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Question: If for hydrogen \[{C_P} - {C_V} = m\] and for nitrogen \[{C_P} - {C_V} = n\] , where \({C_P}\)​ and ...

If for hydrogen CP−CV=m{C_P} - {C_V} = m and for nitrogen CP−CV=n{C_P} - {C_V} = n , where CP{C_P}​ and CV{C_V} refer to specific heats per unit mass respectively at constant pressure and constant volume, the relation between m and n is (molecular weight of hydrogen=2 and molecular weight of nitrogen=14)
(A) n=14mn = 14m
(B) n=7mn = 7m
(C) m=7nm = 7n
(D) m=14nm = 14n

Explanation

Solution

Hint In this question, the specific heats per unit mass is given and we know the relation between molar specific heats of gases which is the amount of heat required by a mole of gas to raise its temperature by a degree . So, we can use the relation between mass and mole to get the result.
Formula Used:
CP−CV=R{C_P} - {C_V} = R

Complete step by step answer
We know the relation CP−CV=R{C_P} - {C_V} = R where CP{C_P} is the specific heat per unit mole at constant pressure and CV{C_V} is the specific heat per unit mole at constant volume. It is for 1 mole of ideal gas. But in the question, specific heats per unit mass is mentioned.
If the molecular weight of a gas is ‘M’ then, 1 mole of gas = M gram that is ‘n’ moles of gas =n×M = n \times M grams
So, for ‘M’ grams, the equation will be CP−CV=RM{C_P} - {C_V} = \dfrac{R}{M}
From the question for hydrogen, we get 2(CP−CV)=R2({C_P} - {C_V}) = R
and for nitrogen, 14(CP−CV)=R14({C_P} - {C_V}) = R
now put the values given in the question to the above equation it becomes,
for hydrogen, 2m=R2m = R and for nitrogen 14n=R14n = R . As we can see, the right side of both the equations are same so equating them as equal,
2m=14n⇒m=7n2m = 14n \Rightarrow m = 7n

Hence, the correct option is C.

Note
The specific heat of gas is at constant pressure is greater than specific heat of gas is at constant volume. The difference between the two is utilized as the heat used in doing work against the external pressure applied to raise 1 mole of gas temperature by 10C{1^0}C at constant pressure. There is a difference in notation of specific heat and molar specific heat. If it is written in small like cP{c_P} or cV{c_V} , then it is specific heats at constant pressure and volume respectively and if it is written in capital like CP{C_P} or CV{C_V} , then it is molar specific heat at constant pressure and volume respectively.