Question

If \[{{C}_{P}}\] and \[{{C}_{V}}\]denotes the specific heats of nitrogen per unit mass at constant pressure and constant volume respectively, then?
(A) \[{{C}_{P}}-{{C}_{V}}=\frac{R}{28}\]
(B) \[{{C}_{P}}-{{C}_{V}}=\frac{R}{14}\]
(C) \[{{C}_{P}}-{{C}_{V}}=R\]
(D) \[{{C}_{P}}-{{C}_{V}}=28R\]

Answer 1

Hint: \[{{C}_{P}}\]refers to specific heat capacity at constant pressure and \[{{C}_{V}}\]refers to specific heat capacity at constant volume. Specific heat capacity of any material or substance is defined as the amount of heat required to increase the temperature of one mole of that substance by \[{{1}^{0}}C\]. Nitrogen is a diatomic gas.

Complete step by step solution:
In order to find out the molar heat capacity we can find it as the result of the product of molar mass with specific hat capacity. The molar mass of the Nitrogen gas is $28$. So, the molar heat capacity at constant pressure will be \[28{{C}_{P}}\]and molar heat capacity at constant volume is \[28{{C}_{V}}\].
From Mayer’s formula we can find our answer:
$\Rightarrow 28{{C}_{P}}-28{{C}_{V}}=R \\
\Rightarrow {{C}_{P}}-{{C}_{V}}=\dfrac{R}{28} \\$

So, the correct option is (a)

Additional Information: In steady state cases the flow of heat takes place at a constant pace. If a system is thermally isolated then it is not possible for heat to move from outside into the system and vice versa.

Note: Nitrogen does not exist alone, it exists in the atmosphere in the form of gas as a diatomic molecule.
The molar mass of nitrogen is 14, and so for diatomic nitrogen it will be just twice, 28 amu. In order to understand in a simple way the meaning of specific heat, take an example of the water. The specific heat of water is 4.2 \[J/{{g}^{0}}C\]. This means that it takes 4.2 joules of energy to raise 1 gram of water by 1 degree Celsius.