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An stoichiometric mixture of hydrogen gas and the air at $25{}^\circ C$ and a total pressure of $1\,atm$, is exploded in a closed rigid vessel. If the process occurs under adiabatic condition, then using the given data answer the questions that follow:
Given:
i.\[{{C}_{p}}=8.3\,cal\,{{\deg }^{-1}}mo{{l}^{-1}}\]
ii.\[{{C}_{p}}=11.3\,cal\,{{\deg }^{-1}}mo{{l}^{-1}};\,\Delta {{H}_{f}}[{{H}_{2}}O(g)]=-57.8\,kcal\]
[Take air as $80$ per ${{N}_{2}}$, $20$ per ${{O}_{2}}$ by volume]
The value of ${{C}_{p}}$ of ${{N}_{2}}$ and ${{H}_{2}}O$ will be:
(in $cal\,{{\deg }^{-1}}mo{{l}^{-1}}$)
A.$8.3,8.3$
B.$8.3,11.3$
C.$11.3,11.3$
D.$11.3,8.3$

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Last updated date: 20th Jun 2024
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Answer
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Hint:Molar heat capacity requires heat to raise a temperature of one mole gas by one Kelvin. In this question, molar heat capacity at constant pressure is used. Here, nitrogen is a diatomic gas and water is the triatomic.
Complete step by step solution:
Here, it is given that the enthalpy of formation of ${{H}_{2}}O$, $\Delta {{H}_{f}}({{H}_{2}}O(g))$ is $-57.8\,kcal$.
The ${{C}_{p}}$ is $8.3$ and $11.3\,cal\,{{\deg }^{-1}}mo{{l}^{-1}}$
As we know, that ${{C}_{p}}$ is the specific heat at constant pressure.
${{N}_{2}}$ is the diatomic gas.
Total degree of freedom $=3\times n$
As ${{N}_{2}}$ is diatomic, total degree of freedom $=3\times 2=6$
${{H}_{2}}O$ is a triatomic gas.
Total degree of freedom $=3\times n\Rightarrow 3\times 3=9$
As ${{N}_{2}}$ is diatomic gas and ${{H}_{2}}O$ is a triatomic gas, therefore, the value of ${{C}_{p}}$ for ${{N}_{2}}$ is less than the value of ${{C}_{p}}$ for ${{H}_{2}}O$.
Therefore, the correct option is (B), that is, $8.3,11.3$

Additional information:
-The standard enthalpy of formation is defined as the enthalpy change during the formation of $1$ mole of substance. It is calculated in kilojoule per mole ($kJ\,mo{{l}^{-1}}$).
Factors that affect the standard enthalpy of formation is:
i.The partial pressure of gas
ii.Temperature of the system
-The concentration of reactant and product
-Heat capacity is defined as the amount of heat required to raise the temperature of a body by $1{}^\circ C$. It is an extreme property, and it is a path function denoted with$(c)$.
-Heat capacity for gas is molar heat capacity. It is the amount of heat required to raise the temperature of $1$ mole of substance by $1{}^\circ C$ or $1K$ .
-It is of two types:
i.Molar heat capacity at constant volume \[({{C}_{V}})\]
ii.Molar heat capacity at constant pressure $({{C}_{P}})$
There are features of heat capacity and they are as follows:
-Heat capacity can be positive, negative, zero or infinite.
-For solid and liquid, ${{C}_{P}}$ is nearly equal to \[{{C}_{V}}\] .
-Heat capacity of gas is more than the heat capacity of solid and liquid.

Note:
-Degree of freedom is the number of ways in which energy is distributed equally.
-Monoatomic gases contain a single atom. For example, $He,Ar$.
-Diatomic gases contain two atoms. For example, ${{H}_{2}},{{O}_{2}}$ .
-Triatomic gases contain three atoms. For example, ${{H}_{2}}O,C{{O}_{2}}$ .