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Question: If \({{C}_{p}},{{C}_{v}}\) denote the specific heats of nitrogen per unit mass at constant pressure ...

If Cp,Cv{{C}_{p}},{{C}_{v}} denote the specific heats of nitrogen per unit mass at constant pressure and constant volume, then,
a)Cp−Cv=R28 b)Cp−Cv=R14 c)Cp−Cv=R2 d)none \begin{aligned} & a){{C}_{p}}-{{C}_{v}}=\dfrac{R}{28} \\\ & b){{C}_{p}}-{{C}_{v}}=\dfrac{R}{14} \\\ & c){{C}_{p}}-{{C}_{v}}=\dfrac{R}{2} \\\ & d)none \\\ \end{aligned}

Explanation

Solution

Let us find the universal relation between the specific heats of any element at constant pressure and constant volume. It will be in terms of molar mass. We need to calculate the molar mass of nitrogen and substitute it in the universal formula. Also, the molar mass depends on the atomicity of the element.

Formula used:
Cp−Cv=Rmolar mass{{C}_{p}}-{{C}_{v}}=\dfrac{R}{molar \, mass}

Complete answer:
Let us first find the universal relation between the specific heats of any element at constant pressure and constant volume,

Cp−Cv=Rmolar mass..(1){{C}_{p}}-{{C}_{v}}=\dfrac{R}{molar \, mass}..(1)
We need to find the molar mass of the element nitrogen,
Molar mass of nitrogen= 2×14=282\times 14=28
Atomicity is very important while calculating the molar mass,
Now, substitute this in equation 1, we get,
Cp−Cv=R28{{C}_{p}}-{{C}_{v}}=\dfrac{R}{28}

So, the correct answer is “Option A”.

Additional Information:
The specific heat capacity of a substance is the heat capacity of a sample of the substance divided by the mass of the sample. In simple terms, it is the amount of energy that must be added in the form of heat to one unit of mass of a substance in order to cost an increase of one unit in temperature. Their specific heat often varies with temperature. The specific heat capacity is different for each state of matter. Liquid water has one of the highest specific heat capacity among common substances. While the substance is undergoing phase transition, such as melting or boiling, its specific heat is technically infinite because the heat goes into changing its state rather than raising the temperature.

Note:
The atomicity of nitrogen is two. The molar mass is the product of atomicity and the atomic weight. Therefore, we must be careful while calculating the molar mass, we need to care of the atomicity of the element. If the same question was asked for chlorine or any other uni-atomic element, the value does change.