and \[{C_P} = {\left( {\dfrac{{\partial H}}{{\partial T}}} \right)_P}\]
Hence, it can be said that \[{C_p}\] is not constant as H changes with respect to T. Therefore option (B) is incorrect.
The equation for an ideal gas is
\[PV = nRT\]
For infinitesimal processes,
$PdV + VdP = nRdT$ . Therefore, option (C) is incorrect.
Hence, the correct option is option (A).

Note:
Note that all of the thermodynamic properties of an ideal gas are summed up in its equation of state, which determines the relationship between its pressure, volume and temperature. The ideal gas law is one of the simplest equations of state. Although reasonably accurate for gases at low pressures and high temperatures, it becomes increasingly inaccurate at higher pressures and lower temperatures.